Chapter 2

2-4. Let \(g\) be a continuous real-valued function on the unit circle \(\left\\{x \in \mathbf{R}^{2}:|x|=1\right\\}\) such that \(g(0,1)=g(1,0)=0\) and \(g(-x)=\) \(-g(x) . \quad\) Define \(f: \mathbf{R}^{2} \rightarrow \mathbf{R}\) by $$ f(x)- \begin{cases}|x| \cdot g\left(\frac{x}{|x|}\right) & x \neq 0 \\ 0 & x=0\end{cases} $$ (a) If \(x \in \mathbf{R}^{2}\) and \(h: \mathbf{R} \rightarrow \mathbf{R}\) is defined by \(h(t)=f(t x)\), show that \(h\) is differentiable. (b) Show that \(f\) is not differentiable at \((0,0)\) unless \(g=0\) Hint: First show that \(D f(0,0)\) would have to be 0 by considering \((h, k)\) with \(k=0\) and then with \(h=0\)

2-6. Let \(f: \mathbf{R}^{2} \rightarrow \mathbf{R}\) be defined by \(f(x, y)=\sqrt{|x y|}\). Show that \(f\) is not differentiable at \((0,0)\).

2-7. Let \(f: \mathbf{R}^{n} \rightarrow \mathbf{R}\) be a function such that \(|f(x)| \leq|x|^{2}\). Show that \(f\) is differentiable at 0 .

2-8. Let \(f: \mathbf{R} \rightarrow \mathbf{R}^{2}\). Prove that \(f\) is differentiable at \(a \in \mathbf{R}\) if and only if \(f^{1}\) and \(f^{2}\) are, and that in this case $$ f^{\prime}(a)=\left(\begin{array}{c} \left(f^{1}\right)^{\prime}(a) \\ \left(f^{2}\right)^{\prime}(a) \end{array}\right) $$

2-9. Two functions \(f, g: \mathbf{R} \rightarrow \mathbf{R}\) are equal up to \(\boldsymbol{n}\) th order at \(a\) if $$ \lim _{h \rightarrow 0} \frac{f(a+h)-g(a+h)}{h^{n}}=0 $$ (a) Show that \(f\) is differentiable at \(a\) if and only if there is a function \(g\) of the form \(g(x)=a_{0}+a_{1}(x-a)\) such that \(f\) and \(g\) are equal up to first order at \(a\). (b) If \(f^{\prime}(a), \ldots, f^{(n)}(a)\) exist, show that \(f\) and the function \(g\) defined by $$ g(x)=\sum_{i=0}^{n} \frac{f^{(i)}(a)}{i !}(x-a)^{i} $$ are equal up to \(n\)th order at \(a\). Hint: The limit $$ \lim _{x \rightarrow a} \frac{f(x)-\sum_{i=0}^{n-1} \frac{f^{(i)}(a)}{i !}(x-a)^{i}}{(x-a)^{n}} $$ may be evaluated by L'Hospital's rule.

2-12. A function \(f: \mathbf{R}^{n} \times \mathbf{R}^{m} \rightarrow \mathbf{R}^{p}\) is bilinear if for \(x, x_{1}, x_{2} \in \mathbf{R}^{n}\), \(y, y_{1}, y_{2} \in \mathbf{R}^{m}\), and \(a \in \mathbf{R}\) we have $$ \begin{aligned} f(a x, y) &=a f(x, y)=f(x, a y) \\ f\left(x_{1}+x_{2}, y\right) &=f\left(x_{1}, y\right)+f\left(x_{2}, y\right) \\\ f\left(x, y_{1}+y_{2}\right) &=f\left(x, y_{1}\right)+f\left(x, y_{2}\right) \end{aligned} $$ (a) Prove that if \(f\) is bilinear, then $$ \lim _{(h, k) \rightarrow 0} \frac{|f(h, k)|}{|(h, k)|}=0 $$ (b) Prove that \(D f(a, b)(x, y)=f(a, y)+f(x, b)\). (c) Show that the formula for \(D p(a, b)\) in Theorem \(2-3\) is a special case of (b).

