Chapter 15

Suppose that the function \(\psi: \mathbb{R}^{2} \rightarrow \mathbb{R}\) is continuously differentiable. Define the function \(g: \mathbb{R}^{2} \rightarrow \mathbb{R}\) by $$g(s, t)=\psi\left(s^{2} t, s\right) \quad \text { for }(s, t) \text { in } \mathbb{R}^{2}$$ Find \(\partial g / \partial s(s, t)\) and \(\partial g / \partial t(s, t)\)

Which of the following mappings \(\mathbf{F}: \mathbb{R}^{2} \rightarrow \mathbb{R}^{2}\) is linear? a. \(\mathbf{F}(x, y)=\left(-y, e^{x}\right)\) for \((x, y)\) in \(\mathbb{R}^{2}\) b. \(\mathbf{F}(x, y)=\left(x-y^{2}, 2 y\right)\) for \((x, y)\) in \(\mathbb{R}^{2}\) c. \(\mathbf{F}(x, y)=17(x, y)\) for \((x, y)\) in \(\mathbb{R}^{2}\)

Define \(\mathbf{F}(x, y)=\left(e^{x y}+2 x, y^{2}+\sin (x-y)\right) \quad\) for \((x, y)\) in \(\mathbb{R}^{2}\) Find the derivative matrix of the mapping \(\mathbf{F}: \mathbb{R}^{2} \rightarrow \mathbb{R}^{2}\) at the points (0,0) and \((\pi, 0)\)

Suppose that the function \(h: \mathbb{R}^{3} \rightarrow \mathbb{R}\) is continuously differentiable. Define the function \(\eta: \mathbb{R}^{3} \rightarrow \mathbb{R}\) by$$\eta(u, v, w)=(3 u+2 v) h\left(u^{2}, v^{2}, u v w\right) \quad \text { for }(u, v, w) \text { in } \mathbb{R}^{3}$$ Find \(D_{1} \eta(u, v, w), D_{2} \eta(u, v, w),\) and \(D_{3} \eta(u, v, w)\)

Define $$ \mathbf{F}(x, y, z)=\left(x y z, x^{2}+y z, 1+3 x\right) $$ for \((x, y, z)\) in \(\mathbb{R}^{3}\) Find the derivative matrix of the mapping \(\mathbf{F}: \mathbb{R}^{3} \rightarrow \mathbb{R}^{3}\) at the points (1,2,3),(0,1,0) and (-1,4,0)

Suppose that the functions \(g: \mathbb{R} \rightarrow \mathbb{R}\) and \(h: \mathbb{R} \rightarrow \mathbb{R}\) have continuous second-order partial derivatives. Define the function \(u: \mathbb{R}^{2} \rightarrow \mathbb{R}\) by $$u(s, t)=g(s-t)+h(s+t) \quad \text { for }(s, t) \text { in } \mathbb{R}^{2} $$ Prove that $$\frac{\partial^{2} u}{\partial t^{2}}(s, t)-\frac{\partial^{2} u}{\partial s^{2}}(s, t)=0$$ for all \((s, t)\) in \(\mathbb{R}^{2}\).

Suppose that the mapping \(\mathbf{F}: \mathbb{R}^{n} \rightarrow \mathbb{R}^{m}\) is continuously differentiable and that the derivative matrix \(\mathbf{D F}(\mathbf{x})\) at each point \(\mathbf{x}\) in \(\mathbb{R}^{n}\) has all its entries equal to 0 . Prove that the mapping \(\mathbf{F}: \mathbb{R}^{n} \rightarrow \mathbb{R}^{m}\) is constant; that is, there is some point \(\mathbf{c}\) in \(\mathbb{R}^{m}\) such that \(\mathbf{F}(\mathbf{x})=\mathbf{c} \quad\) for every \(\mathbf{x}\) in \(\mathbb{R}^{n}\)

Suppose that \(\mathbf{A}\) is an \(m \times n\) matrix. Define the mapping \(\mathbf{F}: \mathbb{R}^{n} \rightarrow \mathbb{R}^{m}\) by \(\mathbf{F}(\mathbf{x})=\mathbf{A} \mathbf{x} \quad\) for every \(\mathbf{x}\) in \(\mathbb{R}^{n}\) Prove that \(\mathbf{D F}(\mathbf{x})=\mathbf{A}\) for all \(\mathbf{x}\) in \(\mathbb{R}^{n}\).

Show that there is no linear mapping \(\mathbf{T}: \mathbb{R}^{2} \rightarrow \mathbb{R}^{2}\) having the property that $$ \mathbf{T}(1,1)=(4,0) \quad \text { and } \quad \mathbf{T}(-2,-2)=(0,1) $$

Suppose that the mapping \(\mathbf{F}: \mathbb{R}^{n} \rightarrow \mathbb{R}^{m}\) is continuously differentiable and that there is a fixed \(m \times n\) matrix \(\mathbf{A}\) so that $$ \mathbf{D F}(\mathbf{x})=\mathbf{A} $$ for every \(\mathbf{x}\) in \(\mathbb{R}^{n}\). Prove that there is some \(\mathbf{c}\) in \(\mathbb{R}^{m}\) so that $$ \mathbf{F}(\mathbf{x})=\mathbf{A} \mathbf{x}+\mathbf{c} $$ for every \(\mathbf{x}\) in \(\mathbb{R}^{n}\) Restate this result for the case when \(n=m=1\).

