Curvilinear coordinates

Let R ⁠ be the major radius (from the center of the hole to the center of the tube) and r ⁠ be the minor radius (the radius of the tube itself), with 0 < r < R⁠. We parametrize the surface using two curvilinear coordinates (φ, θ)⁠:

1. Finding the Tangent Basis Vectors: At any point P(θ, φ) ⁠ on the torus, the tangent space T_PT² ⁠ is a 2D flat plane resting against the donut at P⁠. Its basis is spanned by two partial derivative vectors:
   • Poloidal tangent vector (e_θ⁠): Differentiating with respect to θ ⁠ gives a vector pointing along the small cross-sectional circle of the tube:
     \[ \mathbf{e}_{\theta} = \frac{\partial \mathbf{r}}{\partial\theta} = \left( -r\,\sin\theta\,\sin\phi , -r\,\sin\theta \,\sin\phi ,\ r\,\cos\theta \right) . \]
   • Toroidal tangent vector (e_φ⁠): Differentiating with respect to φ ⁠ gives a vector pointing around the main ring of the donut:
     \[ \mathbf{e}_{\phi} = \frac{\partial \mathbf{r}}{\partial\phi} = \left( - \left( R + r\,\cos\theta \right) \sin\phi , \ \left( R + r\,\cos\theta \right) \cos\phi , \ 0 \right) . \]
2. Why This Example is Illuminating
   • Position-Dependent Geometry: Every point P ⁠ has its own distinct tangent plane T_PT² ⁠ tilted at a different angle in 3D space. A tangent vector at the top of the torus cannot be directly added to a tangent vector on the outer rim without first transporting it.
   • Varying Basis Lengths (Metric Preview): Using the standard ℝ³ ⁠ dot product to measure the lengths of these basis vectors reveals:
     1. \( \displaystyle \quad \| \mathbf{e}_{\theta} \| = +\sqrt{\langle \mathbf{e}_{\theta} \mid \mathbf{e}_{\theta} \rangle} = r \) (constant everywhere).
     2. \( \displaystyle \quad \| \mathbf{e}_{\phi} \| = +\sqrt{\langle \mathbf{e}_{\phi} \mid \mathbf{e}_{\phi} \rangle} = R + r\,\cos\theta . \)
     On the outer equator (θ = 0⁠), ∥e_φ∥ = R + r ⁠ (longest). On the inner equator (θ = π⁠), ∥e_φ∥ = R − r ⁠ ⁠ (shortest). Even though ⟨e_θ , e_φ⟩ = 0 ⁠ everywhere (making it an orthogonal coordinate system), the local stretching of space changes from point to point—which directly motivates why we need the metric tensor g_i,j ⁠ in Theorem 1!
     ClearAll["Global`*"]; (* 1. Define Torus Geometry and Tangent Basis Vectors *) R = 3.0; r = 1.0; pos[th_, ph_] := {(R + r Cos[th]) Cos[ph], (R + r Cos[th]) Sin[ph], r Sin[th]}; eTh[th_, ph_] := {-r Sin[th] Cos[ph], -r Sin[th] Sin[ph], r Cos[th]}; ePh[th_, ph_] := {-(R + r Cos[th]) Sin[ph], (R + r Cos[th]) Cos[ph], 0.0}; (* Helper function to build the tangent plane and basis arrows at any point (th0, ph0) *) tangentElements[th0_, ph0_, labelP_] := Module[ {p0, vTh, vPh, uTh, uPh, corner1, corner2, corner3, corner4}, p0 = pos[th0, ph0]; vTh = eTh[th0, ph0]; vPh = ePh[th0, ph0]; uTh = Normalize[vTh]; uPh = Normalize[vPh]; corner1 = p0 - 1.3 uTh - 1.3 uPh; corner2 = p0 + 1.3 uTh - 1.3 uPh; corner3 = p0 + 1.3 uTh + 1.3 uPh; corner4 = p0 - 1.3 uTh + 1.3 uPh; { (* Translucent Tangent Plane T_P M *) Orange, Opacity[0.38], EdgeForm[Directive[Thick, Darker[Orange]]], Polygon[{corner1, corner2, corner3, corner4}], (* Point P and Label *) Black, PointSize[0.022], Point[p0], Text[Style[labelP, Bold, 14, Black], p0 + {0, 0, 0.45}], (* Right-Angle Orthogonality Marker *) Darker[Gray], Thin, Line[{p0 + 0.25 uTh, p0 + 0.25 uTh + 0.25 uPh, p0 + 0.25 uPh}], (* Poloidal Basis Vector e_theta (Red) *) Red, Arrowheads[0.032], Thickness[0.006], Arrow[{p0, p0 + 0.9 vTh}], Text[Style[Subscript["e", "\[Theta]"], Bold, 15, Red], p0 + 1.1 vTh], (* Toroidal Basis Vector e_phi (Blue, scaled by 0.35 to preserve relative length!) *) Blue, Arrowheads[0.032], Thickness[0.006], Arrow[{p0, p0 + 0.35 vPh}], Text[Style[Subscript["e", "\[Phi]"], Bold, 15, Blue], p0 + 0.43 vPh] } ]; (* 2. Render the Static 3D Scene *) torusDiagram = Show[ ParametricPlot3D[ pos[th, ph], {th, 0, 2 Pi}, {ph, 0, 2 Pi}, PlotStyle -> Directive[LightBlue, Opacity[0.65], Specularity[White, 15]], Mesh -> {16, 24}, MeshStyle -> Directive[Gray, Thin, Opacity[0.35]], Boxed -> False, Axes -> False ], (* Highlight Coordinate Curves *) ParametricPlot3D[pos[th, -Pi/4], {th, 0, 2 Pi}, PlotStyle -> Directive[Red, Dashed, Thick]], ParametricPlot3D[pos[0.35, ph], {ph, 0, 2 Pi}, PlotStyle -> Directive[Blue, Dashed, Thick]], (* Add Two Tangent Spaces: Outer Rim vs. Upper Inner Slope *) Graphics3D[{ tangentElements[0.35, -Pi/4, "P1 (Outer)"], tangentElements[2.35, 3 Pi/4, "P2 (Inner)"] }], PlotRange -> {{-5.2, 5.2}, {-5.2, 5.2}, {-2.6, 2.6}}, ViewPoint -> {1.9, -2.6, 1.8}, ImageSize -> 700, PlotLabel -> Style["Tangent Spaces on a Torus: Stretching of Basis Vectors", Bold, 16] ] (* 3. Automatically Save a High-Res PNG to Your Desktop for Your Website/Slides *) Export[FileNameJoin[{$HomeDirectory, "Desktop", "TorusTangentSpace.png"}], torusDiagram, ImageResolution -> 300]

