Answer to Question #238812 in Electricity and Magnetism for GAYATHRI

Question #238812


1. Verify Stock’s theorem for F~ = x^2i+xyj, where s is the area bonded by

a triangle whose vertices are at (0, 0), (1, −1) and (1, 1), respectively.


1
Expert's answer
2021-09-18T18:28:54-0400

curl F=ijkxyzx2xy0=yk,\text{curl }\vec F=\begin{vmatrix} \vec i& \vec j & \vec k\\ \frac{\partial}{\partial x}& \frac{\partial}{\partial y}&\frac{\partial}{\partial z}\\ x^2 & xy& 0 \end{vmatrix}=y\vec k,

curlFn=y,\text{curl}\vec F\cdot \vec n=y,

ScurlFndS=0101ydydx+0110ydydx=01y2201dx+01y2210dx=0112dx+01(12)dx=0.\underset{S}{\iint }\text{curl}\vec F\cdot \vec ndS =\int_0^1\int_0^1ydydx+\int_0^1\int_{-1}^0ydydx=\int_0^1\frac{y^2}2|_0^1dx+\int_0^1\frac{y^2}2|_{-1}^0dx=\int_0^1\frac 12dx+\int_0^1(-\frac 12)dx=0.


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