- Suppose $f$ is a function of $x$ and $y$ and both $x$ and $y$ are functions of $t$. This means $df = f_x dx+f_ydy$ and $dx=\frac{dx}{dt}dt$ and $dy=\frac{dy}{dt}dt$.
- Use substitution to obtain $df$ in terms of partial derivatives and $dt$. Then state $\frac{df}{dt}$.
- State the matrices $Df(x,y)$ and $D(x,y)(t)$. Then multiply them together to obtain the same answer as above.
- Suppose $w$ is a function of $x$, $y$, and $z$, and also suppose that $x$, $y$, and $z$ are functions of $\phi$. Find $\frac{dw}{dt}$.
- Suppose $\vec r(u,v) = (x,y,z)$. In addition, suppose that $u$ and $v$ are functions of $r$ and $\theta$, so $u=u(r,\theta)$ and $v = v(r\theta)$.
- Find $d\vec r$, and then find $du$ and $dv$. Use substition to obtain $\frac{\partial \vec r}{\partial r}$ and $\frac{\partial \vec r}{\partial \theta}$.
- We can compute $$D\vec r(u,v) = \begin{bmatrix}\frac{\partial x}{\partial u}&?\\ \frac{\partial y}{\partial u}&?\\\frac{\partial z}{\partial u}&?\end{bmatrix}\quad \text{and}\quad D(u,v)(r,\theta) = \begin{bmatrix}\frac{\partial u}{\partial r}&?\\?&?\end{bmatrix}.$$ Fill in the missing pieces of these matrices.
- Multiply the matrices together. In your product matrix you should have 6 entries. Use your result to state general formulas for $\frac{\partial x}{\partial r}$ and $\frac{\partial z}{\partial \theta}$.
- Consider $f(x,y)=x^2-y$.
- Compute the derivative $Df(x,y)$.
- Draw several level curves of $f(x,y)$, making sure your plot is relatively to scale (don't just plot some parabolas down without trying to make them pass through 5 points accurately).
- The derivative above is a vector field. Add the vector field plot to your contour plot.
- State a vector that is normal to the level curve of $f$ that passes through the point $(2,4)$?
- Repeat the above with $f(x,y) = x^2+y^2$.
- Repeat the above with $f(x,y) = 4-y^2$.
- What patterns do you see? State a vector that is normal the level surface $x^2+y^2+z^2=9$ at the point $(-2,2,1)$.
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