     

        ■
     End of Example 1

Step 1: Compute the Tangent Basis Vectors. Taking partial derivatives of r(u, v, z) ⁠ gives the natural basis vectors at any point P(u, v, z)⁠:

Step 2: Construct the Metric Tensor. ⁠ (Part 1: Symmetry) Computing all pairwise Euclidean inner products g_i,j = ⟨e_i ∣ e_j⟩:

Step 3: Verify Positive Definiteness (Part 2). Let v = c¹e_u + c³e_v + c³e_z ⁠ be any tangent vector with components c = [c¹, c², c³]^T⁠. Computing the quadratic form cᵀGc⁠:

Step 4: Compute the Arc Length Element (Part 3). Applying Theorem 1(3), the squared infinitesimal distance\( \displaystyle \quad {\text d}s^2 = \sum_{i,j}\, g_{i,j}\,{\text d}q^i\,{\text d}q^j \quad \) ⁠ is:

In illustrative plot of the skewed coordinate grid, horizontal green dashed lines represent curves of constant v⁠, while blue dashed parabolas represent curves of constant u⁠. At the point P(u = 0, v = 1)    ⇒    (x = 1, y = 1)⁠, the tangent vectors e_u = (1, 0) ⁠ and e_v = (2, 1) ⁠ clearly form an acute angle (α &approx 26.6° ≠ 90°), visually explaining why the off-diagonal metric component g_uv = ⟨e_u ∣ e_v⟩ = 2v ≠ 0. ⁠    ■

• Curves of constant u ⁠ are confocal ellipses\( \displaystyle \quad \frac{x^2}{a^2\,\cosh^2 u} + \frac{y^2}{a^2\,\sinh^2 u} = 1 \quad \) with foci at (±𝑎, 0).
• Curves of constant v ⁠ are confocal hyperbolas\( \displaystyle \quad \frac{x^2}{a^2\,\cos^2 v} - \frac{y^2}{a^2\,\sin^2 v} = 1 \quad \) ⁠ sharing the exact same foci.

