Evaluate $\iint_S \mathbf{r} \cdot d\mathbf{S}$ where S is the surface of a unit sphere $x^2 + y^2 + z^2$ = 1.
By Gauss Divergence theorem:
abla $\cdot \mathbf{r} = 3$. Integral = 3 $\iiint_V dV = 3 * Volume$ of unit sphere = 3 * (4/3 $\pi 1^3) = 4\pi \approx 12.566 \approx 12.57$.
Which of the following vector identities are ALWAYS TRUE for any twice continuously differentiable scalar field $\phi$ and vector field $\mathbf{F}$?
The curl of a gradient is zero, the divergence of a curl is zero, and vector Laplacian identity holds. Divergence of a gradient is the Laplacian
$abla^2$ $\phi, not necessarily zero$.
Work done by force field $\mathbf{F} = x i + y j along$ the line segment from (0,0) to (2,4) is:
Work = $\int_C x dx + y dy = [x^2/2 + y^2/2]_{(0,0)}^{(2,4)} = 4/2 + 16/2 = 2 + 8 = 10$.
Green's Theorem in a plane transforms the line integral $\oint_C (P dx + Q dy) into$:
Green's Theorem states $\oint_C (P dx + Q dy) = \iint_R (Q_x - P_y) dx dy$.
Evaluate $\oint_C (x^2 dy - y^2 dx) around$ the circle $x^2 + y^2$ = 1 using Green's Theorem.
P = -$y^2$, $Q = x^2$. Q_x - P_y = 2x - (-2y) = 2(x + y). Integral = $\iint_R 2(x+y) dx dy$. By symmetry over the unit circle, $\iint x dx dy = 0$ and $\iint y dx dy = 0, so total integral = 0$.
