verify stokes theoremfor vector point function A= (2x-y)i-(2)j where s is hamisphere x^2+y^2+z^2=1

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sciencesolve | Teacher | (Level 3) Educator Emeritus

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The Stoke's theorm needs to express (2x-y)i-(yz2)j-(y2z)k as the curl of a vector field F.

Use curl formula:

curl F=`[[i,j,k],[(del)/(del x),(del)/(del y), (del)/(del z)],[F_x,F_y,F_z]]` = `( (del F_z)/(del y) - (del F_y)/(del z) , (del F_x)/(del z) - (del F_z)/(del x) , (del F_y)/(del x) - (del F_x)/(del y))`

Put `(del F_z)/(del y) - (del F_y)/(del z) ` = 2x - y

`(del F_y)/(del x) - (del F_x)/(del y) = 0 =gt F_y = F_x = 0 =gt F_z = 2x - y`

`` `oint` 2x - y dz = 0 because z is not changing over the boundary curve of the hemisphere.

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