2-13. Define \(I P: \mathbf{R}^{n} \times \mathbf{R}^{n} \rightarrow \mathbf{R}\) by \(I P(x, y)=\langle x, y\rangle\). (a) Find \(D(I P)(a, b)\) and \((I P)^{\prime}(a, b)\). (b) If \(f, g: \mathbf{R} \rightarrow \mathbf{R}^{n}\) are differentiable and \(h: \mathbf{R} \rightarrow \mathbf{R}\) is defined by \(h(t)=\langle f(t), g(t)\rangle\), show that $$ h^{\prime}(a)=\left\langle f^{\prime}(a)^{\mathrm{T}}, g(a)\right\rangle+\left\langle f(a), g^{\prime}(a)^{\mathrm{T}}\right\rangle $$ (Note that \(f^{\prime}(a)\) is an \(n \times 1\) matrix; its transpose \(f^{\prime}(a)^{\mathrm{T}}\) is a \(1 \times n\) matrix, which we consider as a member of \(\mathbf{R}^{n}\).) (c) If \(f: \mathbf{R} \rightarrow \mathbf{R}^{n}\) is differentiable and \(|f(t)|=1\) for all \(t\), show that \(\left\langle f^{\prime}(t)^{\mathrm{T}}, f(t)\right\rangle=0\) (d) Exhibit a differentiable function \(f: \mathbf{R} \rightarrow \mathbf{R}\) such that the function \(|f|\) defined by \(|f|(t)=|f(t)|\) is not differentiable.

2-14 Let \(E_{i}, i=1, \ldots, k\) be Euclidean spaces of various dimensions. A function \(f: E_{1} \times \cdots \times E_{k} \rightarrow \mathbf{R}^{p}\) is called multilinear if for each choice of \(x_{j} \in E_{j}, j \neq i\) the function \(g: E_{i} \rightarrow \mathbf{R}^{p}\) defined by \(g(x)=f\left(x_{1}, \ldots, x_{i-1}, x, x_{i+1}, \ldots, x_{k}\right)\) is a linear transformation. (a) If \(f\) is multilinear and \(i \neq j\), show that for \(h=\left(h_{1}, \ldots, h_{k}\right)\), with \(h_{l} \in E_{l}\), we have $$ \lim _{h \rightarrow 0} \frac{\left|f\left(a_{1}, \ldots, h_{i}, \ldots, h_{j}, \ldots, a_{k}\right)\right|}{|h|}=0 $$ Hint: If \(g(x, y)=f\left(a_{1}, \ldots, x, \ldots, y, \ldots, a_{k}\right)\), then \(g\) is bilinear. (b) Prove that $$ D f\left(a_{1}, \ldots, a_{k}\right)\left(x_{1}, \ldots, x_{k}\right)=\sum_{i=1}^{k} f\left(a_{1}, \ldots, a_{i-1}, x_{i}, a_{i+1}, \ldots, a_{k}\right) $$

2-15. Regard an \(n \times n\) matrix as a point in the \(n\)-fold product \(\mathbf{R}^{n} \times\) \(\therefore \times \mathbf{R}^{n}\) by considering each row as a member of \(\mathbf{R}^{n}\). (a) Prove that det: \(\mathbf{R}^{n} \times \mathbf{R}^{n} \rightarrow \mathbf{R}\) is differentiable and $$ D(\operatorname{det})\left(a_{1}, \ldots, a_{n}\right)\left(x_{1}, \ldots, x_{n}\right)=\sum_{i=1}^{n} \operatorname{det}\left\\{\begin{array}{c} a_{1} \\ \cdot \\ \cdot \\ \cdot \\ x_{i} \\ \cdot \\ \cdot \\ a_{n} \end{array}\right) $$ (b) If \(a_{i j}: \mathbf{R} \rightarrow \mathbf{R}\) are differentiable and \(f(t)=\operatorname{det}\left(a_{i j}(t)\right)\), show that $$ f^{\prime}(t)=\sum_{j=1}^{n} \operatorname{det}\left\\{\begin{array}{cc} a_{11}(t), \ldots, a_{1 n}(t) \\ \cdot & \cdot \\ \cdot & \cdot \\ \cdot & \cdot \\ a_{j 1}^{\prime}(t), \ldots, a_{j n}^{\prime}(t) \\ \cdot & \cdot \\ \cdot & \cdot \\ a_{n 1}(t), \ldots, a_{n n}(t) \end{array}\right) $$ (c) If \(\operatorname{det}\left(a_{i j}(t)\right) \neq 0\) for all \(t\) and \(b_{1}, \ldots, b_{n}: \mathbf{R} \rightarrow \mathbf{R}\) are differentiable, let \(s_{1}, \ldots, s_{n}: \mathbf{R} \rightarrow \mathbf{R}\) be the functions such that \(s_{1}(t), \ldots, 8_{n}(t)\) are the solutions of the equations $$ \sum_{j=1}^{n} a_{j i}(t) s_{j}(t)=b_{i}(t) \quad i=1, \ldots, n $$ Show that \(s_{i}\) is differentiable and find \(s_{i}^{\prime}(t)\).

