Suppose f⃗ is a vector field with div(f⃗ (x, y,z))=−3. use the divergence theorem to calculate the flux of the vector field f⃗ out of the closed, outward-oriented cylindrical surface s of height 7 and radius 5 that is centered about the z-axis with its base in the xy-plane.

The surface is the boundary of a cylindrical region that can be expressed in cylindrical coordinates by the set of points

By the divergence theorem, the flux of across is given by the integral of over :

example 2 – simplify: step 1: completely factor both the numerators and denominators of all fractions. step 2: change the division sign to a multiplication sign and flip (or reciprocate) the fraction after the division sign; essential you need to multiply by the reciprocal. step 3 : cancel or reduce the fractions.

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