20230320

• I-Learn, Class Pictures, Learning Targets, Text Book Practice
• Prep Tasks: Unit 1 - Motion, Unit 2 - Derivatives, Unit 3 - Integration, Unit 4 - Vector Calculus

This is day 5 of Unit 4.

Brain Gains (Rapid Recall, Jivin' Generation)

• 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 $(-1,2)$.
  • Set up and compute the 4 line integrals needed to calculate $\ds \int_C Mdx+Ndy$.
  • Use Green's theorem instead to compute $\ds \int_C Mdx+Ndy$ (so set up and compute $\ds\iint_R N_x-M_ydA$.

Group Problems

1. Compute the work done by $\vec F = (2x-y,2x+4y)$ to move an object counterclockwise once along the triangle with corners $(0,0)$, $(2,0)$, and $(0,3)$.
   1. Set up the single double integral $\iint_R N_x-M_y dA$.
   2. Compute the integral (use use facts about area).
2. Consider the surface $S$ parametrized by $\vec r(u,v) = (u\cos v, u\sin v, u^2)$ for $0\leq u\leq 2$ and $0\leq v\leq 2\pi$.
   1. Draw the surface.
   2. Compute $dS = \left|\dfrac{\partial \vec r}{\partial u}\times\dfrac{\partial \vec r}{\partial v}\right|dudv$.
   3. Set up an integral formula to compute the surface area of $S$.
   4. Set up an integral formula to compute $\bar z$ for this surface.
   5. We would like an orientation $\hat n$ for the surface that points away from the $z$-axis. Does $ \dfrac{\partial \vec r}{\partial u}\times\dfrac{\partial \vec r}{\partial v}$ point towards the $z$-axis, or away from the $z$-axis?
   6. Set up the surface integral that gives the flux of $\vec F = (3yz,-2x+y, z-2x)$ across the surface $S$ in the direction of $\hat n$. Then use software to compute the integral.
3. Find the work done by the vector field $\vec F(x,y,z) = (\frac{-x}{(x^2+y^2+z^2)^{3/2}}, \frac{-y}{(x^2+y^2+z^2)^{3/2}}, \frac{-z}{(x^2+y^2+z^2)^{3/2}})$ on an object that moves from $(1,2,2)$ to $(0,5,12)$.

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