diff --git a/desc/compute/_basis_vectors.py b/desc/compute/_basis_vectors.py
index 305dc087fc..414c014182 100644
--- a/desc/compute/_basis_vectors.py
+++ b/desc/compute/_basis_vectors.py
@@ -602,7 +602,6 @@ def _e_sup_theta_rr(params, transforms, profiles, data, **kwargs):
         / data["sqrt(g)"] ** 2
         + 2 * temp.T * data["sqrt(g)_r"] * data["sqrt(g)_r"] / data["sqrt(g)"] ** 3
     ).T
-
     return data
 
 
@@ -1140,7 +1139,6 @@ def _e_sup_zeta_rz(params, transforms, profiles, data, **kwargs):
         / data["sqrt(g)"] ** 2
         + 2 * temp.T * data["sqrt(g)_r"] * data["sqrt(g)_z"] / data["sqrt(g)"] ** 3
     ).T
-
     return data
 
 
@@ -1184,7 +1182,6 @@ def _e_sup_zeta_t(params, transforms, profiles, data, **kwargs):
             * safediv(data["sqrt(g)_rt"], data["sqrt(g)_r"] ** 2)
         ).T,
     )
-
     return data
 
 
@@ -1235,7 +1232,6 @@ def _e_sup_zeta_tt(params, transforms, profiles, data, **kwargs):
         / data["sqrt(g)"] ** 2
         + 2 * temp.T * data["sqrt(g)_t"] * data["sqrt(g)_t"] / data["sqrt(g)"] ** 3
     ).T
-
     return data
 
 
@@ -1291,7 +1287,6 @@ def _e_sup_zeta_tz(params, transforms, profiles, data, **kwargs):
         / data["sqrt(g)"] ** 2
         + 2 * temp.T * data["sqrt(g)_t"] * data["sqrt(g)_z"] / data["sqrt(g)"] ** 3
     ).T
-
     return data
 
 
@@ -1335,7 +1330,6 @@ def _e_sup_zeta_z(params, transforms, profiles, data, **kwargs):
             * safediv(data["sqrt(g)_rz"], data["sqrt(g)_r"] ** 2)
         ).T,
     )
-
     return data
 
 
@@ -1386,16 +1380,15 @@ def _e_sup_zeta_zz(params, transforms, profiles, data, **kwargs):
         / data["sqrt(g)"] ** 2
         + 2 * temp.T * data["sqrt(g)_z"] * data["sqrt(g)_z"] / data["sqrt(g)"] ** 3
     ).T
-
     return data
 
 
 @register_compute_fun(
-    name="e_phi",
-    label="\\mathbf{e}_{\\phi}",
+    name="e_phi|r,t",
+    label="\\mathbf{e}_{\\phi} |_{\\rho, \\theta}",
     units="m",
     units_long="meters",
-    description="Covariant cylindrical toroidal basis vector",
+    description="Covariant toroidal basis vector in (ρ,θ,ϕ) coordinates",
     dim=3,
     params=[],
     transforms={},
@@ -1407,11 +1400,13 @@ def _e_sup_zeta_zz(params, transforms, profiles, data, **kwargs):
         "desc.geometry.surface.FourierRZToroidalSurface",
         "desc.geometry.core.Surface",
     ],
+    aliases=["e_phi"],
+    # Our usual notation implies e_phi = (∂X/∂ϕ)|R,Z = R ϕ̂, but we need to alias e_phi
+    # to e_phi|r,t = (∂X/∂ϕ)|ρ,θ for compatibility with older versions of the code.
 )
-def _e_sub_phi(params, transforms, profiles, data, **kwargs):
-    # dX/dphi at const r,t = dX/dz * dz/dphi = dX/dz / (dphi/dz)
-    data["e_phi"] = (data["e_zeta"].T / data["phi_z"]).T
-
+def _e_sub_phi_rt(params, transforms, profiles, data, **kwargs):
+    # (∂X/∂ϕ)|ρ,θ = (∂X/∂ζ)|ρ,θ / (∂ϕ/∂ζ)|ρ,θ
+    data["e_phi|r,t"] = (data["e_zeta"].T / data["phi_z"]).T
     return data
 
