Gauss's Theorem, Cartesian Coordinates Calculate both integrals appearing in Gauss's integral theorem for the unit cube
W=[0,1]^(3)
and the vector field
vec(F)(vec(r))=(x^(2),yz,y)^(T)
. Hint: For the integral over the surface of the cube, you need to evaluate 6 individual integrals. Consider how the differential surface element
dvec(A)
should look on each side and note that one component is constant on each side. Example: On the side that coincides with the yz-plane,
dvec(A)=-hat(e)_(x)dydz
and
x=0
.

Question

Gauss's Theorem, Cartesian Coordinates Calculate both integrals appearing in Gauss's integral theorem for the unit cube

W=[0,1]^(3)

and the vector field

vec(F)(vec(r))=(x^(2),yz,y)^(T)

. Hint: For the integral over the surface of the cube, you need to evaluate 6 individual integrals. Consider how the differential surface element

dvec(A)

should look on each side and note that one component is constant on each side. Example: On the side that coincides with the yz-plane,

dvec(A)=-hat(e)_(x)dydz

and

x=0

.

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