Step 1: Natural Tangent Basis Vectors. Taking partial derivatives of r(u, v, z) ⁠ with respect to u, v, z ⁠ yields:

• \( \displaystyle \quad \mathbf{e}_u = \frac{\partial\mathbf{r}}{\partial u} = \left( a\,\sinh (u)\, \cos (v) , \ a\,\sinh (u)\,\cos (v) , \ 0 \right) , \)
• \( \displaystyle \quad \mathbf{e}_v = \frac{\partial\mathbf{r}}{\partial v} = \left( -a\,\cosh (u)\,\sin (v) ,\ a\,\sinh (u)\,\cos (v) , \ 0 \right) , \)
• \( \displaystyle \quad \mathbf{e}_z = \frac{\partial\mathbf{r}}{\partial z} = \left( 0, \ 0, \ 1 \right) . \)

Step 2: Verify Orthogonality (g_i,j = 0 ⁠ for i ≠ j⁠).
Computing the off-diagonal inner products:

Step 3: Scale Factors (\( \displaystyle \quad h_i = \sqrt{g_{i,i}} \) ⁠) and Orthonormal Basis.
Computing the diagonal metric components

Taking square roots gives the three scale factors:

Step 4: Volume Element (dV = h_u h_v h_z dudvdz⁠).
Applying Theorem 2(3), the infinitesimal volume element becomes

Illustrative Plot: Confocal Ellipses, Hyperbolas, and Orthonormal Frame.
The diagram below shows confocal ellipses (u = constant⁠) and confocal hyperbolas (v = constant⁠) for 𝑎 = 1.5⁠. At P(u₀ = 0.7, v₀ = π/4)⁠, the unit vectors\( \displaystyle \quad \hat{\bf e}_u \quad\mbox{and} \quad \hat{\bf e}_v \quad \) are perpendicular and tangent to the intersecting curves.

• Coordinate Geometry: Surfaces of constant σ ⁠ and τ ⁠ are orthogonal families of confocal parabolic cylinders opening upward and downward along the y⁠-axis.
• Scale Factors (from Theorem 2): Computing \( \displaystyle \quad h_i = \left\| \frac{\partial\mathbf{r}}{\partial q^i} \right\| \quad \) ⁠ gives non-constant, coupled scale factors:
  \[ h_{\sigma} = \sqrt{\sigma^2 + \tau^2} , \quad h_{\tau} = \sqrt{\sigma^2 + \tau^2} , \quad h_z = 1 . \]
  Notice that h_σh_τh_z = σ² + τ².

Part 1: Gradient and Laplacian of a Scalar Field.
Consider the scalar potential field f(σ, τ, z) = σ²τ + z². ⁠

1. Gradient (∇f): Applying Theorem 3(1):
   \[ \nabla f = \frac{1}{h_{\sigma}}\, \frac{\partial f}{\partial\sigma}\, \hat{\bf e}_{\sigma} + \frac{1}{h_{\tau}}\, \frac{\partial f}{\partial\tau}\, \hat{\bf e}_{\tau} + \frac{1}{h_z}\, \frac{\partial f}{\partial z} \,\hat{\bf e}_z . \]
   Substituting the partial derivatives \( \displaystyle \quad \frac{\partial f}{\partial\sigma} = 2\sigma\tau , \quad \frac{\partial f}{\partial\tau} = \sigma^2 , \quad \mbox{and} \quad \frac{\partial f}{\partial z} = 2z \quad \) yields:
   \[ \nabla f = \frac{2\sigma\tau}{\sqrt{\sigma^2 + \tau^2}}\, \hat{\bf e}_{\sigma} + \frac{\sigma^2}{\sqrt{\sigma^2 + \tau^2}}\, \hat{\bf e}_{\tau} + 2z\,\hat{\bf e}_z . \]
2. Laplacian (∇²f⁠):
   Applying Theorem 3(4), notice how the scale-factor ratios \( \displaystyle \quad \frac{h_\tau h_z}{h_\sigma} = 1 \quad\mbox{and} \quad \frac{h_\sigma h_z}{h_\tau} = 1 \quad \) simplify the inner derivatives, while ⁠\( \displaystyle \quad \frac{h_\sigma h_\tau}{h_z} = \sigma^2 + \tau^2 : \)
   \[ \nabla^2 f = \frac{1}{\sigma^2 + \tau^2} \left[ \frac{\partial}{\partial\sigma} \left( 1 \cdot 2\sigma\tau \right) + \frac{\partial}{\partial \tau} \left( 1 \cdot \sigma^2 \right) + \frac{\partial}{\partial z} \left(\left( \sigma^2 + \tau^2 \right) \cdot 2z \right) \right] . \]
   Evaluating each derivative gives:
   \[ \nabla^2 f = \frac{2\tau}{\sigma^2 + \tau^2} + 2 . \]

