From b4b13cb865273b32540f44e2a1a0770dea513346 Mon Sep 17 00:00:00 2001 From: Dalton Walker Date: Fri, 24 May 2024 11:02:22 -0400 Subject: [PATCH 1/3] collects all css into a block --- frontend/data/Languages/Python/pip.md | 3 - frontend/data/Linux/SLURM.md | 828 +++++++++--------- .../Deep Learning/Novice/introduction.md | 108 +++ .../data/Mathematics/Calculus/derivatives.md | 142 +++ .../Calculus/partial_derivatives.md | 81 ++ .../Linear Algebra/matrix_multiplication.md | 254 ++++++ .../routes/docs/[...document]/+page.svelte | 376 ++++---- 7 files changed, 1173 insertions(+), 619 deletions(-) delete mode 100644 frontend/data/Languages/Python/pip.md create mode 100644 frontend/data/Machine Learning/Deep Learning/Novice/introduction.md create mode 100644 frontend/data/Mathematics/Calculus/derivatives.md create mode 100644 frontend/data/Mathematics/Calculus/partial_derivatives.md create mode 100644 frontend/data/Mathematics/Linear Algebra/matrix_multiplication.md diff --git a/frontend/data/Languages/Python/pip.md b/frontend/data/Languages/Python/pip.md deleted file mode 100644 index 531f1bf..0000000 --- a/frontend/data/Languages/Python/pip.md +++ /dev/null @@ -1,3 +0,0 @@ -# Pip - -==hey== diff --git a/frontend/data/Linux/SLURM.md b/frontend/data/Linux/SLURM.md index 652b417..00504ed 100644 --- a/frontend/data/Linux/SLURM.md +++ b/frontend/data/Linux/SLURM.md @@ -1,456 +1,428 @@ # SLURM Setup Documentation for a Single Node Cluster - Here are the instructions to set up a single node SLURM cluster on your machine. The following setup works for Ubuntu 22.04.3. ## Step 1: Setup MariaDB Database for Accounting - Before you can start building your SLURM cluster, you must first setup a database for storage of account information and resource limits. This is necessary if you want to enforce resource limitations on certain users or groups within your cluster.
SLURM has built-in support for MySQL. Here we show the steps for setting up a MariaDB database server on your system and creating a SLURM accounting database that your SLURM daemons can later communicate with to store and access information.

- -- Install MariaDB. - - ```shell - sudo apt install mariadb-server - ``` - -- Configure MySQL. - - ```shell - sudo mysql_secure_installation - ``` - - When prompted feel free to remove anonymous users and disable remote root login. Also you may switch to unix_authentication, however, its not required as it should be enabled by default. This means that you can login as the 'root' user in MariaDB only if you are the 'root' user in your system, or if you have sudo privileges. You may choose to set a password if you wish to allow login as root without sudo. - -- Login as root user in MariaDB. Note that you need sudo privileges for this. - - ```shell - sudo mysql -u root - ``` - - Following this you should be logged in as root inside your database. - -- Create SLURM accounting database and a SLURM user with complete access to modify this database. Feel free to set 'password' to something more secure if desired. - - ```sql - CREATE DATABASE slurm_acct_db; - CREATE USER 'slurm'@'localhost' IDENTIFIED BY 'password'; - GRANT ALL ON slurm_acct_db.* TO 'slurm'@'localhost'; - FLUSH PRIVILEGES; - EXIT; - ``` - - You should now be logged out of MariaDB. - -- Enable the MariaDB server to start automatically on system reboot and restart the database server. - - ```shell - sudo systemctl enable mariadb - sudo systemctl restart mariadb - ``` - -- You should now be able to view the slurm user in the users table and check the databases table to see if the slurm_acct_db has been created. - - ```shell - sudo mysql -u root - SELECT User, Host, plugin FROM mysql.user; - SHOW DATABASES; - EXIT; - ``` - -- Optional: Change authentication plugin afterwards for root user. - - ```shell - sudo mysql -u root - ALTER USER 'root'@'localhost' IDENTIFIED WITH unix_socket; - FLUSH PRIVILEGES; - EXIT; - ``` + +* Install MariaDB. + ```shell + sudo apt install mariadb-server + ``` + +* Configure MySQL. + ```shell + sudo mysql_secure_installation + ``` + When prompted feel free to remove anonymous users and disable remote root login. Also you may switch to unix_authentication, however, its not required as it should be enabled by default. This means that you can login as the 'root' user in MariaDB only if you are the 'root' user in your system, or if you have sudo privileges. You may choose to set a password if you wish to allow login as root without sudo. + +* Login as root user in MariaDB. Note that you need sudo privileges for this. + ```shell + sudo mysql -u root + ``` + Following this you should be logged in as root inside your database. + +* Create SLURM accounting database and a SLURM user with complete access to modify this database. Feel free to set 'password' to something more secure if desired. + ```sql + CREATE DATABASE slurm_acct_db; + CREATE USER 'slurm'@'localhost' IDENTIFIED BY 'password'; + GRANT ALL ON slurm_acct_db.* TO 'slurm'@'localhost'; + FLUSH PRIVILEGES; + EXIT; + ``` + You should now be logged out of MariaDB. + +* Enable the MariaDB server to start automatically on system reboot and restart the database server. + + ```shell + sudo systemctl enable mariadb + sudo systemctl restart mariadb + ``` + +* You should now be able to view the slurm user in the users table and check the databases table to see if the slurm_acct_db has been created. + + ```shell + sudo mysql -u root + ``` + ```sql + SELECT User, Host, plugin FROM mysql.user; + SHOW DATABASES; + EXIT; + ``` + +* Optional: Change authentication plugin afterwards for root user. + + ```shell + sudo mysql -u root + ``` + ```sql + ALTER USER 'root'@'localhost' IDENTIFIED WITH unix_socket; + FLUSH PRIVILEGES; + EXIT; + ``` ## Step 2: Setup Munge for Authentication - SLURM uses the MUNGE authentication tool to allow for secure communication between the different SLURM daemons running on our cluster. -- Install MUNGE. - - ```shell - sudo apt install munge - ``` - -- Check if installation is correct. You should see a STATUS: SUCCESS message on running the below command. - - ```shell - munge -n | unmunge | grep STATUS - ``` - - Also you should be able to find a munge key located at /etc/munge/munge.key. If you can't find it, create one using the following command: - - ```shell - sudo /usr/sbin/mungekey - ``` - -- Set appropriate permissions for all munge files. Note that a munge user should have been automatically created after installing MUNGE. You can check this with cat /etc/passwd. - - ```shell - sudo chown -R munge: /etc/munge/ /var/log/munge/ /var/lib/munge/ /run/munge/ - sudo chmod 0700 /etc/munge/ /var/log/munge/ /var/lib/munge/ - sudo chmod 0755 /run/munge/ - sudo chmod 0700 /etc/munge/munge.key - sudo chown -R munge: /etc/munge/munge.key - ``` - -- Modify MUNGE configuration file to allow system logging. - - ```shell - sudo sed -i '/# OPTIONS/aOPTIONS="--syslog"' /etc/default/munge - ``` - -- Enable the munge daemon to start on system reboot and restart it. - ```shell - sudo systemctl enable munge - sudo systemctl restart munge - ``` +* Install MUNGE. + ```shell + sudo apt install munge + ``` + +* Check if installation is correct. You should see a STATUS: SUCCESS message on running the below command. + ```shell + munge -n | unmunge | grep STATUS + ``` + Also you should be able to find a munge key located at /etc/munge/munge.key. If you can't find it, create one using the following command: + ```shell + sudo /usr/sbin/mungekey + ``` + +* Set appropriate permissions for all munge files. Note that a munge user should have been automatically created after installing MUNGE. You can check this with cat /etc/passwd. + ```shell + sudo chown -R munge: /etc/munge/ /var/log/munge/ /var/lib/munge/ /run/munge/ + sudo chmod 0700 /etc/munge/ /var/log/munge/ /var/lib/munge/ + sudo chmod 0755 /run/munge/ + sudo chmod 0700 /etc/munge/munge.key + sudo chown -R munge: /etc/munge/munge.key + ``` + +* Modify MUNGE configuration file to allow system logging. + ```shell + sudo sed -i '/# OPTIONS/aOPTIONS="--syslog"' /etc/default/munge + ``` + +* Enable the munge daemon to start on system reboot and restart it. + ```shell + sudo systemctl enable munge + sudo systemctl restart munge + ``` ## Step 3: Setup SLURM -- Install SLURM. - - ```shell - sudo apt install slurm-wlm - sudo apt install slurmdbd - ``` - - The above should install 3 daemons in total: - - - slurmd - The daemon that runs on each compute node in the cluster. - - slurmctld - The central controller daemon that communicates with each slurmd and managaes all nodes in the cluser. - - slurmdbd - The database daemon that will communicate with our MariaDB database to store accounting information and resource limits. - - Since our setup only includes one node in our cluster, all 3 daemons will be running on the same server. - -- Configure slurmdbd. Create slurmdbd.conf file and set proper permissions and configurations to allow for proper accounting. - - ```shell - sudo touch /etc/slurm/slurmdbd.conf - sudo chown slurm:slurm /etc/slurm/slurmdbd.conf - sudo chmod 600 /etc/slurm/slurmdbd.conf - sudo sh -c "cat > /etc/slurm/slurmdbd.conf" <> /etc/mysql/my.cnf" < /etc/slurm/slurm.conf" <
- Accounts are groups of users. When you set the AllowAccounts parameter for a particular partition, it means that only users belonging to those accounts will be allowed to submit jobs to that partition.

- The line 'AccountingStorageEnforce=limits' is required if you want to set resource limits on certain users within your cluster.

- The line 'GresTypes=gpu' is necessary for allowing GPU support in SLURM. GRES stands for General Resources. So is the line 'SelectType=select/cons_tres' (do NOT set this to cons_res if you want GPU support). - -- Create gres.conf file. This is required if you want to allow GPU support for SLURM. - - ```shell - sudo touch /etc/slurm/gres.conf - sudo chown slurm:slurm /etc/slurm/gres.conf - sudo chmod 664 /etc/slurm/gres.conf - sudo sh -c "cat > /etc/slurm/gres.conf" < /etc/slurm/slurmdbd.conf" <> /etc/mysql/my.cnf" < /etc/slurm/slurm.conf" <
+ Accounts are groups of users. When you set the AllowAccounts parameter for a particular partition, it means that only users belonging to those accounts will be allowed to submit jobs to that partition.

