- Using $\vec F = (2x+y,3x)$ for the curve $x=y^2$ from $(1,-1)$ to $(4,2)$, do the following:
- Set up the flux integral $\ds\int_C Mdy-Ndx$.
- Set up the work integral $\ds\int_C Mdx+Ndy$.
- Find a function $f(x,y)$ so that $Df(x,y) = \begin{bmatrix}2x+3y&3x+4y\end{bmatrix}.$
- Consider a wire that lies along the left half of the circle of radius 7 centered at the origin.
- Give a parametrization for the left half of the circle.
- The centroid of the wire is the $(\bar x,\bar y)$ location that gives the average $x$ value and $y$ value for points along the wire. Without any integrals, what is $\bar y$?
- Set up a formula to compute $\bar x$, so set up the integral $\bar x = \dfrac{\int_C x ds}{\int_C ds}$.
- Set up a formula for the $y$-coordinate of the center of mass if the wire has density $\delta(x,y)=5+x+y$.
- For the function $f(x,y)=3x^2+7xy+5y^2$, compute both $Df$ and $D^2f$.
- Find a function $f(x,y)$ so that $Df(x,y) = \begin{bmatrix}2x+3y&3x+4y\end{bmatrix}.$
|
Sun |
Mon |
Tue |
Wed |
Thu |
Fri |
Sat |