Verify the div. Theorem for the vector field $\underline{A} = (x+y)\underline{i} + (x^2+xy)\underline{j} + z^2\underline{k}$ and a unit radius ball centred at $(1,1,1)$.
The question gives a hint to move the ball so that its centre is at the origin, as then it would be easy to use symmetry arguments for the integrals.
let $x = 1 + X$, $y = 1+Y$, $z = 1+Z$, then in $(X,Y,Z)$ $V$ is the unit ball.
we compute $\underline{A}(X,Y,Z) = (2+X+Y)\underline{i} + (2+3X+X^2+Y+XY)\underline{j} + (1+2Z+Z^2)\underline{k}$, computing $\text {div}(\underline{A}) = 4 + X + 2Z$
then $\displaystyle \int_{V} div A \ dV = \int_V (4+X+2Z) \ dV$ the solutions then claim that $$\int_{V} div A \ dV = \int_V (4+X+2Z) \ dV = 4|V| = \dfrac{16\pi}{3}$$ by symmetry.
I don't see how this is at all by symmetry:
1) It seems as though the lecture has calculated $\displaystyle \int_{V} div A \ dV = \int_V (4+X+2Z) \ dV = \int_V (4) \ dV$ why?
2) Why did we have to move the ball to the origin to use this method?
For verifying the other integral this is what the lecture has:
I'm having troubles understanding which unit normal he used, and what he means by as $S \to S$, $x \to -X$ or ...
Any explanation, please
anyone??
