What is a convex subset of \(\R^n\)?
What is meant by an open subset of \(\R^n\)? What is a closed subset?
Given a \(C^2\) function \(f:\R^n\to\R\), what is the Hessian of \(f\)?
By a multi-index, we mean \(\alpha=(\alpha_1,\dots,\alpha_n)\in \{0,1,2,\dots\}^n\). Define \(\partial^\alpha f\).
What is a local minimum? What is a local maximum?
What is a local extremum? What is a critical point?
What is meant by \(f(h)=o(|h|^m)\) as \(h\to 0\)?
What is meant by \(f(h)=O(|h|^m)\) as \(h\to 0\)?
Recall S01P06. For a \(2\times 2\) matrix \(H\), expand \(\det(I+H)\) as a Taylor series.
Consider the function \[ f(x,y)=\begin{cases} 0 & \text{if } (x,y)=(0,0), \\ e^{\frac{-1}{x^2+y^2}} & \text{otherwise.} \end{cases} \] Find the Taylor series for the nonzero (!) function \(f\) around \((0,0)\).
Find the third-order Taylor polynomial (i.e., with terms of total degree \(\le 3\)) of \[ F(x,y)=\frac{e^{x-2y}\,\sin(x+y)}{1-x^2-y} \] about \((0,0)\), but you are not allowed to compute partials. Instead, use one-variable Taylor polynomials for \(e^t\), \(\sin t\), and \((1-t)^{-1}\).
Let \(f,g\) be \(C^{|\alpha|}\) on an open set in \(\R^n\).
Use S03P04 to formulate a product rule \( \partial^\alpha (fg)=\displaystyle\sum_{\beta + \gamma = \alpha} \text{(something)} (\partial^\beta f)\cdot (\partial^\gamma g). \)
Let \(S\subset \R^n\) be open and convex, and let \(f:S\to\R\) be differentiable with \(\nabla f\equiv 0\) on \(S\). Prove that \(f\) is constant on \(S\).
Let \(f:\R^2\to\R\) be \(C^2\) in a neighborhood of \((0,0)\), with \(f(0,0)=0\) and \(\nabla f(0,0)=\vec{0}\). Suppose the quadratic part of the Taylor expansion of \(f\) at \((0,0)\) is \(Q(x,y)=ax^2+2bxy+cy^2\), so that \[ f(x,y)=ax^2+2bxy+cy^2+o(x^2+y^2). \] Let \(R_\theta\) be the rotation about the origin through angle \(\theta\). Find \(\theta\) so that there exist \(\alpha,\gamma\in\R\) with \[ f(R_\theta(u,v))=\alpha u^2+\gamma v^2+o(u^2+v^2). \]
Use S03P12 to prove a piece of the second partial derivative test. Specifically, suppose \(f:\R^2\to\R\) is \(C^2\) in a neighborhood of \((0,0)\), and assume \(\nabla f(0,0)=\vec{0}\). Assume the determinant of the Hessian of \(f\) at \((0,0)\) is positive, and assume that \(\displaystyle\frac{\partial^2 f}{\partial x^2}(0,0)>0\). Then show that \((0,0)\) is a local minimum of \(f\).
Let \(f\in C^3\) in a neighborhood of \((0,0)\). Define \[ A(r)=\frac{1}{2\pi}\int_0^{2\pi} f(r\cos\theta,r\sin\theta)\,d\theta, \] which is the average value of \(f\) on a circle of radius \(r\). Show that \[ A(r)=f(0,0)+\frac{r^2}{4}\,\Delta f(0,0)+O(r^3). \]
Suppose \(f:\R^2\to\R\) is \(C^2\) and harmonic and \(\displaystyle\frac{\partial^2 f}{\partial x^2}(0,0)\neq 0\).
Then \((0,0)\) is not a local extremum for \(f\).
For a function \(f:\R^2\to\R\) for which second partials exist at \((0,0)\), we have \[ \frac{\partial^2 f}{\partial x \partial y}(0,0)=\frac{\partial^2 f}{\partial y \partial x}(0,0). \]
If \(f:\R^2\to\R\) is differentiable and \(\displaystyle\frac{\partial f}{\partial x}\) vanishes everywhere, then \(f\) is independent of its first input, i.e., for all \(x_1,x_2,y\in\R\), we have \(f(x_1,y)=f(x_2,y)\).
If \(S\subset \R^n\) is open and \(f:S\to\R\) is differentiable with \(|\nabla f(x)|\le M\) for all \(x\in S\), then \( |f(b)-f(a)|\le M|b-a| \) for all \(a,b\in S\).