10. Laplace's Equation (Dedicated to the Physics Majors! ) Laplace's Equation, shown below, is a second-order...

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10. Laplace's Equation (Dedicated to the Physics Majors! ) Laplace's Equation, shown below, is a second-order partial differential equation. In the study of heat conduction, the LaPlace Equation is the steady state heat equation. a a LaPlace's Equation: + = 0 x2 y2 A function which satisfies Laplace's Equation is said to be harmonic. a) Verify that f(x, y) = e*cos(y) is a harmonic function. b) Suppose u(x, y) and v(x, y) are functions which have continuous mixed partial derivatives. Also, assume that u(x, y) and v(x, y) satisfy the Cauchy Riemann Equations: Iv = and == Iv Verify that u(x, y) and v(x, y) are both harmonic functions. 10. Laplace's Equation (Dedicated to the Physics Majors! ) Laplace's Equation, shown below, is a second-order partial differential equation. In the study of heat conduction, the LaPlace Equation is the steady state heat equation. a a LaPlace's Equation: + = 0 x2 y2 A function which satisfies Laplace's Equation is said to be harmonic. a) Verify that f(x, y) = e*cos(y) is a harmonic function. b) Suppose u(x, y) and v(x, y) are functions which have continuous mixed partial derivatives. Also, assume that u(x, y) and v(x, y) satisfy the Cauchy Riemann Equations: Iv = and == Iv Verify that u(x, y) and v(x, y) are both harmonic functions.