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Suppose f(x, y) and g(x, y) are continuously differentiable and dyf(x, y) = dxg(x, y). Suppose this holds in an open convex set C R2. Then there is a p(x, y), which is continuously differentiable, so that dx(x, y) = f(x, y), y(x, y) = g(x, y). This holds for all (x, y) = N. Let u(x, y) be twice continuously differentiable and harmonic in an open convex domain . Then there is a continuously differentiable complex conjugate function v(x, y), defined in , so that dxu(x, y) = dyv(x, y), dyu(x, y) = −dxv(x, y). The functions and v defined above are unique up to a constant.

Suppose f(x, y) and g(x, y) are continuously differentiable and dyf(x, y) = dxg(x, y). Suppose this holds in an open convex set C R2. Then there is a p(x, y), which is continuously differentiable, so that dx(x, y) = f(x, y), y(x, y) = g(x, y). This holds for all (x, y) = N. Let u(x, y) be twice continuously differentiable and harmonic in an open convex domain . Then there is a continuously differentiable complex conjugate function v(x, y), defined in , so that dxu(x, y) = dyv(x, y), dyu(x, y) = −dxv(x, y). The functions and v defined above are unique up to a constant.

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Prove the following assertions: (a) Suppose f(x, y) and g(x, y) are continuously differentiable and dyf(x, y) = dxg(x, y). Suppose this holds in an open convex set NC R². Then there is a (x, y), which is continuously differentiable, so that dxp(x, y) = f(x, y), dyp(x, y) = g(x, y). This holds for all (x, y) = N. (b) Let u(x, y) be twice continuously differentiable and harmonic in an open convex domain . Then there is a continuously differentiable complex conjugate function v(x, y), defined in , so that dxu(x, y) = dyv(x,y), dyu(x, y) = −dxv(x, y) . (c) The functions and v defined above are unique up to a constant.
Transcribed Image Text:Prove the following assertions: (a) Suppose f(x, y) and g(x, y) are continuously differentiable and dyf(x, y) = dxg(x, y). Suppose this holds in an open convex set NC R². Then there is a (x, y), which is continuously differentiable, so that dxp(x, y) = f(x, y), dyp(x, y) = g(x, y). This holds for all (x, y) = N. (b) Let u(x, y) be twice continuously differentiable and harmonic in an open convex domain . Then there is a continuously differentiable complex conjugate function v(x, y), defined in , so that dxu(x, y) = dyv(x,y), dyu(x, y) = −dxv(x, y) . (c) The functions and v defined above are unique up to a constant.
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