+ The line 'AccountingStorageEnforce=limits' is required if you want to set resource limits on certain users within your cluster.

+ The line 'GresTypes=gpu' is necessary for allowing GPU support in SLURM. GRES stands for General Resources. So is the line 'SelectType=select/cons_tres' (do NOT set this to cons_res if you want GPU support). + + +* Create gres.conf file. This is required if you want to allow GPU support for SLURM. + ```shell + sudo touch /etc/slurm/gres.conf + sudo chown slurm:slurm /etc/slurm/gres.conf + sudo chmod 664 /etc/slurm/gres.conf + sudo sh -c "cat > /etc/slurm/gres.conf" < set where + ``` + For example, to modify the name of the user employee_1, we can do: + ```shell + sudo sacctmgr modify user set name=permanent_employee_1 where name=employee_1 + ``` + Similarly, we can delete this user by running the following: + ```shell + sudo sacctmgr remove user where name=employee_1 + ``` + Check out the SLURM accounting page for more info: https://slurm.schedmd.com/accounting.html + +* Set user resource limits. Here we're limiting the amount of resources the user intern_1 can use. We give them 16 CPU threads, 10 GB of memory, and 1 GPU. + ```shell + sacctmgr modify user intern_1 set GrpTRES=cpu=16,mem=100G,gres/gpu=1 + ``` + To unset these limits run the following: + ```shell + sacctmgr modify user intern_1 set GrpTRES=cpu=-1,mem=-1,gres/gpu=-1 + ``` + +* Check the user resource limits. + ```shell + sacctmgr show associations user=ronitagarwala01,test_intern format=User%25,GrpTRES%50,Account,Partition,Cluster%25 + ``` + Note an association is the most basic accounting unit in SLURM. It a set that consists of a user, account, cluster, and optionally a partition. - ```shell - sacctmgr show user - sacctmgr show account - sacctmgr show cluster - ``` - -- Modify and delete users, accounts, clusters. To modify / delete a created user, account or cluster you can use the following format: - - ```shell - sudo sacctmgr modify set where - ``` - - For example, to modify the name of the user employee_1, we can do: - - ````shell - sudo sacctmgr modify user set name=permanent_employee_1 where name=employee_1 - ```shell - Similarly, we can delete this user by running the following: - ```shell - sudo sacctmgr remove user where name=employee_1 - ```` - - Check out the SLURM accounting page for more info: https://slurm.schedmd.com/accounting.html +## Step 5: Submit Batch Jobs to your SLURM Cluster. -- Set user resource limits. Here we're limiting the amount of resources the user intern_1 can use. We give them 16 CPU threads, 10 GB of memory, and 1 GPU. +* In order to submit a job on your SLURM cluster, you must be logged in to your system as one of the users in your SLURM database. A SLURM job consists of a bash script with a bunch of SBATCH directives. Here is an example of our batch job script slurm_tester.sh: + ```bash + #!/bin/sh + #SBATCH --time=200 + #SBATCH --gres=gpu:1 + #SBATCH --output=slurm_output.txt + #SBATCH --job-name=SLURM_Testing + #SBATCH --partition=employee + #SBATCH --account=employee_account_1 + #SBATCH --cpus-per-task=1 + #SBATCH --ntasks=1 + #SBATCH --mem-per-gpu=32G + srun -l python3 slurm_tester.py + ``` + Here we run a simple python script called slurm_tester.py that we have defined in the same directory as slurm_tester.sh. Here is what it looks like: + ```python + import time + time.sleep(20) # Sleep for 20 seconds + print("Hello World!") + ``` + A batch script in slurm consists of a bunch of Sbatch directives, which are comments starting with #SBATCH that tell SLURM about the job requirements - including requested resources and account configurations - for this batch. Not all of these directives need to befined for each job. Here are some descriptions of each directive.

+ * --time: Time requested for job in minutes. If job execution does not complete withinin specified time limit, it will be cancelled.

+ * --gres: Used to specify general resources. Here used to request a single GPU for the batch job.

+ * --output: Specify the output file.

+ * --job-name: Specify job name.

+ * --partition: Specify the partition to submit the job to.

+ * --account: Specify the account to submit the job to.

+ * ntasks: Number of tasks to be launched. A task is an instance of your program that is launched on a compute node. Tasks can run in parallel and use MPI to communicate with one another.

+ * --cpus-per-task: Number of CPU threads required per task.

+ * --mem-per-gpu: Amount of memory per GPU required for the job. Here we set it to 32 GB per GPU. When no unit is specified after the number, the default units are in MB. Note that alternatives include --mem for setting memory allocation and --mem-per-cpu for setting memory per CPU. Note that only one of these three directives can be used at a time. + + More directives can be found in the official SLURM docs: https://slurm.schedmd.com/sbatch.html + + The srun command launches a job step in SLURM, which are sets of possibly parallel tasks within a job. We can have multiple srun statements with our batch script.
+ + To submit our batch job to SLURM, run the following command from your terminal: + ```shell + sbatch slurm_tester.sh + ``` + +* Check the status of your submitted job with squeue. + ```shell + squeue + ``` + Once a job is submitted to SLURM, it either starts executing if there are enough resources available, or its put into a queue to wait for resources to be free. + +* See information about submitted jobs, including job ID, status, name, associated user, account, partition, and allocated resources: + ```shell + sudo sacct --format=JobID,JobName,User,Account,Partition,AllocCPUS,State,ExitCode,AllocTRES%50 + ``` +* Cancel a running job: + ```shell + scancel + ``` + For example, to cancel the job with JobID 32, run: + ```shell + scancel 32 + ``` +* For troubleshooting, try looking at the slurmctld, slurmd, and slurmdbd log files: + ```shell + sudo cat /var/log/slurm/slurmctld.log + sudo cat /var/log/slurm/slurmd.log + sudo cat /var/log/slurm/slurmdbd.log + ``` - ```shell - sacctmgr modify user intern_1 set GrpTRES=cpu=16,mem=100G,gres/gpu=1 - ``` - To unset these limits run the following: +## Optional: Set up a more secure password for slurm database user +* Create password with pwgen: + ```shell + sudo apt install pwgen + DB_SLURM_PASS=$(pwgen -s 16 1) + ``` +* Change slurm user password: + ```shell + sudo mysql -u root -e "ALTER USER 'slurm'@'localhost' IDENTIFIED BY '${DB_SLURM_PASS}';" + ``` +* Store your password in /etc/slurm directory for safe-keeping. Also change permissions so that only slurm user or root / superuser can access it. This is important if you don't want anyone to change the database and thus get access to more resources than they are allocated: + ```shell + echo $DB_SLURM_PASS | sudo tee /etc/slurm/slurm_db_password.txt > dev_null + sudo chmod 600 /etc/slurm/slurm_db_password.txt + ``` +* Change StoragePass in slurmdbd.conf: + ```shell + sudo sed -i "s/StoragePass=.*/StoragePass=$DB_SLURM_PASS/" /etc/slurm/slurmdbd.conf + ``` - ```shell - sacctmgr modify user intern_1 set GrpTRES=cpu=-1,mem=-1,gres/gpu=-1 - ``` +## Additional Resources +* Single Cluster Setup: https://rolk.github.io/2015/04/20/slurm-cluster +* Simple Setup without Accounting: https://github.com/SergioMEV/slurm-for-dummies +* Another simple setup: https://drtailor.medium.com/how-to-setup-slurm-on-ubuntu-20-04-for-single-node-work-scheduling-6cc909574365 +* Setting up GPU Support: https://www.bodunhu.com/blog/posts/set-up-slurm-across-multiple-machines/ +* SLURM Quickstart: https://slurm.schedmd.com/quickstart.html +* SBATCH Directives: https://slurm.schedmd.com/sbatch.html +* SLURM Accounting: https://slurm.schedmd.com/accounting.html +* SLURM Resource Limits: https://slurm.schedmd.com/resource_limits.html +* Useful Ubuntu Discussion Board: https://ubuntuforums.org/showthread.php?t=2404746 +* Niflheim SLURM Setup Guide: https://wiki.fysik.dtu.dk/Niflheim_system/ +* SLURM Man Pages for different SLURM commands: https://slurm.schedmd.com/man_index.html -- Check the user resource limits. - ```shell - sacctmgr show associations user=ronitagarwala01,test_intern format=User%25,GrpTRES%50,Account,Partition,Cluster%25 - ``` - Note an association is the most basic accounting unit in SLURM. It a set that consists of a user, account, cluster, and optionally a partition. -## Step 5: Submit Batch Jobs to your SLURM Cluster. -- In order to submit a job on your SLURM cluster, you must be logged in to your system as one of the users in your SLURM database. A SLURM job consists of a bash script with a bunch of SBATCH directives. Here is an example of our batch job script slurm_tester.sh: - - ```shell - #!/bin/sh - #SBATCH --time=200 - #SBATCH --gres=gpu:1 - #SBATCH --output=slurm_output.txt - #SBATCH --job-name=SLURM_Testing - #SBATCH --partition=employee - #SBATCH --account=employee_account_1 - #SBATCH --cpus-per-task=1 - #SBATCH --ntasks=1 - #SBATCH --mem-per-gpu=32G - srun -l python3 slurm_tester.py - ``` - - Here we run a simple python script called slurm_tester.py that we have defined in the same directory as slurm_tester.sh. Here is what it looks like: - - ```python - import time - time.sleep(20) # Sleep for 20 seconds - print("Hello World!") - ``` - - A batch script in slurm consists of a bunch of Sbatch directives, which are comments starting with #SBATCH that tell SLURM about the job requirements - including requested resources and account configurations - for this batch. Not all of these directives need to befined for each job. Here are some descriptions of each directive.

- - - --time: Time requested for job in minutes. If job execution does not complete withinin specified time limit, it will be cancelled.

- - --gres: Used to specify general resources. Here used to request a single GPU for the batch job.

- - --output: Specify the output file.

- - --job-name: Specify job name.

- - --partition: Specify the partition to submit the job to.

- - --account: Specify the account to submit the job to.