2-20. Find the partial derivatives of \(f\) in terms of the derivatives of \(g\) and \(h\) if (a) \(f(x, y)=g(x) h(y) .\) (b) \(f(x, y)=g(x)^{h(y)}\). (c) \(f(x, y)=g(x)\). (d) \(f(x, y)=g(y)\) (e) \(f(x, y)=g(x+y)\)

2-21. \({ }^{*}\) Let \(g_{1}, g_{2}: \mathbf{R}^{2} \rightarrow \mathbf{R}\) be continuous. Define \(f: \mathbf{R}^{2} \rightarrow \mathbf{R}\) by $$ f(x, y)=\int_{0}^{x} g_{1}(t, 0) d t+\int_{0}^{y} g_{2}(x, t) d t $$ (a) Show that \(D_{2} f(x, y)=g_{2}(x, y)\). (b) How should \(f\) be defined so that \(D_{1} f(x, y)=g_{1}(x, y) ?\) (c) Find a function \(f: \mathbf{R}^{2} \rightarrow \mathbf{R}\) such that \(D_{1} f(x, y)=x\) and \(D_{2} f(x, y)=y .\) Find one such that \(D_{1} f(x, y)=y\) and \(D_{2} f(x, y)=x\)

2-22. \(^{*}\) If \(f: \mathbf{R}^{2} \rightarrow \mathbf{R}\) and \(D_{2} f=0\), show that \(f\) is independent of the second variable. If \(D_{1} f=D_{2} f=0\), show that \(f\) is constant.

2-22. \(^{*}\) If \(f: \mathbf{R}^{2} \rightarrow \mathbf{R}\) and \(D_{2} f=0\), show that \(f\) is independent of the second variable. If \(D_{1} f=D_{2} f=0\), show that \(f\) is constant.

2-24. Define \(f: \mathbf{R}^{2} \rightarrow \mathbf{R}\) by $$ f(x, y)= \begin{cases}x y \frac{x^{2}-y^{2}}{x^{2}+y^{2}} & (x, y) \neq 0 \\\ 0 & (x, y)=0\end{cases} $$ (a) Show that \(D_{2} f(x, 0)=x\) for all \(x\) and \(D_{1} f(0, y)=-y\) for all \(y\). (b) Show that \(D_{1,2} f(0,0) \neq D_{2,1} f(0,0)\)

2-25. \(^{*}\) Define \(f: \mathbf{R} \rightarrow \mathbf{R}\) by $$ f(x)= \begin{cases}e^{-x^{-2}} & x \neq 0 \\ 0 & x=0\end{cases} $$ Show that \(f\) is a \(C^{\infty}\) function, and \(f^{(i)}(0)=0\) for all i. Hint: The limit \(f^{\prime}(0)=\lim _{h \rightarrow 0} \frac{e^{-h^{-2}}}{h}=\lim _{h \rightarrow 0} \frac{1 / h}{e^{h-2}}\) can be evaluated by L'Hospital's rule. It is easy enough to find \(f^{\prime}(x)\) for \(x \neq 0\), and \(f^{\prime \prime}(0)=\lim _{h \rightarrow 0} f^{\prime}(h) / h\) can then be found by L'Hospital's rule.

2-28. Find expressions for the partial derivatives of the following functions: (a) \(F(x, y)=f(g(x) k(y), g(x)+h(y))\). (b) \(F(x, y, z)=f(g(x+y), h(y+z))\). (c) \(F(x, y, z)=f\left(x^{y}, y^{x}, z^{x}\right)\). (d) \(F(x, y)=f(x, g(x), h(x, y))\).

2-29. Let \(f: \mathbf{R}^{n} \rightarrow\) R. For \(x \in \mathbf{R}^{n}\), the limit $$ \lim _{t \rightarrow 0} \frac{f(a+t x)-f(a)}{t} $$ if it exists, is denoted \(D_{x} f(a)\), and called the directional derivative of \(f\) at \(a\), in the direction \(x\). (a) Show that \(D_{e_{i}} f(a)=D_{i} f(a)\). (b) Show that \(D_{t x} f(a)=t D_{x} f(a)\). (c) If \(f\) is differentiable at \(a\), show that \(D_{x} f(a)=D f(a)(x)\) and therefore \(D_{x+y} f(a)=D_{x} f(a)+D_{y} f(a)\).

