47. Harmonic Function A function f is called harmonic when af, af 0. dx? dy? Prove that if f is harmonic, then of dx dx の)。 I dy 3D ду where C is a smooth closed curve in the plane.
47. Harmonic Function A function f is called harmonic when af, af 0. dx? dy? Prove that if f is harmonic, then of dx dx の)。 I dy 3D ду where C is a smooth closed curve in the plane.
Calculus: Early Transcendentals
8th Edition
ISBN:9781285741550
Author:James Stewart
Publisher:James Stewart
Chapter1: Functions And Models
Section: Chapter Questions
Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...
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I need help figuring out number 47. Thanks!
![### Harmonic Function
A function \( f \) is called **harmonic** when
\[
\frac{\partial^2 f}{\partial x^2} + \frac{\partial^2 f}{\partial y^2} = 0.
\]
#### Prove that if \( f \) is harmonic, then
\[
\oint_C \left( \frac{\partial f}{\partial y} \, dx - \frac{\partial f}{\partial x} \, dy \right) = 0
\]
where \( C \) is a smooth closed curve in the plane.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fda21d2aa-bb54-4a64-8baf-73046aeb648e%2Ff4e93384-aa76-4f9f-bbdc-ffa168d66566%2Fnegkdp4_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Harmonic Function
A function \( f \) is called **harmonic** when
\[
\frac{\partial^2 f}{\partial x^2} + \frac{\partial^2 f}{\partial y^2} = 0.
\]
#### Prove that if \( f \) is harmonic, then
\[
\oint_C \left( \frac{\partial f}{\partial y} \, dx - \frac{\partial f}{\partial x} \, dy \right) = 0
\]
where \( C \) is a smooth closed curve in the plane.
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