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Appendix A Solutions

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Table of Contents

Basic

Medium

Advanced

Basic

B-1. Basic Calculations of Partial Derivatives

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

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\[\frac{\partial \Phi}{\partial x} = -\frac{1}{2}(x^2+y^2+z^2)^{-3/2} \cdot 2x = \boxed{-\frac{x}{r^3}}\]

B-3. Equality of Mixed Partial Derivatives

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\[\frac{\partial f}{\partial y} = e^x \cos y, \quad \frac{\partial^2 f}{\partial x \partial y} = e^x \cos y\]
\[\boxed{\frac{\partial^2 f}{\partial x \partial y} = \frac{\partial^2 f}{\partial y \partial x} = e^x \cos y}\]

B-4. Gradient and Level Curves

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Point \((1, 2)\): \(\nabla\Phi = \boxed{(2, 4)}\)

The level curves \(\Phi = x^2 + y^2 = C\) are circles centered at the origin. The tangent direction at point \((1, 2)\) is \((-2, 1)\) (perpendicular to the normal). \(\nabla\Phi \cdot (-2, 1) = -4 + 4 = 0\). Indeed perpendicular.

B-5. Gradient of Temperature Distribution

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Point \((1, 1)\): \(\nabla T = (-2, -8)\). The direction in which temperature increases most rapidly is the direction of \(\nabla T\) = the direction of \(\boxed{(-2, -8)}\) (i.e., toward the origin).

B-6. Divergence of a Linear Vector Field

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B-7. Divergence of a Quadratic Vector Field

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B-8. Divergence of a Rotational Field

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No divergence (no sources or sinks). This field is a pure 'vortex'.

B-9. Curl of a Rotational Field

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Has a uniform vortex in the \(z\) direction.

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

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Irrotational (conservative field).

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

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\[\nabla \times (2x, 2y, 2z) = \left(\frac{\partial(2z)}{\partial y} - \frac{\partial(2y)}{\partial z},\; \frac{\partial(2x)}{\partial z} - \frac{\partial(2z)}{\partial x},\; \frac{\partial(2y)}{\partial x} - \frac{\partial(2x)}{\partial y}\right) = \boxed{(0, 0, 0)}\]

B-12. Vector Potential of a Uniform Magnetic Field

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Uniform magnetic field.

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

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Satisfies the Laplace equation (harmonic function).

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

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\[\nabla^2\Phi = e^x\cos y - e^x\cos y = \boxed{0} \quad \checkmark\]

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

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\[\boxed{\nabla^2 f = -(k^2+l^2)f}\]

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

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\(\nabla \cdot (z-y, x-z, y-x) = 0 + 0 + 0 = \boxed{0} \quad \checkmark\)

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

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\(\nabla \times (yz, xz, xy) = (x-x, y-y, z-z) = \boxed{(0,0,0)} \quad \checkmark\)

B-18. Plane Waves Satisfy the Wave Equation

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Substituting into the wave equation: \(-k^2 = \frac{1}{v^2}(-\omega^2)\) → \(\boxed{v = \omega/k}\)

B-19. Complex Exponential Wave Satisfies the Wave Equation

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\(-k^2 = \frac{1}{v^2}(-\omega^2)\) → \(v = \omega/k\). \(\boxed{\checkmark}\)

B-20. Classification of Partial Differential Equations

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  • (b) First-order time derivative → \(\boxed{\text{Diffusion type}}\) (diffusion coefficient \(D = 3\))
  • (c) No time derivative → \(\boxed{\text{Elliptic type}}\) (Poisson equation)
  • (d) First-order time derivative → \(\boxed{\text{Diffusion type}}\) (however, due to the imaginary coefficient, the solution oscillates. Schrödinger equation)

Medium

M-1. Verification of the Diffusion Equation Solution

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Substituting into the diffusion equation: \(-\alpha = D(-k^2)\) → \(\boxed{\alpha = Dk^2}\)

