exercises

Question 2.131

If \(f\colon\, U \subset {\mathbb R}^{\it n}\rightarrow {\mathbb R}\) is differentiable, prove that \({\bf x}\mapsto f^2({\bf x})+2f({\bf x})\) is differentiable as well, and compute its derivative in terms of \({\bf D}\! f({\bf x})\).

Question 2.132

Prove that the following functions are differentiable, and find their derivatives at an arbitrary point:

  • (a) \(f\colon\, {\mathbb R}^2\rightarrow {\mathbb R},(x,y)\mapsto 2\)
  • (b) \(f\colon\, {\mathbb R}^2\rightarrow {\mathbb R},(x,y)\mapsto x+y\)
  • (c) \(f\colon\, {\mathbb R}^2\rightarrow {\mathbb R},(x,y)\mapsto 2+x+y\)
  • (d) \(f\colon\, {\mathbb R}^2\rightarrow {\mathbb R},(x,y)\mapsto x^2+y^2\)
  • (e) \(f\colon\, {\mathbb R}^2\rightarrow {\mathbb R},(x,y)\mapsto e^{xy}\)
  • (f) \(\,f\colon\, U\rightarrow {\mathbb R},(x,y)\mapsto \sqrt{1-x^2-y^2},\) where \(U=\{(x,y) \mid x^2+y^2<1\}\)
  • (g) \(f\colon\, {\mathbb R}^2\rightarrow {\mathbb R},(x,y)\mapsto x^4-y^4\)

Question 2.133

Verify the first special case of the chain rule for the composition \(f\circ {\bf c}\) in each of the cases:

  • (a) \(f(x,y)=xy,{\bf c}(t)=(e^t,\cos t)\)
  • (b) \(f(x,y)=e^{xy},{\bf c}(t)=(3t^2,t^3)\)
  • (c) \(f(x,y)=(x^2+y^2)\log\,\sqrt{x^2+y^2},{\bf c}(t)=(e^t,e^{-t})\)
  • (d) \(f(x,y)=x\exp (x^2+y^2),{\bf c}(t)=(t,-t)\)

Question 2.134

What is the velocity vector for each path \({\bf c}(t)\) in Exercise 3? [The solution to part (b) only is in the Study Guide to this text.]

Question 2.135

Let \(f\colon\, {\mathbb R}^3\rightarrow {\mathbb R}\) and \(g\colon\, {\mathbb R}^3\rightarrow {\mathbb R}\) be differentiable. Prove that \[ \nabla (fg)=f \nabla \! g+g \nabla \! f. \]

Question 2.136

Let \(f\colon\, {\mathbb R}^3\rightarrow {\mathbb R}\) be differentiable. Making the substitution \[ x=\rho \cos \theta \sin \phi,\quad y=\rho\sin \theta \sin\phi,\quad z=\rho \cos \phi \] (spherical coordinates) into \(f(x,y,z)\), compute \(\partial f/\partial \rho,\partial f/\partial \theta,\) and \(\partial f/\partial \phi\) in terms of \(\partial f/\partial x, \partial f/\partial y, \hbox{and } \partial f/\partial z\).

Question 2.137

Let \(f(u,v)=(\tan\, (u-1)-e^v,u^2-v^2)\) and \(g(x,y)=(e^{x-y},x-y)\). Calculate \(f\circ g\) and \({\bf D}(f\circ g)(1,1)\).

Question 2.138

Let \(f(u,v,w)=(e^{u-w},\cos\,(v\,+\,u)\,+ \sin\,(u\,+\,v\,+w))\) and \(g(x,y)= (e^x,\cos\, (y-x),e^{-y})\). Calculate \(f\circ g\) and \({\bf D}(f\circ g)(0,0)\).

Question 2.139

Find \((\partial /\partial s)(f\circ T)(1,0)\), where \(f(u,v)=\cos\, u\sin\, v\) and \(T{:}\, {\mathbb R}^2\rightarrow {\mathbb R}^2\) is defined by \(T(s,t)=(\cos \,(t^2s), \log\,\sqrt{1+s^2})\).

Question 2.140

Suppose that the temperature at the point \((x,y,z)\) in space is \(T(x,y,z)=x^2+y^2+z^2\). Let a particle follow the right-circular helix \({\sigma}(t)=(\cos t,\sin t,t)\) and let \(T(t)\) be its temperature at time \(t\).

  • (a) What is \(T'(t)\)?
  • (b) Find an approximate value for the temperature at \(t=(\pi /2)+0.01\).