- - ntasks: Number of tasks to be launched. A task is an instance of your program that is launched on a compute node. Tasks can run in parallel and use MPI to communicate with one another.

- - --cpus-per-task: Number of CPU threads required per task.

- - --mem-per-gpu: Amount of memory per GPU required for the job. Here we set it to 32 GB per GPU. When no unit is specified after the number, the default units are in MB. Note that alternatives include --mem for setting memory allocation and --mem-per-cpu for setting memory per CPU. Note that only one of these three directives can be used at a time. - - More directives can be found in the official SLURM docs: https://slurm.schedmd.com/sbatch.html - - The srun command launches a job step in SLURM, which are sets of possibly parallel tasks within a job. We can have multiple srun statements with our batch script.
- - To submit our batch job to SLURM, run the following command from your terminal: - - ```shell - sbatch slurm_tester.sh - ``` - -- Check the status of your submitted job with squeue. - - ```shell - squeue - ``` - - Once a job is submitted to SLURM, it either starts executing if there are enough resources available, or its put into a queue to wait for resources to be free. - -- See information about submitted jobs, including job ID, status, name, associated user, account, partition, and allocated resources: - ```shell - sudo sacct --format=JobID,JobName,User,Account,Partition,AllocCPUS,State,ExitCode,AllocTRES%50 - ``` -- Cancel a running job: - ```shell - scancel - ``` -- For troubleshooting, try looking at the slurmctld, slurmd, and slurmdbd log files: - ```shell - sudo cat /var/log/slurm/slurmctld.log - sudo cat /var/log/slurm/slurmd.log - sudo cat /var/log/slurm/slurmdbd.log - ``` -## Optional: Set up a more secure password for slurm database user -- Create password with pwgen: - ```shell - sudo apt install pwgen - DB_SLURM_PASS=$(pwgen -s 16 1) - ``` -- Change slurm user password: - ```shell - sudo mysql -u root -e "ALTER USER 'slurm'@'localhost' IDENTIFIED BY '${DB_SLURM_PASS}';" - ``` -- Store your password in /etc/slurm directory for safe-keeping. Also change permissions so that only slurm user or root / superuser can access it. This is important if you don't want anyone to change the database and thus get access to more resources than they are allocated: - ```shell - echo $DB_SLURM_PASS | sudo tee /etc/slurm/slurm_db_password.txt > dev_null - sudo chmod 600 /etc/slurm/slurm_db_password.txt - ``` -- Change StoragePass in slurmdbd.conf: - ```shell - sudo sed -i "s/StoragePass=.*/StoragePass=$DB_SLURM_PASS/" /etc/slurm/slurmdbd.conf - ``` -## Additional Resources -- Single Cluster Setup: https://rolk.github.io/2015/04/20/slurm-cluster -- Simple Setup without Accounting: https://github.com/SergioMEV/slurm-for-dummies -- Another simple setup: https://drtailor.medium.com/how-to-setup-slurm-on-ubuntu-20-04-for-single-node-work-scheduling-6cc909574365 -- Setting up GPU Support: https://www.bodunhu.com/blog/posts/set-up-slurm-across-multiple-machines/ -- SLURM Quickstart: https://slurm.schedmd.com/quickstart.html -- SBATCH Directives: https://slurm.schedmd.com/sbatch.html -- SLURM Accunting: https://slurm.schedmd.com/accounting.html -- SLURM Resource Limits: https://slurm.schedmd.com/resource_limits.html -- Useful Ubuntu Discussion Board: https://ubuntuforums.org/showthread.php?t=2404746 -- Niflheim SLURM Setup Guide: https://wiki.fysik.dtu.dk/Niflheim_system/ -- SLURM Man Pages for different SLURM commands: https://slurm.schedmd.com/man_index.html diff --git a/frontend/data/Machine Learning/Deep Learning/Novice/introduction.md b/frontend/data/Machine Learning/Deep Learning/Novice/introduction.md new file mode 100644 index 0000000..e3bedcd --- /dev/null +++ b/frontend/data/Machine Learning/Deep Learning/Novice/introduction.md @@ -0,0 +1,108 @@ +# Introduction + +## Overview + +This series of lessons will build up the knowledge required to effectively apply Deep Learning algorithms to real-world problems. We encourage anyone reading to read through the Mathematics lectures so that you have sufficient background for the following theory. Specifically we recommend: + +1. Mathematics/Calculus/Derivatives +2. Mathematics/Calculus/Partial Derivatives +3. Mathematics/Linear Algebra/Matrix Multiplication + +to get you started. We will cover the basics of Deep Learning in this series, saving more advanced theories for the advanced lecture series. We may briefly discuss specific domains such as Computer Vision, Natural Language Processing, and Deep Reinforcement Learning, but full treatments on those subjects will be saved for their respective series. Feel free to dive into those lectures **after** completing this series first. Pre-maturely investigating those fields without sufficient background is not recommended. + +### Terminology + +Before moving on we should define some common terms that we will use throughout the series. + +1. The inputs to a function are typically referred to as **features**. This can take the form of pixels in an image, tokens in a piece of text, or observations from an environment. Our objective in Machine Learning is to train a model to identify patterns in the features (data) whether those patterns are complex or simple. Deep Learning is typically used when the patterns are complex and require more modeling capacity to capture. +2. In supervised learning problems, each input is mapped to a value (an integer, real number, etc.). We call this value the **label** and can be a class if we are working on a classification problem, a float if we are estimating a value, or some other desired output. The correct label is often referred to as the **ground truth**. + +It would be prudent to also introduce some symbology that will be used throughout the course. First is the mapping symbol: + +$$ f: X \rightarrow Y $$ + +this reads "$f$ maps $X$ to $Y$". The set $X$ is called the **domain** and the set $Y$ is called the **co-domain** of the function $f$. If the name of the function is not required then you will sometimes see the notation $x \mapsto y$ instead of $f(x) = y$. For instance you would write $x \mapsto x^2$ for $f(x) = x^2$ (in Computer Science terms this is the equivalent of an anonymous function). + +Next is the probability symbols $\Pr\{ y \mid x\}$ and $\text{P}(y \mid x)$, both of which read "the probability of $y$ given $x$". We will use the symbol $\Pr$ instead of $\text{P}$ for clarity. The solid line in the middle of the inputs represents a conditional probability, i.e. the probability event $y$ occurs given event $x$ has occurred. Sometimes you will see just a single input $\Pr\{x\}$ which reads the "probability of event $x$". For instance if we model the probability of a coin flip resulting in the event heads or tails we would write that the probability of the event "heads" is + +$$\Pr\{\text{heads}\} = 0.5$$ + +given a fair coin. On the other hand we could write this as a conditional probability given we previously flipped heads, + +$$\Pr\{\text{heads} \mid \text{heads}\} = 0.5.$$ + +Since coin flips should be independent of each other we would drop the "given the event heads" part of the expression and simply write + +$$\Pr\{\text{heads} \mid \text{heads}\} = \Pr\{\text{heads}\} = 0.5.$$ + +If two events $A$ and $B$ are **independent**, then we can reduce + +$$\Pr\{A \mid B \}=\Pr\{A\}$$ + +### Paradigms + +Deep Learning can be applied to many problem paradigms. When thinking of a use case for Deep Learning models, you should check that the features contain complexities that traditional Machine Learning models, such as Random Forest and Naive Bayes, are unable to model. You should always select the simplest model that can correctly model your problem, as more complex models introduce errors (each model has an inductive bias) that require large amounts of data to overcome. Keep in mind that there is no "best" choice for all problems (No free lunch theorem). We will examine this further later. Three main paradigms will be covered in this series, each to different degrees. Our primary focus will be on supervised learning, but we will touch on semi-supervised and reinforcement learning as well. Each will be briefly described below. + +#### Supervised Learning + +In supervised learning, we provide the model with examples of features and labels. In a supervised setting, we have a perfect oracle that can tell the learner which labels are assigned to each input with the aim of generalizing the model to correctly operate on unseen data. The diagram below shows four examples of inputs mapped to outputs. Each of these examples will be provided to the model during training time, with a few examples set aside to test the models performance on new data. + +![Supervised Learning](https://adl-imgs.nyc3.cdn.digitaloceanspaces.com/machine_learning/deep_learning/novice/supervised-learning.svg "supervised-learning") + +For supervised learning to be effective, we need the training data to be properly formed, removing inconsistencies (such as one feature having multiple labels), and insuring that the training set is representative of the real-world problem we are trying to solve. Bad data is worse than no data, since we are can un-learn good weights with bad datum. When working with supervised learning, the data sets are often hand labeled by experts. In this scenario, human labelers can mistakenly introduce bias into their labels, or mis-label features in error. For this reason it is a good idea to test the quality of your training set. + +You should be wary of four major issues when working with supervised learning: + +**Bias-variance tradeoff** + +Suppose we have multiple sets we are using for training, each equally