Brain Gains (Rapid Recall, Jivin' Generation)

  1. Let $\vec F(x,y) = (y^2, 3x)$. Let $C$ be the rectangular curve which starts at $(-1,2)$, then heads to $(3,2)$, then $(3,6)$, then $(-1,6)$, and back to $(-1,2)$.
    1. Set up and compute the 4 line integrals needed to calculate $\ds \int_C Mdx+Ndy$.
    2. Use Green's theorem instead to compute $\ds \int_C Mdx+Ndy$ (so set up and compute $\ds\iint_R N_x-M_ydA$.
  2. Let $\vec F(x,y) = (x^2, 3z, -3y)$. Let $C$ be the curve $\vec r(t) = (3, 4\cos t, 4\sin t)$ for $0\leq t\leq 2\pi$. Let $S$ be the surface parametrized by $\vec r(u,v) = (3, u\cos v, u\sin v)$.
    1. Set up and compute the $\ds \int_C Mdx+Ndy+Pdz$.
    2. Compute the curl of $\vec F$, so compute $(N_z-P_y,P_x-M_z, N_x-M_y)$.
    3. Compute $\ds \iint_S \vec \nabla\times\vec F\cdot \hat n dS$ for the surface $S$ using the orientation $\hat n$ that is compatible with the orientation of $C$.

Group Problems

Set up and compute as many of the integrals as you can from 42.3. Create Mathematica code to perform the computations for you.

  1. Consider the vector field $\vec F = (x,x-z,y+z)$, the surface $S$ parametrized by $\vec r(u,v)=(u^2, u\cos v, u\sin v)$ for $0\leq u\leq 2$ and $0\leq v\leq 2\pi$, and the curve $C$ parametrized by $\vec r(t) = (4,2\cos t, 2\sin t)$ for $0\leq t\leq 2\pi$.
    1. Draw the surface $S$ and curve $C$. How are these two objects related?
    2. Compute $\vec N = \vec r_u\times \vec r_v$ and determine if $\vec N$ points inward toward the $x$-axis, or outwards away from the $x$-axis.
    3. Set up and compute the integral $\ds \int_C Mdx+Ndy+Pdz$, computing the work done by $\vec F$ along $C$.
    4. Set up and compute $\ds \iint_S \vec \nabla \times \vec F\cdot \hat n dS$, computing the flux of the curl of $\vec F$ across $S$ in the direction $\hat n$ outwards away from the $x$-axis.
  2. Consider the vector field $\vec F = (x,x-z,y+z)$, the solid domain $D$ that lies inside the sphere $x^2+y^2+z^2=25$, and the surface $S$ parametrized by $\vec r(u,v)=(5\sin v\cos u, 5 \sin v \sin u, 5 \cos v)$ for $0\leq u\leq 2\pi$ and $0\leq v\leq \pi$.
    1. Draw the surface $S$ and domain $D$. How are these two objects related?
    2. Compute $\vec N = \vec r_u\times \vec r_v$ and determine if $\vec N$ points inward toward the domain $D$ or outwards away from the domain $D$.
    3. Set up and compute $\ds \iint_S \vec F\cdot \hat n dS$ for $\hat n$ pointing outwards, away from the solid inside $S$. This computes the outward flux of $\vec F$ across $S$.
    4. Set up and compute $\ds \iiint_D \vec \nabla \cdot \vec F dV$, the triple integral of the divergence of $\vec F$ over the domain $D$.