Presentations
5.23 - 5.30Pacing Tracker
- The quizzes have included questions for 22 objectives. How many have you passed? What are you plans to master those that you haven't mastered yet?
- We've finished units 1 through 4. Have you started your self-directed learning project for each unit?
- The 6th project can be over any topic from the entire semester. Feel free to get started on this one as soon as you have an idea.
- Remember you can submit only one SDL project per week. Plan ahead and don't let yourself get behind.
Brain Gains
- What is SULI?
Solution
Science Undergraduate Laboratory Internship.
- 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 $(x,y)=(2u+v,3v)$.
- 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 $\vec {PS}\cdot (\vec {PQ}\times \vec {PR})=0$.)
- 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 $\vec w = (0,2,-4)$.
- 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.
- The region in the first quadrant of the $xy$-plane that lies above the line $y=x$ and below the semicircle $y=\sqrt{16-x^2}$.
- The region in the first quadrant that lies below the line that passes through the two points $(4,0)$ and $(0,6)$.
- 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.
- The solid hemiball of radius 3, above the $xy$-plane.
- The solid region in the first octant that lies under the paraboloid $z=9-x^2-y^2$.
- Set up an integral formula to compute each of the following:
- The mass of a disc that lies inside the circle $x^2+y^2=9$ and has density function given by $\delta = x+10$
- The $x$-coordinate of the center of mass (so $\bar x$) of the disc above.
- 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 $2x+3y+6z=6$.
- The $y$-coordinate of the center-of-mass (so $\bar y$) of the same object.
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