1.
As the circulation density is a constant, you can use the fact that you already know the area of this triangular region to compute the double integral quickly !
Section 4.4 Example using Green's Theorem
Let's actually compute this out. We'll verify both parts of Green's Theorem by computing both sides of the equation for each part.
Learning Objectives:
- Compute Circulation and Flux via Green's Theorem
TYPO:Circulation density is \(\frac{\partial N}{\partial x}-\frac{\partial M}{\partial y}\text{.}\) At 2:30 I compute partial of M w.r.t x instead. So the integrand for the double integral should ONLY be y.
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Consider the triangular curve going counterclockwise from (0,0) to (1,0) to (0,1) and back to (0,0). Let \(\vec{F}=2y\hat{i}+x\hat{j}\text{.}\) Use Green's Theorem to compute the counterclockwise circulation two ways:
2.
To verify, you can use our multivariable calculus methods for double integrals to compute it out
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