Find a linear transformation \(\mathbf{T}: \mathbb{R}^{3} \rightarrow \mathbb{R}^{3}\) that has the property that \(\mathbf{T}(1,1,1)=(0,2,0), \quad \mathbf{T}(1,1,-1)=(1,2,0), \quad\) and \(\quad \mathbf{T}(2,0,0)=(1,1,1)\) [Hint: Use linearity to determine \(\mathbf{T}\left(\mathbf{e}_{i}\right)\) for \(\left.i=1,2,3 .\right]\)

Define the mapping \(\mathbf{F}: \mathbb{R}^{2} \rightarrow \mathbb{R}^{2}\) by $$ \mathbf{F}(x, y)=\left(x^{2}-y^{2}, 2 x y\right) \quad \text { for }(x, y) \text { in } \mathbb{R}^{2} $$ a. Find the points \(\left(x_{0}, y_{0}\right)\) in \(\mathbb{R}^{2}\) at which the derivative matrix \(\operatorname{DF}\left(x_{0}, y_{0}\right)\) is invertible. b. Find the points \(\left(x_{0}, y_{0}\right)\) in \(\mathbb{R}^{2}\) at which the differential \(\mathbf{d F}\left(x_{0}, y_{0}\right): \mathbb{R}^{2} \rightarrow \mathbb{R}^{2}\) is an invertible linear mapping.

Suppose that the function \(u: \mathbb{R}^{2} \rightarrow \mathbb{R}\) is harmonic. Let \(a, b, c,\) and \(d\) be real numbers such that $$ a^{2}+b^{2}=1, \quad c^{2}+d^{2}=1, \quad \text { and } \quad a c+b d=0 $$ Define the function \(v: \mathbb{R}^{2} \rightarrow \mathbb{R}\) by $$ v(x, y)=u(a x+b y, c x+d y) $$ for \((x, y)\) in \(\mathbb{R}^{2}\) Prove that the function \(v: \mathbb{R}^{2} \rightarrow \mathbb{R}\) is also harmonic.

Suppose that the functions \(f: \mathbb{R} \rightarrow \mathbb{R}\) and \(g: \mathbb{R} \rightarrow \mathbb{R}\) have continuous second-order partial derivatives. Also suppose that there is a number \(\lambda\) such that and \(\quad g^{\prime \prime}(x)=\lambda g(x) \quad\) for all \(x\) in \(\mathbb{R}\). $$ f^{\prime \prime}(x)=\lambda f(x) \quad \text { and } \quad g^{\prime \prime}(x)=\lambda g(x) $$ Define the function \(u: \mathbb{R}^{2} \rightarrow \mathbb{R}\) by $$ u(x, y)=f(x) g(y) $$ $$ \text { for }(x, y) \text { in } \mathbb{R}^{2} $$ Prove that $$\frac{\partial^{2} u}{\partial x^{2}}(x, y)-\frac{\partial^{2} u}{\partial y^{2}}(x, y)=0$$for every \((x, y)\) in \(\mathbb{R}^{2}\).

Suppose that the mapping \(\mathbf{F}: \mathbb{R}^{n} \rightarrow \mathbb{R}^{n}\) is continuously differentiable. Suppose also that \(\mathbf{F}(\mathbf{0})=\mathbf{0}\) and that the derivative matrix \(\mathbf{D F}(\mathbf{0})\) has the property that there is some positive number \(c\) such that $$ \|\mathbf{D F}(\mathbf{0}) \mathbf{h}\| \geq c\|\mathbf{h}\| $$ for all \(\mathbf{h}\) in \(\mathbb{R}^{n}\). Prove that there is some positive number \(r\) such that $$ \|\mathbf{F}(\mathbf{h})\| \geq c / 2\|\mathbf{h}\| \quad \text { if }\|\mathbf{h}\| \leq r $$

Define \(A=\left(\begin{array}{ll}1 & 0 \\ 0 & 0\end{array}\right) .\) Find all \(2 \times 2\) matrices \(\mathbf{B}\) with the property that $$ \mathbf{A B}=\mathbf{B A} $$

Let \(\mathcal{O}=\left\\{(x, y, z)\right.\) in \(\left.\mathbb{R}^{3} \mid x^{2}+y^{2}+z^{2}>0\right\\}\) and define the function \(u: \mathcal{O} \rightarrow \mathbb{R}\) by $$ u(\mathbf{p})=\frac{1}{\|\mathbf{p}\|} $$ for \(\mathbf{p}\) in \(\mathcal{O}\) Prove that $$ \frac{\partial^{2} u}{\partial x^{2}}(x, y, z)+\frac{\partial^{2} u}{\partial y^{2}}(x, y, z)+\frac{\partial^{2} u}{\partial z^{2}}(x, y, z)=0 $$ for every \((x, y, z)\) in \(\mathcal{O}\).

