Question

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

Answer 1

$$curl \bold A=\begin{vmatrix} \bold i & \bold j & \bold k \\ \\ \dfrac{\partial}{\partial x} & \dfrac{\partial}{\partial y} & \dfrac{\partial}{\partial z} \\ \\ -5y & 4x & z \end{vmatrix}$$

$$=9\bold k$$

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

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

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

$$=\int \int_S9dS=9\int \int_SdS$$

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

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

$$=\int \int_S9dS=9\int \int_SdS=9\pi(4)=36\pi$$

$$∮_C\bold A\cdot d\bold r=36\pi$$