Problems

01

Chain rule: spherical to Cartesian partials

Write the partials $\frac{\partial f}{\partial \rho }$, $\frac{\partial f}{\partial \theta }$, $\frac{\partial f}{\partial \phi }$ in terms of Cartesian coordinates and partials. Here $\rho$, $\theta$, $\phi$ are the spherical coordinate functions.

02

Chain rule: implicit partial differentiation

The set of points $(x,y,z)$ satisfying $x^2+y^2-2z^2+12x-8z-4=0$ is a hyperboloid of one sheet. Near the point $(1,1,1)$, let us consider this equation to determine $z$ as a function of $x$ and $y$.

• (a) Take the partial derivative of both sides of the equation with respect to $x$. (Take $y$ to be constant, and $z(x,y)$ to be a function, so both sides are functions of $x$ and $y$.) Now solve for $\frac{\partial z}{\partial x}$.
• (b) Solve the equation for $z$ in terms of $x$ and $y$ and find the partial derivative $\frac{\partial z}{\partial x}$ directly.
• (c) Make a formal rule that generalizes the above calculation: Let $F(x,y,z)=0$ be an equation that is taken to determine $z$ as a function of $x$ and $y$. Write a formula for $\frac{\partial z}{\partial x}$ in terms of partials of $F$. Indicate where you use the chain rule!