2-32. (a) Let \(f: \mathbf{R} \rightarrow \mathbf{R}\) be defined by $$ f(x)= \begin{cases}x^{2} \sin \frac{1}{x} & x \neq 0 \\ 0 & x=0\end{cases} $$ Show that \(f\) is differentiable at 0 but \(f^{\prime}\) is not continuous at 0 . (b) Let \(f: \mathbf{R}^{2} \rightarrow \mathbf{R}\) be defined by $$ f(x, y)= \begin{cases}\left(x^{2}+y^{2}\right) \sin \frac{1}{\sqrt{x^{2}+y^{2}}} & (x, y) \neq 0 \\ 0 & (x, y)=0\end{cases} $$ Show that \(f\) is differentiable at \((0,0)\) but \(D_{i} f\) is not continuous at \((0,0)\)

2-34. A function \(f: \mathbf{R}^{n} \rightarrow \mathbf{R}\) is homogeneous of degree \(m\) if \(f(t x)=\) \(t^{m} f(x)\) for all \(x .\) If \(f\) is also differentiable, show that $$ \sum_{i=1}^{n} x^{i} D_{i} f(x)=m f(x) $$ Hint: If \(g(t)=f(t x)\), find \(g^{\prime}(1)\)

2-35. If \(f: \mathbf{R}^{n} \rightarrow \mathbf{R}\) is differentiable and \(f(0)=0\), prove that there exist \(g_{i}: \mathbf{R}^{n} \rightarrow \mathbf{R}\) such that $$ f(x)=\sum_{i=1}^{n} x^{i} g_{i}(x) $$ Hint: If \(h_{x}(t)=f(t x)\), then \(f(x)=\int_{0}^{1} h_{x}^{\prime}(t) d t\)

2-36.* Let \(A \subset \mathbf{R}^{n}\) be an open set and \(f: A \rightarrow \mathbf{R}^{n}\) a continuously differentiable \(1-1\) function such that \(\operatorname{det} f^{\prime}(x) \neq 0\) for all \(x\). Show that \(f(A)\) is an open set and \(f^{-1}: f(A) \rightarrow A\) is differentiable. Show also that \(f(B)\) is open for any open set \(B \subset A\).

2-37. (a) Let \(f: \mathbf{R}^{2} \rightarrow \mathbf{R}\) be a continuously differentiable function. Show that \(f\) is not 1-1. Hint: If, for example, \(D_{1} f(x, y) \neq 0\) for all \((x, y)\) in some open set \(A\), consider \(g: A \rightarrow \mathbf{R}^{2}\) defined by \(g(x, y)=\) \((f(x, y), y)\) (b) Generalize this result to the case of a continuously differentiable function \(f: \mathbf{R}^{n} \rightarrow \mathbf{K}^{m}\) with \(m

Problems. 2-36.* Let \(A \subset \mathbf{R}^{n}\) be an open set and \(f: A \rightarrow \mathbf{R}^{n}\) a continuously differentiable 1-1 function such that det \(f^{\prime}(x) \neq 0\) for all \(x\). Show that \(f(A)\) is an open set and \(f^{-1}: f(A) \rightarrow A\) is differentiable. Show also that \(f(B)\) is open for any open set \(B \subset A\). 2-37. (a) Let \(f: \mathbf{R}^{2} \rightarrow \mathbf{R}\) be a continuously differentiable function. Show that \(f\) is not 1-1. Hint: If, for example, \(D_{1} f(x, y) \neq 0\) for all \((x, y)\) in some open set \(A\), consider \(g: A \rightarrow \mathbf{R}^{2}\) defined by \(g(x, y)=\) \((f(x, y), y)\) (b) Generalize this result to the case of a continuously differentiable function \(f: \mathbf{R}^{n} \rightarrow \mathbf{K}^{m}\) with \(m

2-39. Use the function \(f: \mathbf{R} \rightarrow \mathbf{R}\) defined by $$ f(x)= \begin{cases}\frac{x}{2}+x^{2} \sin \frac{1}{x} & x \neq 0 \\ 0 & x=0\end{cases} $$ to show that continuity of the derivative cannot be eliminated from the hypothesis of Theorem \(2-11 .\)