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.
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.
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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Question
![**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.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F77548912-c51c-4c9d-8b51-f3905a3bec75%2Fe8dfd405-99c8-4ea5-9fa6-0a82d0862ca8%2F2kuk6rl_processed.png&w=3840&q=75)
Transcribed Image Text:**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.
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