Trivium
Sketch the surface and contour plot of the graph \(z = f(x,y)\) of a multivariable function \(f\colon \mathbf{R}^2 \to \mathbf{R}.\)
Given a multivariable function \(f\colon \mathbf{R}^2 \to \mathbf{R}\) such that \[ \lim\limits_{(x,y)\to(a,b)} f(x,y) \] doesn’t exist, exhibit two paths in \(\mathbf{R}^2\) terminating at \((a,b)\) along which the outputs of \(f(x,y)\) approach different values.
Compute the partial derivatives of a multivariable function \(f\colon \mathbf{R}^2 \to \mathbf{R}.\)
Given an equation involving the symbols \(x,\) \(y,\) \(z,\) etc, be able to compute the partial derivative of any of them with respect to any other, both in the case that those symbols represent variables and in the case that they represent functions of other variables (e.g. \(t\) or \(s\)). If possible, be able to solve this implicit partial derivative to express the partial derivative explicitly.
Exercises
-
TK TK TK TK TK TK TK TK TK TK TK TK TK TK TK TK TK TK TK TK TK TK TK TK TK
\(\displaystyle y = mx+b\)\(\displaystyle y = mx+b\)
Problems & Challenges
- James Stewart For what values of the number \(r\) is the following function continuous on \(\mathbf{R}^3?\) \[ f(x,y,z) = \begin{cases} \frac{(x+y+z)^r}{x^2+y^2+z^2} \quad&\text{ if } (x,y,z) \neq 0 \\ 0 \quad&\text{ if } (x,y,z) = 0 \end{cases} \]
-
James Stewart Consider Laplace’s equation \[ \frac{\partial^2 u}{\partial x^2} + \frac{\partial^2 u}{\partial y^2} + \frac{\partial^2 u}{\partial z^2} = 0\,. \]
- Show that, when written in cylindrical coordinates, Laplace’s equation becomes \[ \frac{\partial^2 u}{\partial r^2} + \frac{1}{r} \frac{\partial u}{\partial r} + \frac{1}{r^2} \frac{\partial^2 u}{\partial \theta^2} + \frac{\partial^2 u}{\partial z^2} = 0 \,. \]
- Show that, when written in spherical coordinates, Laplace’s equation becomes \[ \frac{\partial^2 u}{\partial \rho^2} + \frac{2}{\rho} \frac{\partial u}{\partial \rho} + \frac{\cot(\varphi)}{\rho^2} \frac{\partial u}{\partial \varphi} + \frac{1}{\rho^2} \frac{\partial^2 u}{\partial \varphi^2} + \frac{1}{\rho^2\sin^2(\varphi)} \frac{\partial^2 u}{\partial \theta^2} = 0 \,. \]