Solution Found!
Average circulation Let S be a small circular disk of
Chapter 13, Problem 44AE(choose chapter or problem)
Average circulation Let S be a small circular disk of radius R centered at the point P with a unit normal vector n. Let C be the boundary of S.
a. Express Ute average circulation of the vector field F on S as a surface integral of \(\nabla \times \mathbf{F}\).
b. Argue that for small R. the average circulation approaches \((\nabla \times \mathbf{F})_{P} \cdot \mathbf{n}\) (the component of \(\nabla \times \mathbf{F}\) in the direction of n evaluated at P) with the approximation improving as \(R \rightarrow 0\).
Text Transcription:
nabla X F
(nabla X F)_p cdot n
nabla X F
R rightarrow 0
Questions & Answers
QUESTION:
Average circulation Let S be a small circular disk of radius R centered at the point P with a unit normal vector n. Let C be the boundary of S.
a. Express Ute average circulation of the vector field F on S as a surface integral of \(\nabla \times \mathbf{F}\).
b. Argue that for small R. the average circulation approaches \((\nabla \times \mathbf{F})_{P} \cdot \mathbf{n}\) (the component of \(\nabla \times \mathbf{F}\) in the direction of n evaluated at P) with the approximation improving as \(R \rightarrow 0\).
Text Transcription:
nabla X F
(nabla X F)_p cdot n
nabla X F
R rightarrow 0
ANSWER:Solution 44AE
(a)
Therefore the value of average circulation of the vector field F is given by,
F