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Learning Reminders

• We are in the 13th week of the semester. If you are on track for an A, then ideally you're finishing your SDL project for the 5th unit, and proposed something for the 6th unit.
• There are only 2 weeks left, which means you can submit at most 2 more SDL projects.
• The final SDL project (6th) can be over any topic from the entire semester. You can use it to expand what we do in the 6th unit, or you may choose to revisit something from a prior unit that you would like to spend more time with.

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Brain Gains (Rapid Recall)

• What is SULI?

• Write the differential of the change-of-coordinates $(x,y) = (2u+3v,4u+5v)$ as a linear combination of partial derivatives.

• Suppose I know $\vec u\times\vec v = (3,-2,-6)$. Compute $\vec v\times \vec u$.

• Compute and simplify $ [(2,0,0)\times(0,3,0)]\cdot (0,0,5)$. What does this number compute?

Group problems

1. Consider the change-of-coordinates $(x,y)=(2u+v,3v)$.
   1. Compute the differential $d(x,y)$ and write it as a linear combination of partial derivatives and as a matrix product.
   2. Compute the Jacobian of this change-of-coordinates (so find the area of the parallelogram formed by the partial derivatives).
2. Let $P=(1,2,0)$, $Q=(0,2,-1)$, and $R=(3,0,4)$.
   1. Find a vector that is orthogonal to both $\vec{PQ}$ and $\vec {PR}$.
   2. Find the area of triangle $\Delta PQR$.
   3. Give an equation of the plane PQR. (Let $S=(x,y,z)$ be any point on the plane PQR. Use $\vec {PS}\cdot (\vec {PQ}\times \vec {PR})=0$.)
3. Use the triple product $|(\vec u\times \vec v)\cdot \vec w|$ to compute the volume of the parallelepiped whose edges are formed by the vectors $\vec u = (1,2,0)$, $\vec v = (-2,3,1)$, and $\vec w = (0,2,-4)$.
4. For each region below, (1) set up a double integral that gives the area of the region. Feel free to use any coordinate system you want. Then set up a formula to compute $\bar x$, the $x$ coordinate of the centroid.
   1. The region in the first quadrant of the $xy$-plane that lies above the line $y=x$ and below the semicircle $y=\sqrt{16-x^2}$.
   2. The region in the first quadrant that lies below the line that passes through the two points $(4,0)$ and $(0,6)$.
5. For each solid region below, (1) set up a triple integral that gives the volume of the region. Then set up a formula to compute $\bar z$, the $z$-coordinate of the centroid of the region.
   1. The solid hemiball of radius 3, above the $xy$-plane.
   2. The solid region in the first octant that lies under the paraboloid $z=9-x^2-y^2$.
6. Set up an integral formula to compute each of the following:
   1. The mass of a disc that lies inside the circle $x^2+y^2=9$ and has density function given by $\delta = x+10$
   2. The $x$-coordinate of the center of mass (so $\bar x$) of the disc above.
   3. 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 $2x+3y+6z=6$.
   4. The $y$-coordinate of the center-of-mass (so $\bar y$) of the same object.

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Problem Set
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