Curvilinear Coordinates
Exercise 1 : Inverting Spherical Coordinates
Invert the transformation $x = r\sin\theta\cos\phi$, $y = r\sin\theta\sin\phi$, $z = r\cos\theta$ to express $(r, \theta, \phi)$ in terms of $(x, y, z)$.
Exercise 2 : Coordinate Examples (Conversions)
Invert the transformation $x = r\sin\theta\cos\phi$, $y = r\sin\theta\sin\phi$, $z = r\cos\theta$ to express $(r, \theta, \phi)$ in terms of $(x, y, z)$.
- Express the point $(x, y, z) = (2, -2, 1)$ in spherical and cylindrical coordinates.
- Express $(r, \theta, \phi) = (3, \pi/3, \pi/4)$ in Cartesian coordinates.
- Express $(s, \phi, z) = (4, 5\pi/6, -2)$ in Cartesian coordinates.
Exercise 3 : Spherical Unit Vectors
Express the spherical unit vectors $\hat{\mathbf{r}}$, $\hat{\boldsymbol{\theta}}$, $\hat{\boldsymbol{\phi}}$ in terms of the Cartesian unit vectors $\hat{\mathbf{x}}$, $\hat{\mathbf{y}}$, $\hat{\mathbf{z}}$, i.e., derive:
$$ \begin{aligned} \hat{\mathbf{r}} &= \sin\theta\cos\phi\,\hat{\mathbf{x}} + \sin\theta\sin\phi\,\hat{\mathbf{y}} + \cos\theta\,\hat{\mathbf{z}}, \\ \hat{\boldsymbol{\theta}} &= \cos\theta\cos\phi\,\hat{\mathbf{x}} + \cos\theta\sin\phi\,\hat{\mathbf{y}} - \sin\theta\,\hat{\mathbf{z}}, \\ \hat{\boldsymbol{\phi}} &= -\sin\phi\,\hat{\mathbf{x}} + \cos\phi\,\hat{\mathbf{y}}. \end{aligned} $$Verify by also finding the inverse relations expressing $\hat{\mathbf{x}}$, $\hat{\mathbf{y}}$, $\hat{\mathbf{z}}$ in terms of $\hat{\mathbf{r}}$, $\hat{\boldsymbol{\theta}}$, $\hat{\boldsymbol{\phi}}$.
Exercise 4 : Cylindrical Unit Vectors
Express the cylindrical unit vectors $\hat{\mathbf{s}}$, $\hat{\boldsymbol{\phi}}$, $\hat{\mathbf{z}}$ in terms of the Cartesian unit vectors:
$$ \begin{aligned} \hat{\mathbf{s}} &= \cos\phi\,\hat{\mathbf{x}} + \sin\phi\,\hat{\mathbf{y}}, \\ \hat{\boldsymbol{\phi}} &= -\sin\phi\,\hat{\mathbf{x}} + \cos\phi\,\hat{\mathbf{y}}, \\ \hat{\mathbf{z}} &= \hat{\mathbf{z}}. \end{aligned} $$Invert these formulas to get $\hat{\mathbf{x}}$, $\hat{\mathbf{y}}$, $\hat{\mathbf{z}}$ in terms of $\hat{\mathbf{s}}$, $\hat{\boldsymbol{\phi}}$, $\hat{\mathbf{z}}$.
Exercise 5 : Unit Vector Relationships
Unit vector relationships:
- Show that $\hat{\mathbf{r}} \cdot \hat{\boldsymbol{\theta}} = 0$ and $\hat{\boldsymbol{\theta}} \cdot \hat{\boldsymbol{\phi}} = 0$ using their Cartesian representations from Problem 1.38.
- Prove that $\frac{\partial \hat{\mathbf{r}}}{\partial \phi} = \sin\theta,\hat{\boldsymbol{\phi}}$.
- Find $\frac{\partial \hat{\boldsymbol{\theta}}}{\partial \theta}$ and $\frac{\partial \hat{\boldsymbol{\phi}}}{\partial \phi}$.
Exercise 6 : Vector Operations in Spherical Coordinates
Vector operations practice:
- Compute $\nabla r$, $\nabla \theta$, and $\nabla \phi$ in spherical coordinates.
- Verify that $\nabla \times (\nabla T) = 0$ for $T = r^2 \sin \theta \cos \phi$.
Exercise 7 : Gradient and Laplacian Practice
For the scalar function $T = r(\cos \theta + \sin \theta \cos \phi)$:
- Compute $\nabla T$ and $\nabla^2 T$ in spherical coordinates.
- Verify $\nabla^2 T$ by converting $T$ to Cartesian coordinates and using $\nabla^2 T = \frac{\partial^2 T}{\partial x^2} + \frac{\partial^2 T}{\partial y^2} + \frac{\partial^2 T}{\partial z^2}$.
