# 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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