-
Stewart
Compute the surface integral \(\iint_S x^2 \,\mathrm{d}S\)
where \(S\) is the unit sphere \(x^2+y^2+z^2=1.\)
-
Stewart
Evaluate \(\iint_S y \,\mathrm{d}S,\)
where \(S\) is the surface \(z=x+y^2\)
for \(0\leq x \leq 1\) and \(0\leq y \leq 2.\)
-
Stewart
Evaluate \(\iint_S z \,\mathrm{d}S,\) where \(S\) is the surface
whose sides \(S_1\) are given by the cylinder \(x^2+y^2=1,\)
whose bottom \(S_2\) is the disk \(x^2+y^2 \leq 1\) in the plane \(z=0,\)
and whose top \(S_3\) is the part of the plane \(z=1+x\) that lines above \(S_2.\)
-
Stewart
Calculate the flux of the vector field
\(\bm{F}(x,y,z) = z\mathbf{i}+y\mathbf{j}+x\mathbf{k}\)
across the unit sphere \(x^2+y^2+z^2=1.\)
-
Stewart
Evaluate \(\iint_S \bm{F}\cdot\mathrm{d}S,\)
where \(\bm{F}(x,y,z) = y\mathbf{i}+x\mathbf{j}+z\mathbf{k}\)
and \(S\) is the boundary of the solid enclosed by
the paraboloid \(z=1-x^2-y^2\) and the plane \(z=0.\)
-
Stewart
The temperature \(u\) in a metal ball is proportional
to the square of the distance from the center of the ball.
Calculate the rate of heat flow across a sphere of radius \(r\)
with center at the center of the ball.
Review Before Next Class
-
Calculate \(\operatorname{curl}\bm{F}\)
for \(\bm{F} = \big\langle \tan(xyz), \sec(xyz), x+y+z \big\rangle.\)
-
Stare at this formula for Green’s Theorem for three minutes,
then write down a physical/geometric interpretation
of each side of the equation.
\[
\oint\limits_C \bm{F}\cdot\mathrm{d}\bm{r}
= \iint\limits_D \operatorname{curl}\bm{F}\cdot\mathbf{k}\,\mathrm{d}A
\]