Spelling corrections

00_intro.ipynb

@@ -63,7 +63,7 @@
    "source": [
     "### PDE Classification - Traditional View\n",
     "\n",
-    "As you may know there are a number of classifications for PDEs, the best known are elliptic, parabolic and hyperbolic.  These classifications are based on a canonincal linear, second order PDE of the form\n",
+    "As you may know there are a number of classifications for PDEs, the best known are elliptic, parabolic and hyperbolic.  These classifications are based on a canonical linear, second order PDE of the form\n",
     "$$\n",
     "    A u_{xx} + 2 B u_{xy} + C u_{yy} + \\cdots = 0.\n",
     "$$"
@@ -85,7 +85,7 @@
     "$$\n",
     "    d \\equiv B^2 - 4 A C,\n",
     "$$\n",
-    "deterimines the classification:\n",
+    "determines the classification:\n",
     " - if $d < 0$ then the equation is **elliptic**,\n",
     " - if $d = 0$ then the equation is **parabolic** and\n",
     " - if $d > 0$ then the equation is **hyperbolic**."
@@ -195,7 +195,7 @@
     "$$\n",
     "    q_t + u q_x = 0\n",
     "$$\n",
-    "where $u \\in \\mathbb R$ describes the transport of the quantity $q$ at a speed $u$.  This is analgous to wave motion."
+    "where $u \\in \\mathbb R$ describes the transport of the quantity $q$ at a speed $u$.  This is analogous to wave motion."
    ]
   },
   {
@@ -279,7 +279,7 @@
     "$$\n",
     "    \\frac{\\text{d}}{\\text{d} t} \\int^{x_2}_{x_1} q(x, t) dx = f(q(x_1, t)) - f(q(x_2, t)). \n",
     "$$\n",
-    "where $x_2 > x_1 \\in \\Omega$ and $\\Omega$ is some domain.  In words this is saying that if we integrate over the interval $[x_1, x_2]$ that the time rate of change on that interval is deterimined by the flux function evaluated at the ends of the interval."
+    "where $x_2 > x_1 \\in \\Omega$ and $\\Omega$ is some domain.  In words this is saying that if we integrate over the interval $[x_1, x_2]$ that the time rate of change on that interval is determined by the flux function evaluated at the ends of the interval."
    ]
   },
   {
@@ -398,7 +398,7 @@
     "$$\n",
     "    \\frac{\\text{D}}{\\text{D} t} (\\rho u) = \\sum forces.\n",
     "$$\n",
-    "Note that the right hand side of the equation no longer has an ordinary or partial derivative but instead has a **total derivative** or **matrial derivative**.  In a *Lagrangian coordinate systems* (our coordinate system moves with the fluid) this is identical to an ordinary derivative in time.  In *Eulerian coordinate systems* (we are fixed in space and the fluid moves past us) this is more complex.  Instead we have\n",
+    "Note that the right hand side of the equation no longer has an ordinary or partial derivative but instead has a **total derivative** or **material derivative**.  In a *Lagrangian coordinate systems* (our coordinate system moves with the fluid) this is identical to an ordinary derivative in time.  In *Eulerian coordinate systems* (we are fixed in space and the fluid moves past us) this is more complex.  Instead we have\n",
     "$$\n",
     "    \\frac{\\text{D}}{\\text{D} t} q = q_t + (u q)_x.\n",
     "$$\n",
@@ -692,7 +692,7 @@
     }
    },
    "source": [
-    "One way to finw the values of the functions $w^{1,2}$ is to evaluate the expression above at $t=0$ to get\n",
+    "One way to find the values of the functions $w^{1,2}$ is to evaluate the expression above at $t=0$ to get\n",
     "$$\n",
     "    w^1(x) r^1 + w^2(x) r^2 = q(x, 0).\n",
     "$$\n",
@@ -710,7 +710,7 @@
     }
    },
    "source": [
-    "Introducing a new variable, $Z_0 \\equiv \\rho_0 c_0$, called the **impedence**, allows us to write the inverse of $R$ as\n",
+    "Introducing a new variable, $Z_0 \\equiv \\rho_0 c_0$, called the **impedance**, allows us to write the inverse of $R$ as\n",
     "$$\n",
     "    R^{-1} = \\frac{1}{2 Z_0} \\begin{bmatrix}\n",
     "                -1 & Z_0 \\\\ 1 & Z_0\n",