133

Question 2.141

Let \(f(x, y, z)=(3y+2, x^2+y^2, x+z^2)\). Let \(\textbf{c}(t)=(\cos(t), \sin(t), t)\).

  • (a) Find the path \(\textbf{p}=f \circ \textbf{c}\) and the velocity vector \(\textbf{p}'(\pi)\).
  • (b) Find \(\textbf{c}(\pi), \textbf{c}'(\pi)\) and \(\textbf{D}f(-1,0, \pi)\).
  • (c) Thinking of \(\textbf{D}f(-1,0, \pi)\) as a linear map, find \(\textbf{D}f(-1,0, \pi) \ (\textbf{c}'(\pi))\).

Question 2.142

Let \(h\colon \mathbb{R}^3 \to \mathbb{R}^5\) and \(g\colon \mathbb{R}^2 \to \mathbb{R}^3\) be given by \(h(x,y,z)= (xyz, e^{xz}, x\sin (y), \frac{-9}{x}, 17)\) and \(g(u,v)=(v^2 + 2u, \pi, 2\sqrt{u})\). Find \(\textbf{D}(h \circ g)(1,1)\).

Question 2.143

Suppose that a duck is swimming in the circle \(x=\cos t,y=\sin t\) and that the water temperature is given by the formula \(T=x^2e^y-xy^3\). Find \(dT/dt\), the rate of change in temperature the duck might feel: (a) by the chain rule; (b) by expressing \(T\) in terms of \(t\) and differentiating.

Question 2.144

Let \(f\colon\, {\mathbb R}^n\to {\mathbb R}^m\) be a linear mapping so that (by Exercise 28, Section 2.3) \({\bf D}\! f({\bf x})\) is the matrix of \(f\). Check the validity of the chain rule directly for linear mappings.

Question 2.145

Let \(f\colon\, {\mathbb R}^2\to {\mathbb R}^2;(x,y)\mapsto (e^{x+y},e^{x-y})\). Let \({\bf c}(t)\) be a path with \({\bf c}(0)=(0,0)\) and \({\bf c}'(0)=(1,1)\). What is the tangent vector to the image of \({\bf c}(t)\) under \(f\) at \(t=0\)?

Question 2.146

Let \(f(x,y)=1/\sqrt{x^2+y^2}\). Compute \(\nabla \! f(x,y)\).

Question 2.147

Write out the chain rule for each of the following functions and justify your answer in each case using Theorem 11.

  • (a) \(\partial h/\partial x,\quad \hbox{ where }\quad h(x,y)=f(x,u(x,y))\)
  • (b) \(d h/d x,\!\quad \hbox{ where}\quad\,\, h(x)=f(x,u(x),v(x))\)
  • (c) \(\partial h/\partial x,\quad \hbox{ where }\quad h(x,y,z)=f(u(x,y,z), v(x,y),w(x))\)

Question 2.148

Verify the chain rule for \(\partial h/\partial x\), where \(h(x,y)=f(u(x,y),v(x,y))\) and \[ f(u,v)=\frac{u^2+v^2}{u^2-v^2}, u(x,y)=e^{-x-y}, v(x,y)=e^{xy}. \]

Question 2.149

  • (a) Let \(y(x)\) be defined implicitly by \(G(x,y(x))=0\), where \(G\) is a given function of two variables. Prove that if \(y(x)\) and \(G\) are differentiable, then \[ \frac{dy}{dx}=-\frac{\partial G/\partial x}{\partial G/\partial y} \hbox{if } \qquad \frac{\partial G}{\partial y}\not=0. \]
  • (b) Obtain a formula analogous to that in part (a) if \(y_1,y_2\) are defined implicitly by \begin{eqnarray*} G_1(x,y_1(x),y_2(x))&=&0,\\ G_2(x,y_1(x),y_2(x))&=&0.\\[-16.7pt] \end{eqnarray*}
  • (c) Let \(y\) be defined implicitly by \[ x^2+y^3+e^y=0. \]

    Compute \(dy/dx\) in terms of \(x\) and \(y\).

Question 2.150

Thermodynamics textsfootnote # use the relationship \[ \Big(\frac{\partial y}{\partial x}\Big) \Big(\frac{\partial z}{\partial y}\Big) \Big(\frac{\partial x}{\partial z}\Big)=-1. \]

Explain the meaning of this equation and prove that it is true. [HINT: Start with a relationship \(F(x,y,z)=0\) that defines \(x=f(y,z),y=g(x,z)\), and \(z=h(x,y)\) and differentiate implicitly.]

