Let surface S be the boundary of the solid object enclosed by x^2+z^2=4, x+y=6, x=0, y=0, and z=0. and, let f(x, y,z)=(3x)i+(x+y+2z)j + (3z)k be a vector field (for example, the velocityfaild of a fluid flow). the solid object has five sides, S1:bottom(xy-plane), S2:left side(xz-plane), S3 rear side(yz-plane), S4:right side, and S5:cylindrical roof. a. Sketch the solid object.
b. Evaluate the flux of F through each side of the object (S1,S2,S3,S4,S5).
c. Find the total flux through surface S.

a. I've attached a plot of the surface. Each face is parameterized by

• with and ;

• with and ;

• with and ;

• with and ; and

• with and .

b. Assuming you want outward flux, first compute the outward-facing normal vectors for each face.

Then integrate the dot product of f with each normal vector over the corresponding face.

c. You can get the total flux by summing all the fluxes found in part b; you end up with 42π - 56/3.

Alternatively, since S is closed, we can find the total flux by applying the divergence theorem.

where R is the interior of S. We have

The integral is easily computed in cylindrical coordinates:

as expected.

measuring things 1 litre of water