**Problem 4:**Suppose the function \( f \) defined on the plane satisfies the Laplace Equation:\[f_{xx} + f_{yy} = 0\]Show that \[\oint_C \frac{\partial f}{\partial y} \, dx - \frac{\partial f}{\partial x} \, dy = 0\]for any simple closed curve \( C \) in the plane.**Explanation:**In this problem, you are asked to verify that for a function \( f \) satisfying the Laplace Equation on the plane, the line integral of a particular expression over any simple closed curve \( C \) is zero. The expression involves the partial derivatives of \( f \) with respect to \( x \) and \( y \). The conclusion is derived from Green's Theorem, given that the conditions of the Laplace Equation imply that the curl of the vector field formed by these partial derivatives is zero.

Suppose that the function defined on the plane satisfies (Laplace Equation)We have to show…

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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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4. Suppose the function f defined on the plane satisfies fax + fyy = 0 (Laplace Equation). Show that fodrody = 0 for any simple closed curve C in the plane.