 
@@ -2434,27 +2429,81 @@ def _e_sub_theta_over_sqrt_g(params, transforms, profiles, data, **kwargs):
         safediv(data["e_theta"].T, data["sqrt(g)"]).T,
         lambda: safediv(data["e_theta_r"].T, data["sqrt(g)_r"]).T,
     )
-
     return data
 
 
 @register_compute_fun(
     name="e_theta_PEST",
-    label="\\mathbf{e}_{\\theta_{PEST}}",
+    label="\\mathbf{e}_{\\vartheta} |_{\\rho, \\phi} = \\mathbf{e}_{\\theta_{PEST}}",
     units="m",
     units_long="meters",
-    description="Covariant straight field line (PEST) poloidal basis vector",
+    description="Covariant poloidal basis vector in (ρ,ϑ,ϕ) coordinates or"
+    " straight field line PEST coordinates. ϕ increases counterclockwise"
+    " when viewed from above (cylindrical R,ϕ plane with Z out of page).",
     dim=3,
     params=[],
     transforms={},
     profiles=[],
     coordinates="rtz",
-    data=["e_theta", "theta_PEST_t"],
+    data=["e_theta", "theta_PEST_t", "e_zeta", "theta_PEST_z", "phi_t", "phi_z"],
+    aliases=["e_vartheta"],
 )
-def _e_sub_theta_pest(params, transforms, profiles, data, **kwargs):
-    # dX/dv at const r,z = dX/dt * dt/dv / dX/dt / dv/dt
-    data["e_theta_PEST"] = (data["e_theta"].T / data["theta_PEST_t"]).T
+def _e_sub_vartheta_rp(params, transforms, profiles, data, **kwargs):
+    # constant ρ and ϕ
+    e_vartheta = (
+        data["e_theta"].T * data["phi_z"] - data["e_zeta"].T * data["phi_t"]
+    ) / (data["theta_PEST_t"] * data["phi_z"] - data["theta_PEST_z"] * data["phi_t"])
+    data["e_theta_PEST"] = e_vartheta.T
+    return data
+
 
+@register_compute_fun(
+    name="e_phi|r,v",
+    label="\\mathbf{e}_{\\phi} |_{\\rho, \\vartheta}",
+    units="m",
+    units_long="meters",
+    description="Covariant toroidal basis vector in (ρ,ϑ,ϕ) coordinates or"
+    " straight field line PEST coordinates. ϕ increases counterclockwise"
+    " when viewed from above (cylindrical R,ϕ plane with Z out of page).",
+    dim=3,
+    params=[],
+    transforms={},
+    profiles=[],
+    coordinates="rtz",
+    data=["e_theta", "theta_PEST_t", "e_zeta", "theta_PEST_z", "phi_t", "phi_z"],
+)
+def _e_sub_phi_rv(params, transforms, profiles, data, **kwargs):
+    # constant ρ and ϑ
+    e_phi = (
+        data["e_zeta"].T * data["theta_PEST_t"]
+        - data["e_theta"].T * data["theta_PEST_z"]
+    ) / (data["theta_PEST_t"] * data["phi_z"] - data["theta_PEST_z"] * data["phi_t"])
+    data["e_phi|r,v"] = e_phi.T
+    return data
+
+
+@register_compute_fun(
+    name="e_rho|v,p",
+    label="\\mathbf{e}_{\\rho} |_{\\vartheta, \\phi}",
+    units="m",
+    units_long="meters",
+    description="Covariant radial basis vector in (ρ,ϑ,ϕ) coordinates or"
+    " straight field line PEST coordinates. ϕ increases counterclockwise"
+    " when viewed from above (cylindrical R,ϕ plane with Z out of page).",
+    dim=3,
+    params=[],
+    transforms={},
+    profiles=[],
+    coordinates="rtz",
+    data=["e_rho", "e_vartheta", "e_phi|r,v", "theta_PEST_r", "phi_r"],
+)
+def _e_sub_rho_vp(params, transforms, profiles, data, **kwargs):
+    # constant ϑ and ϕ
+    data["e_rho|v,p"] = (
+        data["e_rho"].T
+        - data["e_vartheta"].T * data["theta_PEST_r"]
+        - data["e_phi|r,v"].T * data["phi_r"]
+    ).T
     return data
 
