Appendix A Problems¶
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Table of Contents
Basic
- B-1. Basic Calculations of Partial Derivatives
- B-2. Partial Derivatives of \(1/r\)
- B-3. Equality of Mixed Partial Derivatives
- B-4. Gradient and Level Curves
- B-5. Gradient of Temperature Distribution
- B-6. Divergence of a Linear Vector Field
- B-7. Divergence of a Quadratic Vector Field
- B-8. Divergence of a Rotational Field
- B-9. Curl of a Rotational Field
- B-10. Curl of \((yz, xz, xy)\)
- B-11. Verification of \(\nabla \times (\nabla\Phi) = 0\)
- B-12. Vector Potential of a Uniform Magnetic Field
- B-13. Laplacian of \(x^2 - y^2\)
- B-14. Laplacian of \(e^x \cos y\)
- B-15. Laplacian of \(\sin(kx)\sin(ly)\)
- B-16. \(\nabla\cdot(\nabla\times\mathbf{A}) = 0\)
- B-17. \(\nabla\times(\nabla\Phi) = 0\) (\(\Phi = xyz\))
- B-18. Plane Waves Satisfy the Wave Equation
- B-19. Complex Exponential Wave Satisfies the Wave Equation
- B-20. Classification of Partial Differential Equations
Medium
- M-1. Verification of the Diffusion Equation Solution
- M-2. Gradient of Gravitational Potential
- M-3. Zero Divergence of the Coulomb Electric Field
- M-4. Laplacian of \(1/r\)
- M-5. d'Alembert Solution \(g(x - vt)\)
- M-6. Decomposition of Standing Waves
- M-7. Boundary Conditions for String Vibration Modes
Advanced
Basic¶
B-1. Basic Calculations of Partial Derivatives¶
B-2. Partial Derivatives of \(1/r\)¶
B-3. Equality of Mixed Partial Derivatives¶
B-4. Gradient and Level Curves¶
B-5. Gradient of Temperature Distribution¶
Divergence (A.3)¶
B-6. Divergence of a Linear Vector Field¶
B-7. Divergence of a Quadratic Vector Field¶
B-8. Divergence of a Rotational Field¶
rot (A.4)¶
B-9. Curl of a Rotational Field¶
B-10. Curl of \((yz, xz, xy)\)¶
B-11. Verification of \(\nabla \times (\nabla\Phi) = 0\)¶
B-12. Vector Potential of a Uniform Magnetic Field¶
Laplacian (A.5)¶
B-13. Laplacian of \(x^2 - y^2\)¶
B-14. Laplacian of \(e^x \cos y\)¶
B-15. Laplacian of \(\sin(kx)\sin(ly)\)¶
Vector Identity (A.6)¶
B-16. \(\nabla\cdot(\nabla\times\mathbf{A}) = 0\)¶
B-17. \(\nabla\times(\nabla\Phi) = 0\) (\(\Phi = xyz\))¶
B-18. Plane Waves Satisfy the Wave Equation¶
B-19. Complex Exponential Wave Satisfies the Wave Equation¶
B-20. Classification of Partial Differential Equations¶
- (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) \(i\hbar\frac{\partial \psi}{\partial t} = -\frac{\hbar^2}{2m}\frac{\partial^2 \psi}{\partial x^2}\)
Medium¶
M-1. Verification of the Diffusion Equation Solution¶
Gradient (A.2)¶
M-2. Gradient of Gravitational Potential¶
M-3. Zero Divergence of the Coulomb Electric Field¶
M-4. Laplacian of \(1/r\)¶
M-5. d'Alembert Solution \(g(x - vt)\)¶
M-6. Decomposition of Standing Waves¶
M-7. Boundary Conditions for String Vibration Modes¶
Advanced¶
A-1. Identity for \(\nabla\times(\nabla\times\mathbf{E})\)¶
Wave Equation and Second-Order Partial Differential Equation (A.7)¶
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