01_linear_hyperbolic.ipynb

@@ -16,7 +16,7 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 1,
+   "execution_count": null,
    "metadata": {
     "slideshow": {
      "slide_type": "skip"
@@ -45,7 +45,7 @@
     "$$\n",
     "    q_t + A q_x = 0\n",
     "$$\n",
-    "can be transformed into a set of indepdent PDEs as $A$ is diagonalizable and with the new variables $w = R^{-1} q$ we can write\n",
+    "can be transformed into a set of independent PDEs as $A$ is diagonalizable and with the new variables $w = R^{-1} q$ we can write\n",
     "$$\n",
     "    w^p_t + \\lambda^p w^p_x = 0 \\text{ for } p = 1, 2, \\ldots , m.\n",
     "$$\n",
@@ -113,7 +113,7 @@
     "$$\n",
     "    \\ell^p A = \\lambda^p \\ell^p.\n",
     "$$\n",
-    "This allows us to write the chracteristic variables as\n",
+    "This allows us to write the characteristic variables as\n",
     "$$\n",
     "    w^p(x,t) = \\ell^p q(x, t)\n",
     "$$\n",
@@ -184,7 +184,7 @@
     }
    },
    "source": [
-    "Unsurprisinly perhaps $\\mathcal{D}$ directly related to the CFL stability condition.  The most generic version of this stability condition states the following:\n",
+    "Unsurprisingly perhaps $\\mathcal{D}$ directly related to the CFL stability condition.  The most generic version of this stability condition states the following:\n",
     "\n",
     "> The true domain of dependence must be contained in the numerical domain of dependence.\n",
     "\n",
@@ -199,7 +199,7 @@
     }
    },
    "source": [
-    "We can also turn the domain of dependece around and ask what the range of influence of particular point $x_0$ might have sometime in the future.  This is often called the **range of influence**.  \n",
+    "We can also turn the domain of dependence around and ask what the range of influence of particular point $x_0$ might have sometime in the future.  This is often called the **range of influence**.  \n",
     "\n",
     "Try to draw an example of this influence for the same case as before."
    ]
@@ -306,7 +306,7 @@
     "$$\n",
     "    (w^p_r - w^p_\\ell) r^p \\equiv \\alpha^p r^p\n",
     "$$\n",
-    "across the $p$th characteristic.  This implies then that the jump is proportional to the eigenvectors of the system.  This condition is called the **Rankine-Hugoniot jump condition**.  In this case we have written in the case of linear hyperbolic PDEs and we will see an analgous version for nonlinear hyperbolic PDEs."
+    "across the $p$th characteristic.  This implies then that the jump is proportional to the eigenvectors of the system.  This condition is called the **Rankine-Hugoniot jump condition**.  In this case we have written in the case of linear hyperbolic PDEs and we will see an analogous version for nonlinear hyperbolic PDEs."
    ]
   },
   {
@@ -317,7 +317,7 @@
     }
    },
    "source": [
-    "We can also write this condition in terms of the original inital data $q_\\ell$ and $q_r$ such that\n",
+    "We can also write this condition in terms of the original initial data $q_\\ell$ and $q_r$ such that\n",
     "$$\n",
     "    q_r - q_\\ell = \\alpha^1 r^1 + \\cdots + \\alpha^m r^m\n",
     "$$\n",
@@ -336,7 +336,7 @@
     }
    },
    "source": [
-    "We now introduce a notation that we will reuse extenstively in this course.  Define\n",
+    "We now introduce a notation that we will reuse extensively in this course.  Define\n",
     "$$\n",
     "    \\mathcal{W}^p \\equiv \\alpha^p r^p\n",
     "$$\n",
@@ -385,7 +385,7 @@
     }
    },
    "source": [
-    "Note that there are two possible states $q_m$.  The correct one is choosen based on which $\\lambda^p$ is faster.  We will see a more general way to pick which middle state is correct later, also known as **entropy conditions**."
+    "Note that there are two possible states $q_m$.  The correct one is chosen based on which $\\lambda^p$ is faster.  We will see a more general way to pick which middle state is correct later, also known as **entropy conditions**."
    ]
   },
   {
@@ -398,7 +398,7 @@
    "source": [
     "#### Example:  Acoustics\n",
     "\n",
-    "Let us now turn to our usual example, the accoustics equations defined here as\n",
+    "Let us now turn to our usual example, the acoustics equations defined here as\n",
     "$$\n",
     "    \\begin{bmatrix}\n",
     "        p \\\\ u\n",

02_finite_volume.ipynb

@@ -153,7 +153,7 @@
     "$$\n",
     "    f(q_x, x) = -\\beta(x) q_x.\n",
     "$$\n",
-    "Cleary this gives us the heat equation if we substitute this flux function into our general equations\n",
+    "Clearly this gives us the heat equation if we substitute this flux function into our general equations\n",
     "$$\n",
     "    q_t + f(q_x, x)_x = q_t - (\\beta(x) q_x)_x = 0.\n",
     "$$"
@@ -212,7 +212,7 @@
     "For any numerical method we desire that as $\\Delta x, \\Delta t \\rightarrow 0$ that the numerical solution converges to the true solution.  This generally requires the following conditions:\n",
     "\n",
     "1. The method must be consistent:  the approximation is valid locally.\n",
-    "1. The method must be stable:  small errors do not accumalate too fast."
+    "1. The method must be stable:  small errors do not accumulate too fast."
    ]
   },
   {
@@ -262,7 +262,7 @@
    "source": [
     "## Numerical Fluxes\n",
     "\n",
-    "We now will consider a number of different flux defintions and consider their viability."
+    "We now will consider a number of different flux definitions and consider their viability."
    ]
   },
   {