Divergence (Gauss) theorem
Multivariable and vector calculus · Mathematics
Study notes
Q: Find the flux of F = (x^3, y^3, z^3) outward through the unit sphere. div F = 3x^2 + 3y^2 + 3z^2 = 3r^2. Flux = triple integral over ball of 3r^2 dV. Spherical: dV = rho^2 sin phi d(rho) dtheta dphi. = 3 times integral 0 to 2pi dtheta times integral 0 to pi of sin phi dphi times integral 0 to 1 of rho^4 d(rho). = 3 times 2pi times 2 times 1/5 = 12pi/5.