 
diff --git a/desc/compute/_metric.py b/desc/compute/_metric.py
index 42dd5c8218..3a842e9378 100644
--- a/desc/compute/_metric.py
+++ b/desc/compute/_metric.py
@@ -40,17 +40,21 @@ def _sqrtg(params, transforms, profiles, data, **kwargs):
     label="\\sqrt{g}_{PEST}",
     units="m^{3}",
     units_long="cubic meters",
-    description="Jacobian determinant of PEST flux coordinate system",
+    description="Jacobian determinant of (ρ,ϑ,ϕ) coordinate system or"
+    " straight field line PEST coordinates. ϕ increases counterclockwise"
+    " when viewed from above (cylindrical R,ϕ plane with Z out of page).",
     dim=1,
     params=[],
     transforms={},
     profiles=[],
     coordinates="rtz",
-    data=["e_rho", "e_theta_PEST", "e_phi"],
+    data=["sqrt(g)", "theta_PEST_t", "phi_z", "theta_PEST_z", "phi_t"],
 )
 def _sqrtg_pest(params, transforms, profiles, data, **kwargs):
-    data["sqrt(g)_PEST"] = dot(
-        data["e_rho"], cross(data["e_theta_PEST"], data["e_phi"])
+    # Same as dot(data["e_rho|v,p"], cross(data["e_vartheta"], data["e_phi|r,v"])), but
+    # more efficient as it avoids computing radial derivatives of the stream functions.
+    data["sqrt(g)_PEST"] = data["sqrt(g)"] / (
+        data["theta_PEST_t"] * data["phi_z"] - data["theta_PEST_z"] * data["phi_t"]
     )
     return data
 
diff --git a/desc/compute/_surface.py b/desc/compute/_surface.py
index 5e76934ba6..5fb84fa060 100644
--- a/desc/compute/_surface.py
+++ b/desc/compute/_surface.py
@@ -3,6 +3,8 @@
 from .data_index import register_compute_fun
 from .geom_utils import rpz2xyz
 
+# TODO: review when zeta no longer equals phi
+
 
 @register_compute_fun(
     name="x",
@@ -25,6 +27,7 @@
 def _x_FourierRZToroidalSurface(params, transforms, profiles, data, **kwargs):
     R = transforms["R"].transform(params["R_lmn"])
     Z = transforms["Z"].transform(params["Z_lmn"])
+    # TODO: change when zeta no longer equals phi
     phi = transforms["grid"].nodes[:, 2]
     coords = jnp.stack([R, phi, Z], axis=1)
     # default basis for "x" is rpz, the conversion will be done
diff --git a/tests/inputs/master_compute_data_rpz.pkl b/tests/inputs/master_compute_data_rpz.pkl
index b6c10ec359..f230d7a816 100644
Binary files a/tests/inputs/master_compute_data_rpz.pkl and b/tests/inputs/master_compute_data_rpz.pkl differ
diff --git a/tests/inputs/master_compute_data_xyz.pkl b/tests/inputs/master_compute_data_xyz.pkl
index d998fa7d9b..0de07ede29 100644
Binary files a/tests/inputs/master_compute_data_xyz.pkl and b/tests/inputs/master_compute_data_xyz.pkl differ