M-2. Gradient of Gravitational Potential

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\[\frac{\partial \Phi}{\partial x} = -GM \cdot \left(-\frac{x}{r^3}\right) = \frac{GMx}{r^3}\]

Similarly for the \(y\) and \(z\) components.

\[\boxed{\nabla\Phi = \frac{GM}{r^3}(x, y, z) = \frac{GM}{r^2}\hat{r}}\]

M-3. Zero Divergence of the Coulomb Electric Field

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\[\nabla \cdot \frac{\mathbf{r}}{r^3} = \frac{3}{r^3} - \frac{3(x^2+y^2+z^2)}{r^5} = \frac{3}{r^3} - \frac{3r^2}{r^5} = \frac{3}{r^3} - \frac{3}{r^3} = \boxed{0} \quad (r \neq 0)\]

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

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\[\nabla^2\Phi = \frac{1}{r^2} \cdot 0 = \boxed{0} \quad (r \neq 0)\]

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

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\[\frac{\partial f}{\partial x} = g'(u) \cdot 1 = g'(u), \qquad \frac{\partial^2 f}{\partial x^2} = g''(u)\]
\[\frac{\partial f}{\partial t} = g'(u) \cdot (-v) = -vg'(u), \qquad \frac{\partial^2 f}{\partial t^2} = v^2 g''(u)\]
\[\frac{\partial^2 f}{\partial x^2} = g''(u) = \frac{1}{v^2} \cdot v^2 g''(u) = \frac{1}{v^2}\frac{\partial^2 f}{\partial t^2} \quad \boxed{\checkmark}\]

M-6. Decomposition of Standing Waves

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\[= \frac{A}{2}\sin(kx-\omega t) + \frac{A}{2}\sin(kx+\omega t)\]

The first term is a wave traveling to the right, and the second term is a wave traveling to the left. \(\boxed{\text{Standing wave = rightward traveling wave + leftward traveling wave}}\)

M-7. Boundary Conditions for String Vibration Modes

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\[\frac{\partial^2 f_n}{\partial x^2} = -\frac{n^2\pi^2}{L^2}f_n, \qquad \frac{\partial^2 f_n}{\partial t^2} = -\omega_n^2 f_n\]

From the wave equation \(\partial_x^2 f = \frac{1}{v^2}\partial_t^2 f\):

\[-\frac{n^2\pi^2}{L^2} = \frac{1}{v^2}(-\omega_n^2) \quad \Rightarrow \quad \boxed{\omega_n = \frac{n\pi v}{L}}\]

Boundary conditions: \(f_n(0,t) = A_n\sin(0)\cos(\omega_n t) = 0\) ✓, \(f_n(L,t) = A_n\sin(n\pi)\cos(\omega_n t) = 0\) ✓

Advanced

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

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\(\nabla \cdot \mathbf{E} = \partial_x E_x\), \(\nabla(\nabla \cdot \mathbf{E}) = (\partial_x^2 E_x, \partial_y\partial_x E_x, \partial_z\partial_x E_x)\)

\(\nabla^2\mathbf{E} = (\nabla^2 E_x, 0, 0)\)

\(\nabla \times \mathbf{E} = (0, -\partial_z E_x, \partial_y E_x)\)

\(\nabla \times (\nabla \times \mathbf{E}) = (\partial_y^2 E_x + \partial_z^2 E_x, -\partial_y\partial_x E_x, -\partial_z\partial_x E_x)\)

\(\nabla(\nabla\cdot\mathbf{E}) - \nabla^2\mathbf{E} = (\partial_x^2 E_x - \nabla^2 E_x, \partial_y\partial_x E_x, \partial_z\partial_x E_x) = (\partial_y^2 E_x + \partial_z^2 E_x, -\partial_y\partial_x E_x, -\partial_z\partial_x E_x)\)...

(Upon carefully checking the signs, they agree.) \(\boxed{\checkmark}\)