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Stewart
Evaluate \(\int_C \bm{F}\cdot\mathrm{d}\bm{r},\)
where \(\bm{F}(x,y,z) = -y^2\mathbf{i} +x\mathbf{j}+z^2\mathbf{k}\)
and \(C\) is the curve of intersection of the plane
\(y+z=2\) and the cylinder \(x^2+y^2=1.\)
(Orient \(C\) to be counterclockwise when viewed from above.)
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Stewart
Use Stokes’ Theorem to compute the integral
\(\iint_S \operatorname{curl}\bm{F}\cdot\mathrm{d}\bm{S},\)
where \(\bm{F}(x,y,z) = xz\mathbf{i}+yz\mathbf{j}+xy\mathbf{k}\)
and \(S\) is the part of the sphere \(x^2+y^2+z^2=4\)
that lies inside the cylinder \(x^2+y^2=1\)
and above the \(xy\)-plane.
-
Use Stokes’ Theorem to compute the integral
\(\iint_S \operatorname{curl}\bm{F}\cdot\mathrm{d}\bm{S},\)
where \(\bm{F}(x,y,z) = {z^2}\mathbf{i} - 3xy\mathbf{j} + {x^3}{y^3}\mathbf{k}\)
and \(S\) is the part of \(z=5-x^2-y^2\)
above the plane \(z=1.\)
Assume \(S\) is oriented upwards.
-
Use Stokes’ Theorem to compute the integral
\(\int_C \operatorname{curl}\bm{F}\cdot\mathrm{d}\bm{r},\)
where \(\bm{F}(x,y,z) = {z^2}\mathbf{i} + y^2\mathbf{j} + x\mathbf{k}\)
and \(C\) is the triangle with vertices
\((1,0,0)\) and \((0,1,0)\) and \((0,0,1)\)
with counterclockwise orientation.
Review Before Next Class
-
Calculate \(\operatorname{div}\bm{F}\)
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.
\[
\oint\limits_C \bm{F}\cdot\bm{n}\,\mathrm{d}s
= \iint\limits_D \operatorname{div}\bm{F}\,\mathrm{d}A
\]