representative of the real-world data we are modeling. An algorithm (recall that an algorithm is the device we use to train/fit a machine learning model) is biased toward a specific feature if the model is consistently wrong in the labeling of said feature. A learning algorithm has high variance if the model predicts different labels for the same feature. The prediction error of a model is the sum of the bias error and variance error. Unfortunately there is a tradeoff between bias and variance. If the model is too flexible (has too many trainable parameters), it will fit each piece of training data differently, increasing the variance. If the model is too rigid (not enough parameters), it will not fit the training data very well, increasing the bias. It is our job as Machine Learning engineers to find the model with just enough complexity to fit the data, but make it low enough in variance to predict unseen data. + +**Modeling Capacity** + +If the complexity of the prediction function is high, meaning we have numerous parameters that we are using to model the interaction between the features and labels, then we need large quantities of good data to optimize the weights of the "true" function (the predictor). On the other hand, when our prediction function is simple (say for instance a linear model), then we can often learn from a small amount of data. + +**Feature Dimensionality** + +Richard E. Bellman coined the phrase "curse of dimensionality" when referring to the ratio of data complexity to quantity. When out feature space has relatively few dimensions, then we do not need as much data. As the complexity of our input features grows, so too does the need for data. Take for example Natural Language Processing. Setting aside the problem of modeling capacity, it would be impossible to train a deep neural net to model human language off of only a few examples. Foundational models such as GPT-4 and Mixtral-8x7b require trillions of examples to properly model the complex patterns found in human language. When analyzing your training set, consider how much training data you have in comparison to the dimensionality of the features. As the dimensionality increases, the extra dimensions can throw off the algorithm, increasing the variance. + +If you do not need as many dimensions to model your problem, then consider using a dimensionality reduction algorithm such as PCA or t-SNE. + +**Output Noise** + +Since humans are error-prone, hand labeled data should be checked for mislabeled features. Data that is mislabeled will introduce modeling errors, since the model will not know which label is correct. In this situation, it is better to remove the mislabeled points. No data is better than malformed data. + +**Overfitting** + +Trying to model the training data too closely can lead to overfitting. When you overfit a model it is not able to generalize to unseen data. A common example is when you use a higher-degree polynomial for your function. The higher degree polynomial will have more degrees of freedom to fit your training data, but will perform poorly on new data. Using a training set of examples that we feed to the model to learn from and a testing set to test the model at the end, we can test for overfitting. If we achieve a high accuracy on the training set (say 97%) and only a (34%) accuracy on the test set, then we have clearly tuned our model to only predict things it has seen before. Recall that a test set is not shown to the model until training has finished and it is ready to be evaluated. The test set contains data points that are representative of the problem we are trying to solve, but have been withheld from the model so we can test performance on new data. + +**Redundant Data** + +If the input features are highly correlated, then certain models perform worse. For instance Random Forest models are good at detecting interactions between features, but highly correlated data can mask these interactions[^1]. + +**Presence of Interactions and Non-linearities** + +Models that are linear perform well on data with linearities. If your data is complex, it likely is non-linear and will need some sort of mechanism to handle that interaction. Deep learning models with activation functions can handle non-linear data. We will cover that more when discussing activation functions. + +#### Semi-supervised Learning + +In semi-supervised learning you have a large number of un-labeled examples and a few labeled examples. This is a challenging paradigm, as it is far easier to work in the supervised learning domain where every example is labeled. Unfortunately, this is not often the case since labeling data is expensive and time consuming. Quickly hand labeling data leads to errors which is worse than having a set of high quality data. If data is abundant, yet labeling it all is too difficult then semi-supervised algorithms are ideal. + +![Semi-supervised Learning](https://adl-imgs.nyc3.cdn.digitaloceanspaces.com/machine_learning/deep_learning/novice/semi-supervised.svg "semi-supervised-learning") + +In the diagram above we can see a few points are labeled purple and others are pink. These points are the few examples of labeled data, where the white circles represent un-labeled examples that are more abundant. If we attempted to train a supervised model on the few examples that are labeled, we would likely produce a horizontal line that would cleanly separate the two sets. The problem is that line will mis-classify the points in the top set as being pink. + +A better algorithm would be a clustering algorithm that (such as $k$-means) that would label nearby points. Keep in mind that the problem here is not the model complexity; a linear model would fail as would a non-linear model since the non-labeled data is not being used to train the model. The model separates the two sets in a way that is consistent with the train examples it is fed. + +We will not cover semi-supervised learning in this series in any real detail. We plan to add more information on semi-supervised methods in another series in the future. + +![Reinforcement Learning](https://adl-imgs.nyc3.cdn.digitaloceanspaces.com/machine_learning/deep_learning/novice/rl-loop.svg "reinforcement-learning") + +In Reinforcement Learning, an agent (a robot, an animal, or even you) interacts with an environment (a game, a system, or the world) through actions. The environment responds to these actions by providing the agent a reward signal and transitioning the state of the environment through the environment's mechanics. + +An example of this is a self-driving car. In this scenario the agent is the car and the environment is the world. At every time interval the car makes a decision and takes an action. This action could be to continue driving straight, stop, make a left turn, etc. After the agent's action is taken, the environment provides some evolution of the world. For instance if the car drives straight, its location is now updated with a new location. The reward signal drives the agent toward a goal. In our scenario a reward signal can be the amount of gas expended at each time step (a reward can be negative, sometimes called a penalty), with a big payoff at the end when it reaches its destination. Deep Reinforcement Learning is out of the scope of this series and will be covered in a future series. + +## What To Expect + +We will focus mainly on the supervised learning case, deviating when appropriate. We will start by discussing learning algorithms and basic modeling functions, building up to more complex model architectures later in the series. We will see how each of these functions can be decomposed into building blocks that form graphs. Then we will cover loss functions and model optimizations. + +[^1]: [Stack Exchange Post](https://datascience.stackexchange.com/questions/24452/in-supervised-learning-why-is-it-bad-to-have-correlated-features) diff --git a/frontend/data/Mathematics/Calculus/derivatives.md b/frontend/data/Mathematics/Calculus/derivatives.md new file mode 100644 index 0000000..d914188 --- /dev/null +++ b/frontend/data/Mathematics/Calculus/derivatives.md @@ -0,0 +1,142 @@ +# Derivatives + +_The derivative is a way of showing that one quantity depends on another, even if it's not immediately obvious how._ + +## Introduction + +We are often interested in quantifying and measuring change in a system. The system under measure can be something as simple as a single variable function such as a line $y=mx+b$ or as complex as our universe. In either case we ask the question, "how did a small change of some quantity result in a change in another?". This commonly happens in real life, where small decisions throughout the day can produce different timelines that are the accumulation of minute decisions people in a large and complex system make. Indeed, you could argue that all of science focuses on the study of change, of cause and effect. While we won't study macro systems where there are countless variables contributing to outcomes, we will focus on smaller systems of one or a handful of independent variables. + +One way to capture the effect one variable has on another is through derivatives. Before we move on to formal definitions of derivatives and look at some of their properties, lets introduce some necessary symbology. Since we are mainly discussing change we will look at some different symbols used to represent change. The first is the Greek letter delta, which typically represents a small change in some quantity. Suppose we are working with the variable $x$, then $\Delta x$ is a change in that quantity. This change can be positive or negative. Given two values for $x$, say $x_1$ and $x_2$, we write $\Delta x = x_1 - x_2$ for the change