- Let $f(x,y)=2xy^2$. Suppose $x=3\cos t$ and $y=e^{2t}$.
- Compute $df$ in terms of $x,y,dx,dy$.
- Compute $dx$ and $dy$ in terms of $t$ and $dt$.
- Use substitution to compute $df$ in terms of $t$ and $dt$. Then state $\frac{df}{dt}$.
- State the derivative $Df(x,y)$.
- State the derivative $D(x,y)(t)$.
- Multiply the matrices $Df(x,y)$ and $D(x,y)(t)$ to obtain the derivative $\frac{df}{dt}$.
- We need to learn to do the above symbolically. So 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 $\vec r(u,v) = (x,y,z)$. In addition, suppose that $u$ and $v$ are functions of $r$ and $\theta$, so $u=u(r,\theta)$ and $v = v(r\theta)$.
- 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.
- Muliply 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 $\frac{\partial z}{\partial \theta}$.
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