1. Divergence (∇ • F⁠):
   Applying Theorem 3(2), the scale-factor products inside the derivatives simplify cleanly:
   \[ F_{\sigma} h_\tau h_z = \sigma , \quad F_{\tau} h_\sigma h_z = \tau , \quad F_z h_\sigma h_\tau = \sigma\tau \left( \sigma^2 + \tau^2 \right) . \]
   Substituting these into the divergence formula gives:
   \begin{align*} \nabla \bullet \mathbf{F} &= \frac{1}{\sigma^2 + \tau^2} \left[ \frac{\partial}{\partial\sigma} \left( \sigma \right) + \frac{\partial}{\partial\tau} \left( \tau \right) + \frac{\partial}{\partial z}\left( \sigma\tau \left( \sigma^2 + \tau^2 \right) \right) \right] \\ &= \frac{}{\sigma^2 + \tau^2} . \end{align*}
2. Curl (∇ × F⁠):
   Applying Theorem 3(3), compute the scaled components h_iF_i:
   \[ h_\sigma F_{\sigma} = \sigma , \quad h_\tau F_\tau = \tau , \quad h_z F_z = \sigma \tau . \]
   Expanding the determinant:
   \[ \nabla \times \mathbf{F} = \frac{1}{\sigma^2 + \tau^2} \begin{vmatrix} \sqrt{\sigma^2 + \tau^2}\,\hat{\bf e}_\sigma & \sqrt{\sigma^2 + \tau^2}\,\hat{\bf e}_\tau & \hat{\bf e}_z \\ \frac{\partial}{\partial \sigma} & \frac{\partial}{\partial \tau} &\frac{\partial}{\partial z} \\ \sigma & \tau & \sigma\tau \end{vmatrix} . \]
   Evaluating each component of the determinant yields:
   • \( \displaystyle \quad \hat{\bf e}_\sigma \quad \) component: \( \displaystyle \quad \frac{\sqrt{\sigma^2 + \tau^2}}{\sigma^2 + \tau^2} \left( \frac{\partial}{\partial \tau} \left( \sigma \right) - \frac{\partial}{\partial z} \left( \sigma\tau \right) \right) = \frac{\sigma}{\sqrt{\sigma^2 + \tau^2}} . \)
   • \( \displaystyle \quad \hat{\bf e}_\tau \quad \) component: \( \displaystyle \quad \frac{\sqrt{\sigma^2 + \tau^2}}{\sigma^2 + \tau^2} \left( \frac{\partial}{\partial z} \left( \sigma \right) - \frac{\partial}{\partial \sigma} \left( \sigma\tau \right) \right) = - \frac{\tau}{\sqrt{\sigma^2 + \tau^2}} . \)
   • \( \displaystyle \quad \hat{\bf e}_z \quad \) component: \( \displaystyle \quad \frac{1}{\sigma^2 + \tau^2} \left( \frac{\partial}{\partial z} \left( \tau \right) - \frac{\partial}{\partial \tau} \left( \sigma \right) \right) = 0 . \)
   Combining all three terms gives
   \[ \nabla \times \mathbf{F} = \frac{\sigma}{\sqrt{\sigma^2 + \tau^2}}\, \hat{\bf e}_\sigma - \frac{\tau}{\sqrt{\sigma^2 + \tau^2}}\, \hat{\bf e}_\tau . \]

1. Vector addition
2. Brannon, R.M., Curvilinear Analysis in a Euclidean Space, Utah, 2004.
3. Curvilinear coordinates