Question #281717

use Stokes theorem evaluate ∮A.dr, where A=-5yi+4xj+zk and C is cirvle x²+y²=4,z=1


1
Expert's answer
2021-12-23T04:54:40-0500
curlA=ijkxyz5y4xzcurl \bold A=\begin{vmatrix} \bold i & \bold j & \bold k \\ \\ \dfrac{\partial}{\partial x} & \dfrac{\partial}{\partial y} & \dfrac{\partial}{\partial z} \\ \\ -5y & 4x & z \end{vmatrix}

=9k=9\bold k

n=0,0,1\bold n=\langle0, 0, 1\rangle

curlAn=9curl \bold A\cdot \bold n=9

CAdr=ScurlAdS∮_C\bold A\cdot d\bold r=\int\int _Scurl \bold A\cdot d\bold S

=S9dS=9SdS=\int \int_S9dS=9\int \int_SdS

The double integral in the latter formula is the area of the circle. Therefore, the integral is

CAdr=ScurlAdS∮_C\bold A\cdot d\bold r=\int\int _Scurl \bold A\cdot d\bold S

=S9dS=9SdS=9π(4)=36π=\int \int_S9dS=9\int \int_SdS=9\pi(4)=36\pi

CAdr=36π∮_C\bold A\cdot d\bold r=36\pi


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