Exercises

01

Chain rule, several parameters

Let $f(x,y,z)=xy+z$ and $x=s^2$, $y=st$, and $z=s{t}^2$.

• (a) Generalize the chain rule above to the case of 3 variables and compute the partials $\frac{\partial f}{\partial s}$ and $\frac{\partial f}{\partial t}$.
• (b) Write out $f$ in terms of $s$ and $t$ and compute the same partials directly.

02

Polar partials

Write $\frac{\partial f}{\partial r}$ in terms of $\frac{\partial f}{\partial x}$ and $\frac{\partial f}{\partial y}$ and $x$ and $y$.

03

Chain rule: 3 variables, 3 parameters

Find $\frac{\partial u}{\partial s}$ at the point $(r,s,t)=(2,1,0)$ given that:

$$u=x^{4}y+y^2{z}^3,x=rs{e}^t,y=r{s}^2{e}^{-t},z=r^{2}s\sin (t).$$