in $x$ between $x_1$ and $x_2$. If this change is infinitesimal, then we would instead write $dx$. Think of the symbol $dx$ as mean "very small", whereas $\Delta x$ can be any change between two acceptable values of $x$. + +## Definition + +To define a derivative we should first state that derivatives measure the amount of change one variable has on another. This change is expressed as a ratio between the variable that is changing and the one which caused the change: + +$$\frac{dy}{dx}$$ + +In this fraction the numerator is the variable that changes with respect to $x$. The variable that we are measuring the effect of is the variable on the denominator. Note that there are many different ways that Mathematicians represent derivatives. It is common in differential equations to treat the derivative as a multiplier: + +$$ \frac{d}{dx} y $$ + +where the variable on the right is the one that is dependent on $x$. Given a function $y=f(x)$, it is common to see an apostrophe to represent a derivative: $f'(x)$. Some textbooks will also use an uppercase $D$, $D_x f(x)$, but this is not nearly as common as the others and is mainly used in Physics and Differential Equations. We will stick to the first way, using $dx$ and $dy$ since this is the most intuitive way. + +The formal definition of a derivative is the limit definition. An interesting historical note is that the letter $h$ is used in the definition through German convention. Mathematicians still use it today, although $\Delta x$ is becoming more common. + +Given some function $f(x)$, we can measure a small change in the function's value given a small change in $x$ by examining the ratio of the function difference $f(x + \Delta x) - f(x)$ to the change in $x$ given by the calculation $x + \Delta x - x$: + +$$ \frac{f(x+\Delta x) - f(x)}{x + \Delta x - x} = \frac{f(x+\Delta x) - f(x)}{\Delta x}$$ + +Derivatives measure "instantaneous" change, meaning $\Delta x$ cannot be arbitrarily large, it needs to be infinitesimally small. We can achieve this by taking the limit as $\Delta x$ approaches zero: + +$$\lim_{\Delta x \rightarrow 0} \frac{f(x+\Delta x) - f(x)}{\Delta x} = \frac{df}{dx}$$ + +If this limit exists, then the derivative exists. There is another requirement that must be satisfied, namely that the domain must contain an open ball around $x$ that contains $x$. Our goal is not to teach Real Analysis, but it is a caveat that needs to be stated in case there are any Mathematicians reading this. Note that if a function $f$ has a derivative at $x=a$, then $f$ is continuous at $x=a$. + +If this limit exists for every point in the function's domain, then the function is **differentiable**. + +## Rules of Differentiation + +I will not exhaustively define all of the rules of differentiation here, but will include some basic ones for quick reference. The goal of this document is to provide some intuition behind quantifying change as we build up to partial derivatives and onto Deep Learning. + +### Constants + +Since derivatives the change in one variable in a relation to another, the derivative of a constant must be zero. If $f(x) = c$, then + +$$ \frac{df}{dx} = \frac{d}{dx} c = 0$$ + +This is fairly straightforward to prove from the definition of a limit. If you feel uneasy with proofs you can feel free to skip them whenever they show up in this document, although I would encourage you to read them and even attempt them, it is the best way to learn Mathematics. + +If $f(x) = c$ is a constant function, then any change in that function's domain yields the same value. For instance if $f(x) = 5$ for all values of $x$ on the real number line, then $f(x) = 5$ and $f(x + \Delta x) = 5$, hence + +$$ +\begin{aligned} +\frac{df}{dx} &= \lim_{\Delta x \to 0} \frac{f(x + \Delta x) - f(x)}{\Delta x} \\ +&= \lim_{\Delta x \to 0} \frac{c - c}{\Delta x} \\ +&= \lim_{\Delta x \to 0} \frac{0}{\Delta x} \\ +&= \lim_{\Delta x \to 0} 0 \\ +&= 0 +\end{aligned} +$$ + +### Polynomials + +The first rule that is often taught is the derivative of a polynomial. This will be the only rule we derive, just to expose the reader to how these rules come into existence. + +Before deriving the formula, we need to quickly talk about the Binomial Theorem and polynomial expansions. You will often run into expressions of the form $(x+y)^n$ where $n$ is some integer. The expansion of this expression for the first few values of $n$ are as follows: + +$$ +\begin{aligned} +(x+y)^0 &= 1 \\ +(x+y)^1 &= (x+y) \\ +(x+y)^2 &= x^2 + 2xy + y^2 \\ +(x+y)^3 &= x^3 + 3x^2y + 3xy^2 + y^3 \\ +(x+y)^4 &= x^4 + 4x^3y + 6x^2y^2 + 4xy^3 + y^4 \\ +(x+y)^5 &= x^5 + 5x^4y + 10x^3y^2 + 10x^2y^3 + 5xy^4 + y^5 +\end{aligned} +$$ + +Do you notice a pattern forming? Notice that there is no $y$ term in the first quantity of the expansion, but $y$ is present in all of the following quantities of the summation. We can write a general expansion of this as + +$$(x+y)^n = x^n + nx^{n-1}y + \frac{n(n-1)}{2}x^{n-2}y^2 + \cdots + \frac{n(n-1)}{2}x^2y^{n-2} + nxy^{n-1} + y^n$$ + +Feel free to verify this equation with a few values of $n$. With this formula we can define the derivative of a polynomial: + +Let $f(x) = x^n$, + +$$ +\begin{aligned} +\frac{df}{dx} &= \lim_{\Delta x \rightarrow 0} \frac{f(x+\Delta x) - f(x)}{\Delta x} \\ +&= \lim_{\Delta x \rightarrow 0} \frac{(x + \Delta x)^n - x^n}{\Delta x} \\ +&= \lim_{\Delta x \rightarrow 0} \frac{x^n + nx^{n-1}\Delta x + \frac{n(n-1)}{2}x^{n-2}\Delta x^2 + \cdots - x^n}{\Delta x} \\ +&= \lim_{\Delta x \rightarrow 0} \frac{nx^{n-1}\Delta x + \frac{n(n-1)}{2}x^{n-2}\Delta x^2 + \cdots}{\Delta x} \\ +&= \lim_{\Delta x \rightarrow 0} nx^{n-1} + \frac{n(n-1)}{2}x^{n-2}\Delta x + \cdots \\ +&= nx^{n-1} +\end{aligned} +$$ + +Line one is the standard definition of a derivative using the limit definition and line two substitutes the functions in place of $f$. We used the binomial expansion to expand $(x+\Delta x)^n$ on line three. On line four we canceled the two $x^n$ terms and finally on line five we divided all of the terms by $\Delta x$. Finally, on line six we have the result after all of the quantities with $\Delta x$ values are taken to zero (multiplying any amount by zero yields zero). + +Keep in mind that coefficients can be factored out using standard limit rules, hence + +$$ \frac{d}{dx}cx^n = cnx^{n-1}.$$ + +## Trigonometric Functions + +For the following functions we will show the derivation of the derivative of $\sin(x)$, the rest are computed in a similar manner, using the appropriate trigonometry identities. + +Recall the angle sum identity for $\sin(x)$: + +$$\sin(x + h) = \sin(x)\cos(h) + \cos(x)\sin(h),$$ + +we can use this and the limit definition to compute the derivative of sine. + +$$ +\begin{aligned} +\frac{d \sin(x)}{dx} &= \lim_{\Delta x \to 0} \frac{f(x + \Delta x) - f(x)}{\Delta x} \\ +&= \lim_{\Delta x \to 0} \frac{\sin(x + \Delta x) - \sin(x)}{\Delta x}\\ +&= \lim_{\Delta x \to 0} \frac{(\sin(x)\cos(\Delta x) + \cos(x)\sin(\Delta x)) - \sin(x)}{\Delta x} \\ +&= \lim_{\Delta X \to 0} \frac{\sin(x)\cos(\Delta x - 1) + \cos(x)\sin(\Delta x)}{\Delta x} \\ +&= \lim_{\Delta x \to 0} \left(\sin(x)\cdot \frac{\cos(\Delta x - 1)}{\Delta x} \right) + \lim_{\Delta x \to 0} \left(\cos(x)\cdot \frac{\sin(\Delta x - 1)}{\Delta x} \right) \\ +&= \sin(x) \cdot \lim_{\Delta x \to 0} \frac{\cos(\Delta x - 1)}{\Delta x} + \cos(x) \cdot \lim_{\Delta x \to 0} \frac{\sin(\Delta x)}{\Delta x} \\ +&= \sin(x) \cdot 0 + \cos(x) \cdot 1 \\ +&= \cos(x) +\end{aligned} +$$ + +We will list the rest of the identities below. + +$$ +\begin{aligned} +\frac{d}{dx} \cos(x) &= -\sin(x) \\ +\frac{d}{dx} \tan(x) &= \sec^2(x) \\ +\frac{d}{dx} \cot(x) &= - \csc^2(x) \\ +\frac{d}{dx} \sec(x) &= \sec(x)\tan(x) \\ +\frac{d}{dx} \csc(x) &= -\csc(x)\cot(x) +\end{aligned} +$$ + +![Epsilon-Delta Limit](https://adl-imgs.nyc3.cdn.digitaloceanspaces.com/calculus/derivatives/epsilon-delta.svg "epsilon-delta-limit") + +![Product Rule](https://adl-imgs.nyc3.cdn.digitaloceanspaces.com/calculus/derivatives/product-rule.svg "product-rule") + +![Chain Rule](https://adl-imgs.nyc3.cdn.digitaloceanspaces.com/calculus/derivatives/chain-rule.svg "chain-rule") diff --git a/frontend/data/Mathematics/Calculus/partial_derivatives.md b/frontend/data/Mathematics/Calculus/partial_derivatives.md new file mode 100644 index 0000000..cba5baa --- /dev/null +++ b/frontend/data/Mathematics/Calculus/partial_derivatives.md @@ -0,0 +1,81 @@ +# Partial Derivatives + +_Generalization of derivatives to multivariate functions_ + +## Introduction + +Partial derivatives extend the idea of instantaneous rate of change to functions of more than one variable. While functions of a single variable such as exponential functions $e^x$, polynomial functions $y=c +a_1x + \cdots + a_nx^n$, and trignometric functions $y=sin(x)$ are common, in the Machine Learning world it is rare for an estimator to have only a single independent variable. In this document we will describle how to extend the concept of differentiation to functions of more than one variable and examine some of the more pertinent properties. + +## Multivariate Functions + +Before we go define differentiation in multiple dimensions, we will first review multivariate functions. If you are comfortable with this theory already then you can move on to the next section (the outline will link you to the next section). + +Real-valued functions of more than one variable, or **real multivariate functions**, are defined in a similar manner to real-valued functions of a single variable, or **real univariate functions**: + +--- + +Suppose $D$ is a set of $n$-tuples of real numbers $(x_1, x_2, \ldots, x_n)$. A real multivariate function $f$ on $D$ is a rule that assigns a unique real number + +$$w=f(x_1, x_2, \ldots, x_n)$$ + +to each element in $D$. The set $D$ is the function's **domain**. The set of $w$-values taken on by $f$ if the function's **range**. The symbol $w$ if the **dependent variable** of $f$, and $f$ is said to be a function of the $n$ **independent variables** $x_1, x_2, \ldots, x_n$. We also call the $x_j$'s the function's **input variables** and call $w$ the function's **output variables**. + +--- + +**Note**: when dealing with real-valued functions, we consider the domain to be the set of input variables that yield a real result (unless otherwise stated) + +It is useful to define the interior and boundary of a real-valued space. Before moving on we will introduce some supporting symbology. To denote the space of real-valued singletons, $x$, we write $\mathbb{R}$. For pairs of real-valued points we write $\mathbb{R}^2$. Following this trend, the set of all $n$-dimensional real-valued point is denoted $\mathbb{R}^n$. + +--- + +A point $\vec{x} = (x_1, x_2, \ldots x_n)$ in $R$ is called an **interior point** of $R$ if there exists some infinitesimal $\delta > 0$ such that all points $\delta$ from $\vec{x}$ are contained within $R$. Below is an illustration of this in two dimensions over some region $R$. + +![Interior Point Illustration](https://adl-imgs.nyc3.cdn.digitaloceanspaces.com/interior-point.svg "interior point illustration") + +Notice that no matter how small we choose $\delta$, all of the points in the disc are contained within $R$. Note that in two dimensions, the set of all points $\delta$ away from $\vec{x}$ is referred to as a "disc" and in three dimensions it is referred to as a "ball". When discussing regions of more than three dimensions, it is common to still refer to this set as a ball. + +If no matter how small we select $\delta$ to be, our ball still contains points outside of $R$, then $\vec{x}$ is called a **boundary point**. + +![Boundary Point Illustration](https://adl-imgs.nyc3.cdn.digitaloceanspaces.com/boundary-point.svg "boundary point illustration") + +Since $\vec{x}$ is in $R$, a ball will always contain _some_ points of $R$. The set of all interior points is referred to as the **interior** and the set of all boundary points is called the **boundary**. + +A region is **open** if it consists entirely of interior points (the set is equal to is interior) and **closed** if it contains its interior and boundary. + +Examples of open sets are any open interval $(a, b)$ where $a, b \in \mathbb{R}$ (if you are not familiar with the symbol $\in$, it means "member of", thus both $a$ and $b$ are real numbers). The interval $[a, b]$ is closed since you cannot select a $\delta$ such that the interval is entirely contained within the interval. Intervals with one $>$ or $<$ condition are open, for instance consider $(a, b]$, point $b$ is a boundary point since it will always contain points outside of the interval no matter what radius you select, whereas $a$ is an interior point, hence the interval is open. + +## Limits + +We say that a function $f(x_1, x_2)$ approaches the **limit** $L$ as $(x_1, x_2)$ approaches $(a, b)$, and write + +$$\lim_{(x_1, x_2) \rightarrow (a, b)} f(x_1, x_2) = L$$ + +if, for every number $\epsilon > 0$, there exists a corresponding number $\delta > 0$ such that for all $(x_1, x_2)$ in the domain of $f$, + +$$\lvert f(x_1, x_2) - L \rvert < \epsilon \,\,\, \text{whenever} \,\,\, 0 < \sqrt{(x_1 - a)^2 + (x_2 - b)^2} < \delta$$ + +![Limit Illustration](https://adl-imgs.nyc3.cdn.digitaloceanspaces.com/limit-def.svg "limit illustration") + +--- + +A function $f(x_1, x_2)$ is **continuous at the point $(a, b)$** if + +1. $f$ is defined at $(a, b)$, +2. $\lim_{(x_1, x_2) \rightarrow (a,b)} f(x_1, x_2)$ exists, +3. $\lim_{(x_1, x_2) \rightarrow (a,b)} f(x_1, x_2) = f(a, b)$. + +A function is **continuous** if it is continuous at every point of its domain. + +--- + +Composite functions are continuous at a point $(a, b)$ if the outer function is a continuous univariate function and the inner function is continuous at $(a, b)$; in other words, if $f$ is continuous at $(a, b)$ and $g$ is a univariate continuous function, then + +$$h(x_1, x_2) = g \circ f (x_1, x_2) = g(f(x_1, x_2))$$ + +is continuous at $(a, b)$. + +For example $f(x_1, x_2) = x_1 - x_2$ is continuous at $(a, b)$ and $e^{x_1}$ is a continuous univariate function, hence + +$$e^{x_1 - y_1}$$ + +is continuous at $(a, b)$. diff --git a/frontend/data/Mathematics/Linear Algebra/matrix_multiplication.md b/frontend/data/Mathematics/Linear Algebra/matrix_multiplication.md new file mode 100644 index 0000000..5ed34d8 --- /dev/null +++ b/frontend/data/Mathematics/Linear Algebra/matrix_multiplication.md @@ -0,0 +1,254 @@ +# Matrix Multiplication + +_Matrix multiplication is used in many applications and is a fundamental operation in linear algebra. It has a rich and beautiful theory, which is often hidden behind the mechanical process of multiplying rows and columns._ + +## Introduction + +The rules behind matrix multiplication may seem arcane at first, but they are easy to memorize. We will give an intuitive answer at the end of this document once the mechanics of matrix multiplication are explained. + +## Definition + +First we will provide the formal definition of matrix multiplication and in the next section we will work through a practical example. + +If you need a refresher on any of the notation below, see the "Matrix Basics" document for more information. + +If $A = [a_{ij}]$ is an $m \times n$ matrix and $B=[b_{ij}]$ is an $n \times p$ matrix, then the **product AB** is an $m \times p$ matrix + +$$ AB = [c_{ij}]$$ + +where + +$$c_{ij} = \sum_{k=1}^n a_{ik}b_{kj} = a_{i1}b_{1j} + a_{i2}b_{2j} + \cdots + a_{in}b_{nj} $$ + +## Practical Example + +We will start with two small matrices $A$ and $B$: + +$$ +A = \begin{bmatrix} +-1 & 3 \\ +4 & -2 \\ +5 & 0 +\end{bmatrix} +$$ + +and + +$$ +B = \begin{bmatrix} +-3 & 2 \\ +-4 & 1 +\end{bmatrix}. +$$ + +To find the product $AB$ we will set it equal to $C$ and look at how each element of $C$ is computed: + +$$ +AB = \begin{bmatrix} +-1 & 3 \\ +4 & -2 \\ +5 & 0 +\end{bmatrix}\begin{bmatrix} +-3 & 2 \\ +-4 & 1 +\end{bmatrix} = \begin{bmatrix} +c_{11} & c_{12} \\ +c_{21} & c_{22} \\ +c_{31} & c_{32} +\end{bmatrix} +$$ + +The first value $c_{11}$ is computed by taking the first row in $A$ and the first column $B$, + +$$c_{11} = (-1)(-3) + (3)(-4) = -9.$$ + +An easy way to remember this is that you take the row of the value being computed (in this case 1) and take the corresponding row from the first matrix in the product $A$. In this case, that is $[-1, 3]$. We then look at the column of the value to be computed (again 1) and take the corresponding column from the second matrix, $B$: $[-3; -4]$. + +After performing the computation we have the following matrix: + +$$ +AB = \begin{bmatrix} +-1 & 3 \\ +4 & -2 \\ +5 & 0 +\end{bmatrix}\begin{bmatrix} +-3 & 2 \\ +-4 & 1 +\end{bmatrix} = \begin{bmatrix} +-9 & c_{12} \\ +c_{21} & c_{22} \\ +c_{31} & c_{32} +\end{bmatrix} +$$ + +Lets compute the value of the second position, $c_{12}$. We will take the first row of $A$ and the second column of $B$ and take their inner product (note this is just an easy way to describe this calculation and matrix multiplication is not comprised of a series of inner products). + +$$c_{12} = (-1)(2) + (3)(1) = 1$$ + +That completes the first row of $C$: + +$$ +AB = \begin{bmatrix} +-1 & 3 \\ +4 & -2 \\ +5 & 0 +\end{bmatrix}\begin{bmatrix} +-3 & 2 \\ +-4 & 1 +\end{bmatrix} = \begin{bmatrix} +-9 & 1 \\ +c_{21} & c_{22} \\ +c_{31} & c_{32} +\end{bmatrix} +$$ + +The second row can be computed by taking the second row of $A$ and the first and second columns of $B$ for each value of $C$ respectively, + +$$c_{21} = (4)(-3) + (-2)(-4) = -4$$ + +and + +$$c_{22} = (4)(2) + (-2)(1) = 6. $$ + +This yields the following matrix $C$: + +$$ +AB = \begin{bmatrix} +-1 & 3 \\ +4 & -2 \\ +5 & 0 +\end{bmatrix}\begin{bmatrix} +-3 & 2 \\ +-4 & 1 +\end{bmatrix} = \begin{bmatrix} +-9 & 1 \\ +-4 & 6 \\ +c_{31} & c_{32} +\end{bmatrix} +$$ + +To compute the last row of $C$, we will repeat this process with the last row of $A$ and the first and second columns of $B$. + +$$c_{31} = (5)(-3) + (0)(-4) = -15$$ + +and + +$$c_{32} = (5)(2) + (0)(1).$$ + +This completes the multiplication and gives + +$$ +AB = \begin{bmatrix} +-1 & 3 \\ +4 & -2 \\ +5 & 0 +\end{bmatrix}\begin{bmatrix} +-3 & 2 \\ +-4 & 1 +\end{bmatrix} = \begin{bmatrix} +-9 & 1 \\ +-4 & 6 \\ +-15 & 10 +\end{bmatrix} +$$ + +### Matrix Multiplication Requirements + +For two matrices to be multiplied, it is necessary that their inner dimensions agree. If the dimensions of $A$ are $m \times n$, then the first dimension of $B$ must be $n$. In the above example we had $3 \times 2$ and $2 \times 2$, so we could multiply the matrices. + +### Matrix Multiplication In More Dimensions + +The pattern described above works for matrices of any dimensions (as long as the two multiplied matrices inner dimensions agree). Taking the inner product of the corresponding row of the first matrix and the column of the second work for all multipliable matrices. + +$$ +\begin{bmatrix} +a_{11} & a_{12} & \cdots & a_{1n} \\ +a_{21} & a_{22} & \cdots & a_{2n} \\ +\vdots & \vdots & & \vdots \\ +a_{i1} & a_{i2} & \cdots & a_{in} \\ +\vdots & \vdots & & \vdots \\ +a_{m1} & a_{m2} & \cdots & a_{mn} +\end{bmatrix} +\begin{bmatrix} +b_{11} & b_{12} & \cdots & b_{1j} & \cdots & b_{1p} \\ +b_{21} & b_{22} & \cdots & b_{2j} & \cdots & b_{2p} \\ +\vdots & \vdots & & \vdots & & \vdots \\ +b_{n1} & b_{n2} & \cdots & b_{nj} & \cdots & b_{np} \\ +\end{bmatrix} += +\begin{bmatrix} +c_{11} & c_{12} & \cdots & c_{1j} & \cdots & c_{1p} \\ +c_{21} & c_{22} & \cdots & c_{2j} & \cdots & c_{2p} \\ +\vdots & \vdots & & \vdots & & \vdots \\ +c_{i1} & c_{i2} & \cdots & c_{ij} & \cdots & c_{ip} \\ +\vdots & \vdots & & \vdots & & \vdots \\ +c_{m1} & c_{m2} & \cdots & c_{mj} & \cdots & c_{mp} \\ +\end{bmatrix} +$$ + +where + +$$c_{ij} = a_{i1}b_{1j} + a_{i2}b_{2j} + \cdots a_{in}b_{nj}$$ + +## Properties + +There are a few properties of matrix multiplication that are good to remember: + +- $A(BC) = (AB)C$ (associative property) +- $A(B+C) = AB + AC$ (distributive property) +- $(A+B)C = AC + BC$ (distributive property) +- $c(AB) = (cA)B = A(cB)$ + +These properties follow from the definition, and can be proven without any additional mechanics. Since matrix multiplication is associative, we usually drop any parentheses, and simply write $ABC$. + +Note that $AB \neq BA$. This is easily shown with an example: + +Let + +$$ +A = \begin{bmatrix} +1 & 3 \\ +2 & -1 +\end{bmatrix} +$$ + +and + +$$ +B = \begin{bmatrix} +2 & -1 \\ +0 & 2 +\end{bmatrix} +$$ + +then $AB$ equals + +$$ +AB = \begin{bmatrix} +1 & 3 \\ +2 & -1 +\end{bmatrix}\begin{bmatrix} +2 & -1 \\ +0 & 2 +\end{bmatrix}=\begin{bmatrix} +2 & 5 \\ +4 & -4 +\end{bmatrix} +$$ + +yet + +$$ +BA = \begin{bmatrix} +2 & -1 \\ +0 & 2 +\end{bmatrix}\begin{bmatrix} +1 & 3 \\ +2 & -1 +\end{bmatrix}=\begin{bmatrix} +0 & 7 \\ +4 & -2 +\end{bmatrix} +$$ + +Sometimes $AB=BA$, but in general this isn't true. diff --git a/frontend/src/routes/docs/[...document]/+page.svelte b/frontend/src/routes/docs/[...document]/+page.svelte index 8b899bd..51ec127 100755 --- a/frontend/src/routes/docs/[...document]/+page.svelte +++ b/frontend/src/routes/docs/[...document]/+page.svelte @@ -178,7 +178,7 @@ grid-template-rows: 1fr 320px; align-items: start; justify-content: center; - background-color: var(--background-1); + background-color: var(--background-3); width: 100%; height: calc(100vh - 2.5rem - 4rem); max-height: calc(100vh - 2.5rem - 4rem); @@ -207,58 +207,15 @@ overflow: scroll; } + :global(*)::-webkit-scrollbar { + display: none; + } + :global([data-heading-rank='1']) { height: calc(100vh - 2.5rem - 4rem); max-height: calc(100vh - 2.5rem - 4rem); } - :global(#document section) { - width: 100%; - max-width: 100%; - display: flex; - flex-direction: column; - align-items: center; - } - - /*noinspection ALL*/ - :global(#document div.document-code-block-container) { - display: grid; - grid-template-rows: 3rem 1fr; - background: var(--background-9); - border-radius: 7px; - padding: 10px; - margin-top: 20px; - margin-bottom: 20px; - width: 100%; - } - - :global(#document div.document-code) { - max-width: calc(100vw - 640px); - word-wrap: pre-wrap; - overflow: hidden; - } - - :global(#document .language-shell) { - background: var(--background-2); - } - - :global(#document code) { - word-wrap: break-word; - max-width: calc(100vw - 640px); - } - - /*noinspection ALL*/ - :global(#document div.document-code-heading-container) { - display: flex; - flex-direction: row; - gap: 17px; - margin-left: 6px; - } - - :global(#document span.document-code-prompt) { - color: var(--font-5); - } - :global(.document-code-img) { height: 2rem !important; filter: invert(52%) sepia(6%) saturate(194%) hue-rotate(202deg) @@ -271,15 +228,17 @@ cursor: pointer; } - :global(#document a) { - color: var(--highlight); - font-family: var(--f-Regular), sans-serif; - font-size: 1.8rem; - align-self: start; - - } - #document { + & :global(section) { + width: 100%; + max-width: 100%; + display: flex; + flex-direction: column; + align-items: center; + } + + // List Styling + & :global(li::marker) { content: counters(item, ".", upper-roman)"."; text-align: center; @@ -292,6 +251,20 @@ line-height: 1.5; } + :global(li) { + padding: 0 2rem; + font-size: 2rem; + color: var(--font-2); + } + + :global(ul), + :global(ol) { + color: var(--font-2); + margin-left: 20px; + font-family: var(--f-Regular), sans-serif; + font-size: 2rem; + } + & :global(em) { font-style: italic; } @@ -330,166 +303,193 @@ height: auto; } - & :global(pre) { - padding: 1.25em 1em; + // Text Styling + + & :global(p) { + padding: 1rem 2rem; + color: var(--font-2); + line-height: 1.5; + font-family: var(--f-Regular), sans-serif; + font-size: 2rem; + align-self: start; + } + + & :global(a) { + color: var(--highlight); + font-family: var(--f-Regular), sans-serif; + font-size: 1.8rem; + align-self: start; } - } + :global(h1) { + font-family: var(--f-ExtraBold), sans-serif; + font-size: 4rem; + font-weight: 100; + padding: 3rem 0; + } - :global(#document h1) { - - font-family: var(--f-ExtraBold), sans-serif; - font-size: 4rem; - font-weight: 100; - padding: 3rem 0; - } + :global(h2) { + font-family: var(--f-Bold), sans-serif; + font-size: 3rem; + opacity: 0.8; + padding: 2rem 0; + } - :global(#document h2) { - font-family: var(--f-Bold), sans-serif; - font-size: 3rem; - opacity: 0.8; - padding: 2rem 0; - } + :global(h3) { + font-family: var(--f-Bold), sans-serif; + font-size: 2.3rem; + opacity: 0.8; + padding: 1.3rem 0; + } - :global(#document h3) { - font-family: var(--f-Bold), sans-serif; - font-size: 2.3rem; - opacity: 0.8; - padding: 1.3rem 0; - } + :global(h4) { + font-family: var(--f-Bold), sans-serif; + font-size: 2rem; + opacity: 0.8; + padding: 1rem 0; + } - :global(#document h4) { - font-family: var(--f-Bold), sans-serif; - font-size: 2rem; - opacity: 0.8; - padding: 1rem 0; - } + :global(h5) { + font-family: var(--f-Bold), sans-serif; + font-size: 1.5rem; + opacity: 0.8; + padding: 1rem 0; + } - :global(#document h5) { - font-family: var(--f-Bold), sans-serif; - font-size: 1.5rem; - opacity: 0.8; - padding: 1rem 0; - } + :global(h6) { + font-family: var(--f-Bold), sans-serif; + font-size: 1.3rem; + opacity: 0.8; + padding: 1rem 0; + } - :global(#document h6) { - font-family: var(--f-Bold), sans-serif; - font-size: 1.3rem; - opacity: 0.8; - padding: 1rem 0; - } - :global(.katex) { - overflow: auto hidden; - position: relative; - } + // Code Block Styling - // :global(.katex-display) { - // margin: 0 !important; - // } + & :global(pre) { + padding: 1.25em 1em; + } - :global(.katex-display) { - overflow: auto hidden; - } + :global(code) { + word-wrap: break-word; + max-width: calc(100vw - 640px); + } - :global(.katex-display span) { - color: var(--font-2); - font-family: 'KaTeX_Main', serif; - font-size: 2.5rem; - line-height: 3; - // text-overflow: ellipsis; - text-align: start; - } + :global(div.document-code-block-container) { + display: grid; + grid-template-rows: 3rem 1fr; + background: var(--background-9); + border-radius: 7px; + padding: 10px; + margin-top: 20px; + margin-bottom: 20px; + width: 100%; + } - :global(p:has(.katex)) { - text-align: center; - - } + :global(div.document-code) { + max-width: calc(100vw - 640px); + word-wrap: pre-wrap; + overflow: hidden; + } - :global(#document p) { - padding: 1rem 2rem; - color: var(--font-2); - line-height: 1.5; - font-family: var(--f-Regular), sans-serif; - font-size: 2rem; - align-self: start; - } + :global(span.document-code-prompt) { + color: var(--font-5); + } - :global(#document p:has( > span.katex)) { - align-self: center; - } + :global(div.document-code-heading-container) { + display: flex; + flex-direction: row; + gap: 17px; + margin-left: 6px; + } - :global(#document :is(th tr)) { - color: var(--font-2); - } + :global(.language-shell) { + background: var(--background-2); + } - :global(#document li) { - padding: 0 2rem; - font-size: 2rem; - color: var(--font-2); - } + // Math Styling - :global(#document ul), - :global(#document ol) { - color: var(--font-2); - margin-left: 20px; - font-family: var(--f-Regular), sans-serif; - font-size: 2rem; - } + :global(.katex) { + overflow: auto hidden; + position: relative; + } + :global(.katex-display) { + overflow: auto hidden; + } -// ====================== TABLE ====================== + :global(.katex-display span) { + color: var(--font-2); + font-family: 'KaTeX_Main', serif; + font-size: 2.5rem; + line-height: 3; + // text-overflow: ellipsis; + text-align: start; + } - :global(thead) { - background-color: var(--background-2); - font-family: var(--f-Medium), sans-serif; - font-size: 2rem; - } + :global(p:has(.katex)) { + text-align: center; + + } - :global(th) { - padding: 10px; - color: var(--font-3); - text-align: center; - } + :global(p:has( > span.katex)) { + align-self: center; + } - :global(tr) { - color: var(--font-2); - font-family: var(--f-Regular), sans-serif; - font-size: 1.8rem; - } + :global(code .code-line span) { + text-overflow: ellipsis; + overflow: hidden; + white-space: pre-wrap; + } - :global(td) { - border: solid 1px var(--background-2); - color: var(--font-2); - font-family: var(--f-Medium), sans-serif; - font-size: 2rem; - text-align: center; - } + :global(code .code-line span) { + text-overflow: ellipsis; + overflow: hidden; + white-space: pre-wrap; + } - :global(code .code-line span) { - text-overflow: ellipsis; - overflow: hidden; - white-space: pre-wrap; - } + :global(li > code), + :global(p > code) { + font-family: var(--f-Mono), monospace; + font-size: 2rem; + line-height: 1.5em; + padding: 5px; + border-radius: 10px; + background-color: var(--background-2); + } - :global(code .code-line span) { - text-overflow: ellipsis; - overflow: hidden; - white-space: pre-wrap; - } + // Table Styling - :global(li > code), - :global(p > code) { - font-family: var(--f-Mono), monospace; - font-size: 2rem; - line-height: 1.5em; - padding: 5px; - border-radius: 10px; - background-color: var(--background-2); - } + :global(:is(th tr)) { + color: var(--font-2); + } + + :global(thead) { + background-color: var(--background-2); + font-family: var(--f-Medium), sans-serif; + font-size: 2rem; + } + + :global(th) { + padding: 10px; + color: var(--font-3); + text-align: center; + } + + :global(tr) { + color: var(--font-2); + font-family: var(--f-Regular), sans-serif; + font-size: 1.8rem; + } + + :global(td) { + border: solid 1px var(--background-2); + color: var(--font-2); + font-family: var(--f-Medium), sans-serif; + font-size: 2rem; + text-align: center; + } - :global(*)::-webkit-scrollbar { - display: none; } /* ====================== OUTLINE ====================== */ From ee69e487610729d8285e14b028478fbd21a6b54c Mon Sep 17 00:00:00 2001 From: Dalton Walker Date: Fri, 24 May 2024 11:46:16 -0400 Subject: [PATCH 2/3] fixes some of the math formatting --- frontend/src/app.html | 17 +++++- .../routes/docs/[...document]/+page.svelte | 50 +++++++----------- frontend/static/css/fonts.css | 12 +++++ frontend/static/fonts/STIX-Italic.woff2 | Bin 0 -> 174452 bytes frontend/static/fonts/STIX.woff2 | Bin 0 -> 158184 bytes 5 files changed, 46 insertions(+), 33 deletions(-) create mode 100644 frontend/static/fonts/STIX-Italic.woff2 create mode 100644 frontend/static/fonts/STIX.woff2 diff --git a/frontend/src/app.html b/frontend/src/app.html index c863e9f..3cea0da 100755 --- a/frontend/src/app.html +++ b/frontend/src/app.html @@ -5,7 +5,22 @@ - + + + + + %sveltekit.head% diff --git a/frontend/src/routes/docs/[...document]/+page.svelte b/frontend/src/routes/docs/[...document]/+page.svelte index 51ec127..1ac97ee 100755 --- a/frontend/src/routes/docs/[...document]/+page.svelte +++ b/frontend/src/routes/docs/[...document]/+page.svelte @@ -178,7 +178,7 @@ grid-template-rows: 1fr 320px; align-items: start; justify-content: center; - background-color: var(--background-3); + background-color: var(--background-1); width: 100%; height: calc(100vh - 2.5rem - 4rem); max-height: calc(100vh - 2.5rem - 4rem); @@ -309,7 +309,7 @@ padding: 1rem 2rem; color: var(--font-2); line-height: 1.5; - font-family: var(--f-Regular), sans-serif; + font-family: 'KaTeX_Main', sans-serif; font-size: 2rem; align-self: start; } @@ -322,7 +322,7 @@ } :global(h1) { - font-family: var(--f-ExtraBold), sans-serif; + font-family: var(--f-STIX), sans-serif; font-size: 4rem; font-weight: 100; padding: 3rem 0; @@ -407,35 +407,6 @@ background: var(--background-2); } - // Math Styling - - :global(.katex) { - overflow: auto hidden; - position: relative; - } - - :global(.katex-display) { - overflow: auto hidden; - } - - :global(.katex-display span) { - color: var(--font-2); - font-family: 'KaTeX_Main', serif; - font-size: 2.5rem; - line-height: 3; - // text-overflow: ellipsis; - text-align: start; - } - - :global(p:has(.katex)) { - text-align: center; - - } - - :global(p:has( > span.katex)) { - align-self: center; - } - :global(code .code-line span) { text-overflow: ellipsis; overflow: hidden; @@ -458,6 +429,21 @@ background-color: var(--background-2); } + // Math Styling + + :global(.katex) { + font-size: 1.1em; + } + + :global(.katex-display) { + font-size: 2rem; + } + + :global(span.katex span) { + color: var(--font-1); + // font-family: 'STIX', serif; + } + // Table Styling :global(:is(th tr)) { diff --git a/frontend/static/css/fonts.css b/frontend/static/css/fonts.css index 36235eb..27e39b3 100755 --- a/frontend/static/css/fonts.css +++ b/frontend/static/css/fonts.css @@ -1,4 +1,6 @@ :root { + --f-STIX: 'STIX'; + /* ========= MONOSPACE ========= */ --f-Mono: 'FiraCode.VF'; @@ -24,6 +26,16 @@ /* Fonts */ +@font-face { + font-family: 'STIX'; + src: url('../fonts/STIX.woff2'); +} + +@font-face { + font-family: 'STIX-Italic'; + src: url('../fonts/STIX-Italic.woff2'); +} + /* ===================================== INTER ===================================== */ @font-face { font-family: 'Inter.Thin'; diff --git a/frontend/static/fonts/STIX-Italic.woff2 b/frontend/static/fonts/STIX-Italic.woff2 new file mode 100644 index 0000000000000000000000000000000000000000..25ea77f10ca1cd6cf4d0a2ec7117d25d88a53070 GIT binary patch literal 174452 zcmV)LK)JtnPew8T0RR910;zNW6aWAK2K4{{0;vE10wG8M00000000000000000000 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zGl7hsrTxflHwv6?WSz-|3v3vs8wuHPFB=BJhC58)?mV{|)^RU$)cfpXaHSM>U>ovr z(~4HiiNoTsXvLhE6NklNF(>B40jc5OA?V7BAidmRyo%y`0R^@7YdA1C+;Kja~73dg}d@xY>k`q=~Eb zs=z4vd0<>pPy9WlYodhjt=R%zp||^M8eOcE4gZADUN-`Pa;1%B=M(}&meT@u7(p?dAf;kDTBWDp&Hvk31pol$ZrB9? literal 0 HcmV?d00001 From 3b83038bf0b27e68678f04b2028c5f9b7e1da7d4 Mon Sep 17 00:00:00 2001 From: Dalton Walker Date: Fri, 24 May 2024 14:26:32 -0400 Subject: [PATCH 3/3] finished math reformatting --- .../data/Mathematics/Calculus/derivatives.md | 18 ++++++++++++------ frontend/package.json | 1 + frontend/src/app.html | 12 +++++++++--- .../routes/docs/[...document]/+page.server.ts | 1 + .../src/routes/docs/[...document]/+page.svelte | 9 ++++++--- 5 files changed, 29 insertions(+), 12 deletions(-) diff --git a/frontend/data/Mathematics/Calculus/derivatives.md b/frontend/data/Mathematics/Calculus/derivatives.md index d914188..237d684 100644 --- a/frontend/data/Mathematics/Calculus/derivatives.md +++ b/frontend/data/Mathematics/Calculus/derivatives.md @@ -24,11 +24,15 @@ The formal definition of a derivative is the limit definition. An interesting hi Given some function $f(x)$, we can measure a small change in the function's value given a small change in $x$ by examining the ratio of the function difference $f(x + \Delta x) - f(x)$ to the change in $x$ given by the calculation $x + \Delta x - x$: -$$ \frac{f(x+\Delta x) - f(x)}{x + \Delta x - x} = \frac{f(x+\Delta x) - f(x)}{\Delta x}$$ +$$ +\frac{f(x+\Delta x) - f(x)}{x + \Delta x - x} = \frac{f(x+\Delta x) - f(x)}{\Delta x} +$$ Derivatives measure "instantaneous" change, meaning $\Delta x$ cannot be arbitrarily large, it needs to be infinitesimally small. We can achieve this by taking the limit as $\Delta x$ approaches zero: -$$\lim_{\Delta x \rightarrow 0} \frac{f(x+\Delta x) - f(x)}{\Delta x} = \frac{df}{dx}$$ +$$ +\lim_{\Delta x \rightarrow 0} \frac{f(x+\Delta x) - f(x)}{\Delta x} = \frac{df}{dx} +$$ If this limit exists, then the derivative exists. There is another requirement that must be satisfied, namely that the domain must contain an open ball around $x$ that contains $x$. Our goal is not to teach Real Analysis, but it is a caveat that needs to be stated in case there are any Mathematicians reading this. Note that if a function $f$ has a derivative at $x=a$, then $f$ is continuous at $x=a$. @@ -42,7 +46,9 @@ I will not exhaustively define all of the rules of differentiation here, but wil Since derivatives the change in one variable in a relation to another, the derivative of a constant must be zero. If $f(x) = c$, then -$$ \frac{df}{dx} = \frac{d}{dx} c = 0$$ +$$ +\frac{df}{dx} = \frac{d}{dx} c = 0 +$$ This is fairly straightforward to prove from the definition of a limit. If you feel uneasy with proofs you can feel free to skip them whenever they show up in this document, although I would encourage you to read them and even attempt them, it is the best way to learn Mathematics. @@ -135,8 +141,8 @@ $$ \end{aligned} $$ -![Epsilon-Delta Limit](https://adl-imgs.nyc3.cdn.digitaloceanspaces.com/calculus/derivatives/epsilon-delta.svg "epsilon-delta-limit") +![Epsilon-Delta Limit](https://adl-imgs.nyc3.cdn.digitaloceanspaces.com/calculus/derivatives/epsilon-delta.svg 'epsilon-delta-limit') -![Product Rule](https://adl-imgs.nyc3.cdn.digitaloceanspaces.com/calculus/derivatives/product-rule.svg "product-rule") +![Product Rule](https://adl-imgs.nyc3.cdn.digitaloceanspaces.com/calculus/derivatives/product-rule.svg 'product-rule') -![Chain Rule](https://adl-imgs.nyc3.cdn.digitaloceanspaces.com/calculus/derivatives/chain-rule.svg "chain-rule") +![Chain Rule](https://adl-imgs.nyc3.cdn.digitaloceanspaces.com/calculus/derivatives/chain-rule.svg 'chain-rule') diff --git a/frontend/package.json b/frontend/package.json index 15678e4..a90d87c 100644 --- a/frontend/package.json +++ b/frontend/package.json @@ -73,6 +73,7 @@ "rehype-document": "^7.0.3", "rehype-format": "^5.0.0", "rehype-katex": "^7.0.0", + "rehype-mathjax": "^6.0.0", "rehype-mermaidjs": "^2.0.0", "rehype-prism-plus": "^2.0.0", "rehype-raw": "^7.0.0", diff --git a/frontend/src/app.html b/frontend/src/app.html index 3cea0da..6b34511 100755 --- a/frontend/src/app.html +++ b/frontend/src/app.html @@ -5,22 +5,28 @@ - + + + - + > --> %sveltekit.head% diff --git a/frontend/src/routes/docs/[...document]/+page.server.ts b/frontend/src/routes/docs/[...document]/+page.server.ts index 8a13ecc..ca7ba71 100755 --- a/frontend/src/routes/docs/[...document]/+page.server.ts +++ b/frontend/src/routes/docs/[...document]/+page.server.ts @@ -5,6 +5,7 @@ import remarkRehype from 'remark-rehype'; import rehypeStringify from 'rehype-stringify'; import remarkMath from 'remark-math'; import rehypeKatex from 'rehype-katex'; +import rehypeMathjaxChtml from 'rehype-mathjax'; import remarkGfm from 'remark-gfm'; import rehypePrism from 'rehype-prism-plus'; import rehypeSectionize from '@hbsnow/rehype-sectionize'; diff --git a/frontend/src/routes/docs/[...document]/+page.svelte b/frontend/src/routes/docs/[...document]/+page.svelte index 1ac97ee..c0a136f 100755 --- a/frontend/src/routes/docs/[...document]/+page.svelte +++ b/frontend/src/routes/docs/[...document]/+page.svelte @@ -229,6 +229,10 @@ } #document { + :global(img:not(.img-logo)) { + border-radius: 2rem; + } + & :global(section) { width: 100%; max-width: 100%; @@ -309,7 +313,7 @@ padding: 1rem 2rem; color: var(--font-2); line-height: 1.5; - font-family: 'KaTeX_Main', sans-serif; + font-family: var(--f-Regular), sans-serif; font-size: 2rem; align-self: start; } @@ -322,7 +326,7 @@ } :global(h1) { - font-family: var(--f-STIX), sans-serif; + font-family: var(--f-ExtraBold), sans-serif; font-size: 4rem; font-weight: 100; padding: 3rem 0; @@ -441,7 +445,6 @@ :global(span.katex span) { color: var(--font-1); - // font-family: 'STIX', serif; } // Table Styling