Write down a statement of the chain rule for multivariable functions.
What is meant by the G\^ateaux derivative of a function \(f : \R^n \to \R^m\)?
Compare this to S01P16.
What is meant by the Fr\'echet derivative of a function \(f : \R^n \to \R^m\)?
For a smooth function \(f : \R^3 \to \R\), consider the level set \(f^{-1}(0) = \{ (x,y,z) \in \R^3 : f(x,y,z) = 0 \}\). For a point \((x,y,z) \in f^{-1}(0)\), define the tangent space to \(f^{-1}(0)\) at the point \((x,y,z)\), denoted \(T_{(x,y,z)} f^{-1}(0)\).
Let \(g:\R^2\to\R\) be \(C^1\) with \(\nabla g(2,-1)=(4,7)\). Define \(f:\R^2\to\R^2\) via \[ f(x,y)=(x+y,\;x-2y). \] Then compute \(\nabla(g\circ f)(1,1)\).
Consider a “change of variables” where we relate \((x,y)\) to \((u,v)\) via \[ x = u + v, \qquad y = v. \] Verify that \(\displaystyle\frac{\partial y}{\partial u} = 0\) and yet \(\displaystyle\frac{\partial u}{\partial y} \neq 0\).
This is the so-called second fundamental confusion of calculus, cf. \textsection 10.3 of Penrose's The Road to Reality.
Use the chain rule to rewrite the Laplacian \(\Delta = \displaystyle\frac{\partial^2}{\partial x^2} + \displaystyle\frac{\partial^2}{\partial y^2}\) in terms of polar coordinates, i.e., in terms of \(\displaystyle\frac{\partial}{\partial r}\) and \(\displaystyle\frac{\partial}{\partial \theta}\).
Suppose \(L : \R^n \to \R\) is linear. Show that there exists \(\vec{v} \in \R^n\) so that \(L(\vec{a}) = \vec{v} \cdot \vec{a}\).
Use this to explain S01P02.
Recall S01P14. Suppose \(u, v : \R^2 \to \R\) are smooth functions satisfying \[ u_x(x,y)=v_y(x,y)\qquad \text{and} \qquad u_y(x,y)=-v_x(x,y), \] for all \((x,y) \in \R^2\). Show that \(\Delta u = 0\) and \(\Delta v = 0\).
Let \(\Phi:\R^2\to\R^2\) be a rotation by \(\theta\) and a translation by \((a,b)\), i.e., \[ \Phi(x,y)=\bigl(x\cos\theta - y\sin\theta + a,\; x\sin\theta + y\cos\theta + b\bigr). \] If \(u\) is \(C^2\) and \(\Delta u=0\), show that \(\Delta(u\circ \Phi) = 0\).
Find some other map \(\Psi : \R^2 \to \R^2\) which isn't just a rotation or a translation but with the property that \(\Delta u = 0\) implies \(\Delta (u \circ \Psi) = 0\).
Suppose \(f : \R^n \to \R\) is continuously differentiable and \(f\) is positively homogeneous of degree \(\mathbf{k}\), meaning \(f(\lambda \cdot \vec{v}) = \lambda^k f(\vec{v})\) for all \(\vec{v} \in \R^n\) and \(\lambda > 0\). Prove \[ k\,f(x_{1},\ldots ,x_{n})=\sum _{i=1}^{n}x_{i}{\frac {\partial f}{\partial x_{i}}}(x_{1},\ldots ,x_{n}). \] This is the first half of Euler's homogeneous function theorem.
For a positive integer \(k\), consider the function \(f_k : \R^2 \to \R\) defined in polar coordinates via \(f_k(r,\theta) = r^k \cos(k\theta)\).
Note that \(f_k\) is homogeneous of degree \(k\), and use the formula in S02P07 to check \(\Delta f_k = 0\).
Then write \(f_k\) in Cartesian coordinates to find some harmonic polynomials of degree \(k\).
Suppose \(f : \R^3 \to \R\) is a smooth function, and \(f(0,0,0) = 0\) and \(\nabla f(0,0,0) \neq 0\). Explain why for \(v \in T_{(0,0,0)} f^{-1}(0)\) we have \(v \cdot \nabla f(0,0,0) = 0\).
If all directional derivatives vanish at \(a\), then the function is differentiable at \(a\).
Let \(\gamma:\R\to\R^2\) be a \(C^2\) curve parametrized by arclength, meaning \(|\gamma'(s)| \equiv 1\). Then \(\gamma'(s)\cdot \gamma''(s)=0\). Hint: use S01P05.
Let \(f:\R^n\to\R^m\) be differentiable at \(a\), and let \(g:\R^m\to\R\) be differentiable at \(f(a)\). Then \[ \nabla(g\circ f)(a)=Df(a)\left( \left(\nabla g\right)(f(a)) \right). \]
If \(f : \R^n \to \R^m\) is differentiable at \(a\) and \(g : \R^m \to \R\) has all directional derivatives at \(f(a)\), then \[ D_v (g \circ f)(a) = (D_{Df(a)(v)} g)(f(a)). \]