Problem Set 7

  1. Easy
    Let \(f:\mathbb{R}^{2}\to\mathbb{R}^{2}\) be the map \(f(x,y) = (x^2-y^2, 2xy)\).
    1. Calculate \(Df\) and \(\det Df\)
    2. Let \(S = \{(x,y)\in\mathbb{R}^{2}: x^2+y^2\leq 1, x\geq 0, y\geq 0\}\). Make a sketch of \(f(S)\) by showing where some of the coordinate curves get mapped.

  2. Easy
    Suppose that \(f:\mathbb{R}^{3}\to\mathbb{R}^{2}\) is a function such that \(f(0,0,0) = (1,2)\) and: \[ Df_{(0,0,0)} = \left(\begin{array}{ccc} 1 & 2 & 3 \\ 0 & 0 & 1\end{array}\right) \] Let \(g:\mathbb{R}^{2}\to\mathbb{R}^{2}\) be the map \(g(x,y) = (x+2y+1, 3xy)\). Find \(D(g\circ f)_{(0,0,0)}\)

  3. Easy
    Suppose that \(f(x,y,z,t)\), \(x(t)\), \(y(x,t,s)\), and \(z(y,x)\). Use the chain rule to find an expression for \(\frac{\partial f}{\partial t}\) and \(\frac{\partial f}{\partial s}\).

  4. Easy
    Let \(F(\varphi,\theta) = (x(\varphi,\theta),y(\varphi,\theta),z(\varphi,\theta))\) be the spherical polar coordinate system on the unit sphere \(S^{2} = \{(x,y,z): x^2+y^2+z^2=1\}\).
    1. Sketch the coordinate curves \(\theta = \pi/4\) and \(\varphi = \pi/2\)
    2. Compute the derivative to the coordinate curves from part (1) at the point \((\varphi,\theta) = (\pi/2,\pi/4)\). Add these arrows to your plot.
    3. Prove that \(\partial_{\varphi}F \times \partial_{\theta}F\) is parallel to \(\nabla(x^2+y^2+z^2)\).

  5. Medium
    Let \(f,g: \mathbb R^2 \to \mathbb R \) and \(h: \mathbb R^3 \to \mathbb R\) be \(C^1\) functions and define \(\displaystyle F(x,y) = \int_{f(x,y)}^{g(x,y)} h(x,y,t) \mathrm dt\). Compute \(\displaystyle \frac{\partial F}{\partial x}, \frac{\partial F}{\partial y} \).

  6. Medium
    Let \(\mathbf G : \mathbb R^2 \to \mathbb R^2\) be a \(C^1 \) function satisfying \(\mathbf G(\mathbf 0) =\mathbf 0\). Define a function \(\mathbf H(\mathbf x) = \mathbf G^{\circ n}(\mathbf x) \) where \(\mathbf G^{\circ n}\) denotes the \(n\)-fold composition of \(\mathbf G\) with itself. If \(\mathrm d\mathbf G(\mathbf 0) = \begin{pmatrix} 1 & 1 \\ 0 & 1\end{pmatrix} \) determine \(\mathrm d \mathbf H(\mathbf 0) \).

  7. Hard
    Let \( f: \mathbb R^n \to \mathbb R\) be a \(C^\infty \) function (infinitely differentiable). Fix a point \(\mathbf a \in \mathbb R^n\) and set \( \gamma: \mathbb R \to \mathbb R^n, \gamma(t) = \mathbf a t \).
    Define the function \( g(t) = f(\gamma(t))\). Show that \( g^{(k)}(t) = \left[ \mathbf a \cdot \nabla \right]^k f(\gamma(t)) \) where \(\nabla = \left( \partial_1, \ldots, \partial_n \right) \) is thought of as an operator.

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