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Appendix A Problems

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Table of Contents

Basic

Medium

Advanced

Basic

B-1. Basic Calculation of Partial Derivatives

Given \(f(x, y) = x^3 + 2xy^2 - y\), find \(\frac{\partial f}{\partial x}\) and \(\frac{\partial f}{\partial y}\).

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B-2. Partial Derivative of \(1/r\)

Given \(\Phi(x, y, z) = \frac{1}{\sqrt{x^2 + y^2 + z^2}}\), find \(\frac{\partial \Phi}{\partial x}\). (Hint: Let \(r = \sqrt{x^2 + y^2 + z^2}\), so that \(\Phi = 1/r\))

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B-3. Commutativity of Mixed Partial Derivatives

Given \(f(x, y) = e^{x}\sin y\), compute \(\frac{\partial^2 f}{\partial x \partial y}\) and \(\frac{\partial^2 f}{\partial y \partial x}\), and verify that they are equal.

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B-4. Gradient and Level Curves

Find the gradient \(\nabla\Phi\) of \(\Phi(x, y) = x^2 + y^2\), and calculate the gradient vector at the point \((1, 2)\). Verify that this vector is perpendicular to the level curve \(\Phi = \text{const}\).

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B-5. Gradient of a Temperature Distribution

A temperature distribution on a two-dimensional plane is given by \(T(x, y) = 100 - x^2 - 4y^2\). Find the temperature gradient \(\nabla T\) at the point \((1, 1)\), and determine the direction in which the temperature increases most rapidly.

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B-6. Divergence of a Linear Vector Field

Calculate the divergence \(\nabla \cdot \mathbf{V}\) of \(\mathbf{V} = (x, y, z)\).

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B-7. Divergence of a Quadratic Vector Field

Calculate the divergence of \(\mathbf{V} = (x^2, y^2, z^2)\).

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B-8. Divergence of a Curl Field

Show that for any vector field \(\mathbf{A}\), the divergence of the curl \(\nabla \times \mathbf{A}\) is zero. That is, prove that

\[ \nabla \cdot (\nabla \times \mathbf{A}) = 0 \]

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B-9. Curl of a Rotational Field

Calculate the curl \(\nabla \times \mathbf{V}\) of \(\mathbf{V} = (-y, x, 0)\). Does this field have "vorticity"?

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B-10. Curl of \((yz, xz, xy)\)

Calculate the curl of \(\mathbf{V} = (yz, xz, xy)\).

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B-11. Verification of \(\nabla \times (\nabla\Phi) = 0\)

Compute the curl of \(\mathbf{V} = \nabla\Phi\) (where \(\Phi = x^2 + y^2 + z^2\)) and verify that \(\nabla \times (\nabla\Phi) = 0\).

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B-12. Vector Potential of a Uniform Magnetic Field

For the vector potential \(\mathbf{A} = \frac{B}{2}(-y,\, x,\, 0)\) corresponding to a uniform magnetic field \(\mathbf{B} = B\hat{z}\), calculate the Laplacian of each component \(\nabla^2 A_x\), \(\nabla^2 A_y\), \(\nabla^2 A_z\). Also, verify that \(\nabla \times \mathbf{A} = \mathbf{B}\) holds.

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B-13. Laplacian of \(x^2 - y^2\)

Compute the Laplacian \(\nabla^2\Phi\) of \(\Phi(x, y) = x^2 - y^2\). Does this function satisfy Laplace's equation \(\nabla^2\Phi = 0\)?

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B-14. Laplacian of \(e^x \cos y\)

Verify that \(\Phi(x, y) = e^x \cos y\) satisfies the Laplace equation.

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B-15. Laplacian of \(\sin(kx)\sin(ly)\)

For the two-dimensional scalar field \(f(x,y) = \sin(kx)\sin(ly)\), compute the Laplacian \(\nabla^2 f\).

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B-16. \(\nabla\cdot(\nabla\times\mathbf{A}) = 0\)

For \(\mathbf{A} = (xy, yz, zx)\), compute \(\nabla \cdot (\nabla \times \mathbf{A})\) and verify that it equals zero.

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B-17. \(\nabla\times(\nabla\Phi) = 0\) (\(\Phi = xyz\))

For \(\Phi = xyz\), compute \(\nabla \times (\nabla\Phi)\) and verify that it equals zero.