- Test the gradient theorem $\int_a^b (\nabla T)\cdot d\mathbf{l} = T(b) - T(a)$ along the path from $(0,0,0)$ to $(0,0,2)$ shown in the figure.
Exercise 8 : Divergence in Spherical Coordinates
For the vector field $\mathbf{v} = (r \cos \theta) \hat{\mathbf{r}} + (r \sin \theta) \hat{\boldsymbol{\theta}} + (r \sin \theta \cos \phi) \hat{\boldsymbol{\phi}}$:
- Compute $\nabla \cdot \mathbf{v}$.
- Verify the divergence theorem using the inverted hemispherical bowl of radius $R$ resting on the $xy$-plane and centered at the origin.
Exercise 9 : Divergence and Curl Examples
Divergence and curl computations:
- Compute $\nabla \cdot (r^n \hat{\mathbf{r}})$ for arbitrary $n$.
- For $\mathbf{A} = f(r)\hat{\mathbf{r}}$, show that $\nabla \times \mathbf{A} = 0$.
Exercise 10 : Radial Functions — Laplacian & Curl
Challenging problem: For an arbitrary scalar function $f(r)$ and vector function $\mathbf{g}(r)\hat{\mathbf{r}}$:
- Show that $\nabla^2 f(r) = \frac{1}{r^2}\frac{d}{dr}\left(r^2\frac{df}{dr}\right)$.
- Prove that $\nabla \times [\mathbf{g}(r)\hat{\mathbf{r}}] = 0$.
- Find the most general form of $\mathbf{v}(r,\theta,\phi)$ such that $\nabla \times \mathbf{v} = 0$ in spherical coordinates.
Exercise 11 : Gradients in Cylindrical Coordinates
Compute $\nabla s$, $\nabla \phi$, and $\nabla z$ in cylindrical coordinates.
Exercise 12 : Cylindrical Divergence & Curl
For $\mathbf{v} = s(2 + \sin^2\phi),\hat{\mathbf{s}} + s\sin\phi\cos\phi,\hat{\boldsymbol{\phi}} + 3z,\hat{\mathbf{z}}$:
- Compute $\nabla \cdot \mathbf{v}$ in cylindrical coordinates.
- Compute $\nabla \times \mathbf{v}$.
- Verify the divergence theorem using the quarter-cylinder defined by $0 \leq s \leq 2$, $0 \leq \phi \leq \pi/2$, $0 \leq z \leq 5$.
Exercise 13 : Curl in Cylindrical Coordinates
Compute $\nabla \times (\phi \hat{\mathbf{z}})$ in cylindrical coordinates.
Exercise 14 : Divergence Theorem Applications
Verify the divergence theorem $\int_{\mathcal{V}} (\nabla\cdot\mathbf{v}),d\tau = \oint_{\mathcal{S}} \mathbf{v}\cdot d\mathbf{a}$ for:
- $\mathbf{v}_1 = r^2 \hat{\mathbf{r}}$ over a sphere of radius $R$ centered at the origin.
- $\mathbf{v}_2 = (1/r^2) \hat{\mathbf{r}}$ over the same volume. Compare with Problem 1.16.
Exercise 15 : Volume and Surface Elements
Volume and surface elements:
- Derive the spherical volume element $d\tau = r^2\sin\theta,dr,d\theta,d\phi$ from the Jacobian determinant.
- Derive the cylindrical surface element $da = s,d\phi,dz$ for a surface of constant $s$.
- Find the area element on a sphere of radius $R$ (constant $r$ surface).
Exercise 16 : Coordinate Representations of a Field
Practical applications:
- Express the vector field $\mathbf{F} = -y\hat{\mathbf{x}} + x\hat{\mathbf{y}}$ in both cylindrical and spherical coordinates.
- Compute $\nabla \cdot \mathbf{F}$ and $\nabla \times \mathbf{F}$ in all three coordinate systems.
- What physical field does $\mathbf{F}$ represent?
Exercise 17 : Laplacian in Spherical Symmetry
Derive the expression for the Laplacian $\nabla^2 f(r)$ in spherical coordinates for functions depending only on $r$, and apply it to $f(r)=1/r$.
Exercise 18 : Poisson’s Equation (Radial)
Solve Poisson’s equation $\nabla^2 V(r) = -\rho(r)/\epsilon_0$ for a spherically symmetric charge density $\rho(r)=\rho_0$ for $r<R$ and $0$ for $r>R$, matching boundary conditions at $r=R$.
Exercise 19 : Divergence Theorem on a Cone
Use the divergence theorem to compute the flux of $\mathbf{F}=r^n\hat{\mathbf{r}}$ outward through the surface of a right circular cone of height $h$ and base radius $a$ (centered on the z-axis).