Brain Gains (Rapid Recall, Jivin' Generation)

  • Consider the surface $\vec r(u,v) = (2u+3v,4u+5v,6u+7v)$. The differential is $$d\vec r=\begin{pmatrix}2\\4\\6\end{pmatrix}du+\begin{pmatrix}3\\5\\7\end{pmatrix}dv.$$ Find the area of the parallelogram with edges $(2,4,6)du$ and $(3,5,7)dv$.

Solution

The magnitude of the cross product of these two vectors is $$\begin{align*} |(2,4,6)du\times(3,5,7)dv| &=\left|(2,4,6)\times(3,5,7)\right|dudv\\ &=\left|(4(7)-5(6),3(6)-2(7),2(5)-3(4))\right|dudv\\ &=\left|(-2,4,-2)\right|dudv\\ &=\sqrt{4+16+4}dudv\\ &=\sqrt{24}dudv \end{align*}$$

  • Draw the surface $S$ with parametrization $\vec r(u,v) = (4\cos u, v, 4\sin u)$ for $0\leq u\leq \pi$ and $2\leq v\leq 5$.
  • For a vector field $\vec F(x,y)=(M,N)$ that is continuously differentiable everywhere, Green's theorem states $$\oint_C Mdx+Ndy = \iint_R N_x-M_ydA,$$ where $R$ is region inside a simple closed piecewise smooth curve $C$ that traverses around the boundary of $R$ in a counterclockwise fashion. Let $\vec F = (3x+4y,6x+7y)$ and let $C$ be the curve which starts at $(1,-3)$, travels right to $(5,-3)$, up to $(5,7)$, left to $(1,7)$, and then back down to $(1,-3)$. Find the work done by $\vec F$ along $C$.

Solution

Note that $N_x-M_y=6-4=2$. By Green's theorem, the requested integral is the same as $\iint_R2dA=2A$, twice the area of the rectangle $R$ inside $C$. The width of this rectangle is 4, and the height is 10, so the answer is $W=2A = 2(40)=80$.

  • For the surface $\vec r(u,v) = (u,v,9-u^2-v^2)$, compute the normal vector $\vec n = \frac{\partial \vec r}{\partial u}\times \frac{\partial \vec r}{\partial v}$.

Solution

The partial derivatives are $$\frac{\partial \vec r}{\partial u} = (1,0,-2u),\quad \frac{\partial \vec r}{\partial v} = (0,1,-2v), $$ The normal vector $\vec n$ is the cross product of these two, so $$\vec n = (2u, 2v,1).$$

  • Give an equation of the tangent plane to the surface $\vec r(u,v) = (u,v,9-u^2-v^2)$ at the point $\vec r(2,1)$.
  • Set up an iterated double integral to compute the surface area $\ds S = \iint_S |\vec r_u\times \vec r_v|dudv$ of the portion of the surface $\vec r(u,v) = (u,v,9-u^2-v^2)$ for $0\leq u\leq 3$ and $-3\leq v\leq 3$.

Solution

The area of the parallelogram formed by $$\frac{\partial \vec r}{\partial u}du = (1,0,-2u)du\quad \text{and}\quad \frac{\partial \vec r}{\partial v}dv = (0,1,-2v)dv$$ is the magnitude of their cross product. The cross product of these two vectors is $$\vec n = (2u, 2v,1)dudv.$$ The magnitude is $$d\sigma = \sqrt{4u^2+4v^2+1}dudv.$$ The surface area is $$\sigma = \iint_S d\sigma = \int_{0}^{3}\int_{-3}^{3}\sqrt{4u^2+4v^2+1}dvdu.$$

Group Problems

  1. Compute the work done by $\vec F = (-3y,3x)$ to move an object counterclockwise once along the circle $\vec r(t) = (5\cos t, 5\sin t).$
    1. Compute this work using $ \oint_{C} M dx+Ndy$. [Check $150\pi$.]
    2. Compute this work using $\iint_R N_x-M_y dA$. (You can set up the integral in rectangular, or polar, or just use facts about area.)
  2. 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).
  3. Draw each curve or surface given below.
    1. $\vec r(t) = (3\cos t,3\sin t)$ for $0\leq t\leq 2\pi$.
    2. $\vec r(u,v) = (3\cos u,3\sin u,v)$ for $0\leq u\leq 2\pi$ and $0\leq v\leq 5$.
    3. $\vec r(u,v) = (4\cos u,v, 3\sin u)$ for $0\leq u\leq \pi$ and $0\leq v\leq 7$.
    4. $\vec r(t) = (3\cos t,3\sin t,4t)$ for $0\leq t\leq 6\pi$. (Check: Helix)
    5. $\vec r(u,v) = (u\cos v,u\sin v,u)$ for $0\leq v\leq 2\pi$ and $0\leq u\leq 4$. (Check: Cone)
    6. $\vec r(u,v) = (u\cos v,u\sin v,v)$ for $0\leq v\leq 6\pi$ and $2\leq u\leq 4$. (Check: Spiral stair case)
    7. $\vec r(t) = (0,t,9-t^2)$ for $0\leq t\leq 3$.
    8. $\vec r(u,v) = (u\cos v,u\sin v,9-u^2)$ for $0\leq v\leq 2\pi$ and $0\leq u\leq 3$.
  4. Consider the surface parametrized by $\vec r(u,v) = (u\cos v, u\sin v, u^2)$ for $0\leq u\leq 3$ 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 $\bar z$ for this surface.