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B-18. Showing that a Plane Wave Satisfies the Wave Equation

Verify that \(f(x, t) = A\sin(kx - \omega t)\) satisfies the wave equation \(\frac{\partial^2 f}{\partial x^2} = \frac{1}{v^2}\frac{\partial^2 f}{\partial t^2}\), and show that \(v = \omega/k\).

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B-19. Verifying that a Complex Exponential Wave Satisfies the Wave Equation

Verify that \(f(x, t) = e^{i(kx - \omega t)}\) satisfies the wave equation. (where \(i\) is the imaginary unit)

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B-20. Classification of Partial Differential Equations

Classify the following partial differential equations as hyperbolic, parabolic (diffusion-type), or elliptic.

  • (a) \(\frac{\partial^2 u}{\partial t^2} = 4\frac{\partial^2 u}{\partial x^2}\)
  • (b) \(\frac{\partial u}{\partial t} = 3\frac{\partial^2 u}{\partial x^2}\)
  • (c) \(\frac{\partial^2 u}{\partial x^2} + \frac{\partial^2 u}{\partial y^2} = -\rho(x,y)\)
  • (d) \(i\hbar\frac{\partial \psi}{\partial t} = -\frac{\hbar^2}{2m}\frac{\partial^2 \psi}{\partial x^2}\)

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Medium

M-1. Verification of the Diffusion Equation Solution

For the diffusion equation

\[ \frac{\partial u}{\partial t} = D \frac{\partial^2 u}{\partial x^2} \]

substitute the solution of the form \(u(x, t) = e^{ikx - \alpha t}\) and express \(\alpha\) in terms of \(D\) and \(k\).

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M-2. Gradient of the Gravitational Potential

Compute the gradient of \(\Phi(x, y, z) = -\frac{GM}{r}\) (where \(r = \sqrt{x^2 + y^2 + z^2}\)), and show that \(\nabla\Phi = \frac{GM}{r^3}(x, y, z) = \frac{GM}{r^2}\hat{r}\). (Gravitational potential from Chapter 1)

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M-3. Zero Divergence of the Coulomb Electric Field

For \(\mathbf{E} = \frac{q}{4\pi\varepsilon_0}\frac{\mathbf{r}}{r^3}\) (where \(\mathbf{r} = (x, y, z)\), \(r = |\mathbf{r}|\)), show that \(\nabla \cdot \mathbf{E} = 0\) in the region where \(r \neq 0\). (Hint: \(\frac{\partial}{\partial x}\frac{x}{r^3} = \frac{1}{r^3} - \frac{3x^2}{r^5}\))

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M-4. Laplacian of \(1/r\)

Confirm that \(\Phi(r) = 1/r\) (\(r = \sqrt{x^2 + y^2 + z^2}\), \(r \neq 0\)) satisfies \(\nabla^2\Phi = 0\) by using the spherical coordinate Laplacian \(\nabla^2\Phi = \frac{1}{r^2}\frac{d}{dr}(r^2\frac{d\Phi}{dr})\). (Related to Chapter 1, Exercise 1.3)

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M-5. d'Alembert Solution \(g(x - vt)\)

Show that \(f(x, t) = g(x - vt)\) (where \(g\) is an arbitrary twice-differentiable function) satisfies the wave equation, using the chain rule.

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M-6. Decomposition of a Standing Wave

Confirm that the wave equation solution \(f(x, t) = A\sin(kx)\cos(\omega t)\) is a standing wave. Using the product-to-sum formula for trigonometric functions, show that it can be written as a superposition of a rightward-traveling wave and a leftward-traveling wave.

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M-7. Boundary Conditions for String Vibration Modes

Confirm that the solution satisfying the boundary conditions \(f(0, t) = f(L, t) = 0\) of the wave equation describing the vibration of a string (length \(L\), both ends fixed) is \(f_n(x, t) = A_n \sin(n\pi x/L)\cos(\omega_n t)\) (\(\omega_n = n\pi v/L\)). (Preparation for string vibration modes in Chapter 14)

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Advanced

A-1. Identity for \(\nabla\times(\nabla\times\mathbf{E})\)

For the case where the electric field has the form \(\mathbf{E} = (E_x(x,y,z),\, 0,\, 0)\), show by directly computing each side component by component that the vector identity

\[ \nabla \times (\nabla \times \mathbf{E}) = \nabla(\nabla \cdot \mathbf{E}) - \nabla^2 \mathbf{E} \]

holds.

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