Rapid Recall
- What is SULI?
Solution
Science Undergraduate Laboratory Internship. See https://www.energy.gov/science/wdts/workforce-development-teachers-and-scientists
- Write the differential of the change-of-coordinates $(x,y) = (2u+3v,4u+5v)$ as a linear combination of partial derivatives.
Solution
The solution is $$\begin{pmatrix}dx\\dy\end{pmatrix} = \begin{pmatrix}2\\4\end{pmatrix} du+\begin{pmatrix}3\\5\end{pmatrix} dv.$$
- Suppose I know $\vec u\times\vec v = (3,-2,-6)$. Compute $\vec v\times \vec u$.
Solution
We have $\vec v\times \vec u = - \vec u\times \vec v = (-3,2,6)$.
- Compute and simplify $ [(2,0,0)\times(0,3,0)]\cdot (0,0,5)$. What does this number compute?
Solution
We get $$ [(2,0,0)\times(0,3,0)]\cdot (0,0,5) =[(0,0,6)]\cdot (0,0,5) = 30. $$ This is the volume of the parallelepiped (box in this particular situation) whose edges are the three vectors we began with.
Group problems
- Consider the change-of-coordinates $(x,y)=(2u,3v)$.
- Compute the differential $d(x,y)$ and write it as a linear combination of partial derivatives and as a matrix product.
- Compute the Jacobian of this change-of-coordinates (so find the area of the parallelogram formed by the partial derivatives).
- Let $P=(1,2,0)$, $Q=(0,2,-1)$, and $R=(3,0,4)$.
- Find a vector that is orthogonal to both $\vec{PQ}$ and $\vec {PR}$.
- Find the area of triangle $\Delta PQR$.
- Give an equation of the plane PQR. (Let $S=(x,y,z)$ be any point on the plane PQR. Use $\vec {PS}\cdot (\vec {PQ}\times \vec {PR})=0$.)
- A wire lies along the curve $C$ parametrized by $\vec r(t) = (t^2+1, 3t, t^3)$ for $-1\leq t\leq 2$.
- Compute $ds$. (Remember - a little distance equals the product of the speed and a little time.)
- Set up an integral to find $\bar x$, then $\bar y$, then $\bar z$, for the centroid of $C$.
- Set up an integral formula to compute the $z$-coordinate of the center-of-mass (so $\bar z$) of the solid object in the first octant (all variables positive) that lies under the plane $2x+3y+6z=6$ (Draw the region as well).
- Draw the region described the bounds of each integral.
- $\ds\int_{0}^{3}\int_{0}^{\pi}\int_{0}^{5}rdzdrd\theta$
- $\ds\int_{-1}^{1}\int_{0}^{1-y^2}\int_{0}^{x}dzdxdy$
- $\ds\int_{0}^{2}\int_{0}^{1-y/2}\int_{6+z}^{6-z}dxdzdy$
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