Suppose that the continuously differentiable mapping \(\mathbf{F}: \mathbb{R}^{2} \rightarrow \mathbb{R}^{2}\) is represented in component functions as $$ \mathbf{F}(x, y)=(\psi(x, y), \varphi(x, y)) \quad \text { for }(x, y) \text { in } \mathbb{R}^{2} $$ Define the function \(g: \mathbb{R}^{2} \rightarrow \mathbb{R}\) by $$ g(x, y)=\frac{1}{2}\left[(\psi(x, y))^{2}+(\varphi(x, y))^{2}\right] $$ $$ \text { for }(x, y) \text { in } \mathbb{R}^{2} $$ a. Show that $$ \mathbf{D} g\left(x_{0}, y_{0}\right)=\left[\mathbf{D} \mathbf{F}\left(x_{0}, y_{0}\right)\right]^{\mathbf{T}} \mathbf{F}\left(x_{0}, y_{0}\right) $$ b. Use (a) to prove that if \(\left(x_{0}, y_{0}\right)\) is a minimizer of the function \(g: \mathbb{R}^{2} \rightarrow \mathbb{R}\) and the matrix \(\mathbf{D F}\left(x_{0}, y_{0}\right)\) is invertible, then $$ \mathbf{F}\left(x_{0}, y_{0}\right)=\mathbf{0} $$

Suppose that the functions \(f: \mathbb{R}^{2} \rightarrow \mathbb{R}, g: \mathbb{R}^{2} \rightarrow \mathbb{R},\) and \(h: \mathbb{R}^{2} \rightarrow \mathbb{R}\) are continu- ously differentiable. Express the following two limits in terms of partial derivatives of these functions: a. \(\lim _{t \rightarrow 0} \frac{f(g(1+t, 2), h(1+t, 2))-f(g(1,2), h(1,2))}{t}\) b. \(\lim _{t \rightarrow 0} \frac{f(g(1,2)+t, h(1,2))-f(g(1,2), h(1,2))}{t}\)

Find the \(2 \times 2\) matrix associated with the mapping in the plane that rotates points \(90^{\circ}\) counterclockwise about the origin.

Suppose that the function \(g: \mathbb{R}^{n} \rightarrow \mathbb{R}\) is continuously differentiable. For points \(\mathbf{x}\) and \(\mathbf{p}\) in \(\mathbb{R}^{n},\) the Directional Derivative Theorem asserts that if \(\psi(t)=g(\mathbf{x}+t \mathbf{p})\) for \(t\) in \(\mathbb{R},\) then $$ \psi^{\prime}(t)=\langle\nabla g(\mathbf{x}+t \mathbf{p}), \mathbf{p}\rangle $$ for every \(t\) in \(\mathbb{R}\) Show that this formula is a special case of the Chain Rule.

Let \(\mathbf{A}\) be an \(n \times n\) matrix and suppose that \(\mathbf{B}\) and \(\mathbf{B}^{\prime}\) are two \(n \times n\) matrices with the property that $$ \mathbf{A B}=\mathbf{I}_{n}=\mathbf{B}^{\prime} \mathbf{A} $$ Show that \(\mathbf{B}=\mathbf{B}^{\prime}\) by verifying that $$ \mathbf{B}=\mathbf{I}_{n} \mathbf{B}=\left(\mathbf{B}^{\prime} \mathbf{A}\right) \mathbf{B}=\mathbf{B}^{\prime}(\mathbf{A} \mathbf{B})=\mathbf{B}^{\prime} \mathbf{I}_{n}=\mathbf{B}^{\prime} $$

Suppose that the mapping \(\mathbf{T}: \mathbb{R}^{n} \rightarrow \mathbb{R}^{n}\) has the property that there is another mapping \(\mathbf{S}: \mathbb{R}^{n} \rightarrow \mathbb{R}^{n}\) such that $$ \mathbf{T}(\mathbf{S}(\mathbf{x}))=\mathbf{S}(\mathbf{T}(\mathbf{x}))=\mathbf{x} $$ for all \(\mathbf{x}\) in \(\mathbb{R}^{n}\). Prove that \(\mathbf{T}: \mathbb{R}^{n} \rightarrow \mathbb{R}^{n}\) is invertible and that its inverse is the mapping \(\mathbf{S}: \mathbb{R}^{n} \rightarrow \mathbb{R}^{n}\).

Let \(\mathbf{A}\) be an \(n \times n\) matrix. Show that for each pair of indices \(i\) and \(j\) such that \(1 \leq i, j \leq n\) $$ \left\langle\mathbf{A} \mathbf{e}_{i}, \mathbf{e}_{j}\right\rangle=\left\langle\mathbf{e}_{i}, \mathbf{A}^{\mathbf{T}} \mathbf{e}_{j}\right\rangle $$ Use this and the linearity of the scalar product to show that for any two points \(\mathbf{u}\) and \(\mathbf{v}\) in \(\mathbb{R}^{n}\), $$ \langle\mathbf{A} \mathbf{u}, \mathbf{v}\rangle=\left\langle\mathbf{u}, \mathbf{A}^{\mathrm{T}} \mathbf{v}\right\rangle $$