- 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)
- The gravitational vector field and electrostatic field are directly related to the radial field $\ds\vec F(x,y,z) = \frac{\left(-x,-y,-z\right)}{(x^2+y^2+z^2)^{3/2}}$.
- We say that a vector field is conservative when the work done by the field is independent of the path traveled. Show that this vector field is conservative, by finding a potential for $\vec F$.
- Compute the work done by $\vec F$ to move an object from $(1,2,-2)$ to $(0,-3,4)$ along ANY path that avoids the origin.
- 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 $0\leq t\leq 2\pi$.
- Draw the surface $S$ and curve $C$. How are these two objects related?
- 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.
- Set up and compute the integral $\ds \int_C Mdx+Ndy+Pdz$, computing the work done by $\vec F$ along $C$.
- 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 $y$-axis.
- 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 $0\leq v\leq \pi$.
- Draw the surface $S$ and domain $D$. How are these two objects related?
- 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 $D$.
- 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 $S$.
- 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 $D$.
- 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
- 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 $(0,3)$.
- Set up the single double integral $\iint_R N_x-M_y dA$.
- Compute the integral (use use facts about area).
- 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 $0\leq v\leq 2\pi$.
- Draw the surface.
- Compute $dS = \left|\dfrac{\partial \vec r}{\partial u}\times\dfrac{\partial \vec r}{\partial v}\right|dudv$.
- Set up an integral formula to compute the surface area of $S$.
- Set up an integral formula to compute $\bar z$ for this surface.
- 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?
- 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.
- 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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