Question 2.151

Dieterici’s equation of state for a gas is \[ P(V-b)e^{a/RVT}=RT, \] where \(a,b\), and \(R\) are constants. Regard volume \(V\) as a function of temperature \(T\) and pressure \(P\) and prove that \[ \frac{\partial V}{\partial T}=\Big(R+\frac{a}{TV}\Big)\Big/ \Big(\frac{RT}{V-b}-\frac{a}{V^2}\Big). \]

Question 2.152

This exercise gives another example of the fact that the chain rule is not applicable if \(f\) is not differentiable. Consider the function \[ f(x,y)=\left\{ \begin{array}{lc} \displaystyle \frac{xy^2}{x^2+y^2}& (x,y)\not=(0,0)\\[10pt] 0 & (x,y)=(0,0). \end{array}\right. \]

134

Show that

  • (a) \(\partial f/ \partial x\) and \(\partial f/\partial y\) exist at (0, 0).
  • (b) If \({\bf g}(t)=(at,bt)\) for constants \(a\) and \(b\), then \(f\circ {\bf g}\) is differentiable and \((f\circ {\bf g})'(0)= ab^2/(a^2+b^2)\), but \(\nabla \! f(0,0)\, { \cdot}\,{\bf g}'(0)=0.\)

Question 2.153

Prove that if \(f\colon\, U\subset {\mathbb R}^n\to {\mathbb R}\) is differentiable at \({\bf x}_0 \in U\), there is a neighborhood \(V\) of \({\bf 0}\in {\mathbb R}^n\) and a function \(R_1\colon\, V\to {\mathbb R}\) such that for all \({\bf h}\in V\), we have \({\bf x}_0+{\bf h}\in U\), \[ f({\bf x}_0+{\bf h})=f({\bf x}_0)+[{\bf D}\! f({\bf x}_0)]{\bf h}+R_1({\bf h}) \] and \[ \frac{R_1({\bf h})}{ \| {\bf h} \| }\to 0 {\rm as} {\bf h}\to {\bf 0}. \]

Question 2.154

Suppose \({\bf x}_0\in {\mathbb R}^n\) and \(0\leq r_1<r_2\). Show that there is a \(C^1\) function \(f\colon\, {\mathbb R}^n\to {\mathbb R}\) such that \(f({\bf x})=0\) for \(\| {\bf x}-{\bf x}_0 \| \geq r_2; 0 <f({\bf x})<1\) for \(r_1< \| {\bf x}-{\bf x}_0 \| <r_2\); and \(f({\bf x})=1\) for \(\| {\bf x}-{\bf x}_0 \| \leq r_1\). [HINT: Apply a cubic polynomial with \(g(r_1^2)=1\) and \(g(r^2_2)= g'(r^2_2)= g'(r_1^2)=0\) to \(\| {\bf x}-{\bf x}_0 \| ^2\) when \(r_1< \| {\bf x}-{\bf x}_0 \| <r_2.\)]

Question 2.155

Find a \(C^1\) mapping \(f\colon\, {\mathbb R}^3\to {\mathbb R}^3\) that takes the vector \({\bf i}+{\bf j}+{\bf k}\) emanating from the origin to \({\bf i}-{\bf j}\) emanating from \((1, 1, 0)\) and takes \({\bf k}\) emanating from \((1, 1, 0)\) to \({\bf k}-{\bf i}\) emanating from the origin.

Question 2.156

What is wrong with the following argument? Suppose \(w=f(x,y,z)\) and \(z=g(x,y)\). By the chain rule, \[ \frac{\partial w}{\partial x}=\frac{\partial w}{\partial x}\frac{\partial x}{\partial x}+\frac{\partial w}{\partial y}\frac{\partial y}{\partial x}+\frac{\partial w}{\partial z}\frac{\partial z}{\partial x}=\frac{\partial w}{\partial x}+\frac{\partial w}{\partial z}\frac{\partial z}{\partial x}. \]

Hence, \(0=(\partial w/\partial z)(\partial z/\partial x)\), and so \(\partial w/\partial z=0\) or \(\partial z/\partial x=0\), which is, in general, absurd.

Question 2.157

Prove rules (iii) and (iv) of Theorem 10. (HINT: Use the same addition and subtraction tricks as in the one-variable case and Theorem 8.)

Question 2.158

Show that \(h\colon\, {\mathbb R}^n\to {\mathbb R}^m\) is differentiable if and only if each of the \(m\) components \(h_i\colon\, {\mathbb R}^n\to {\mathbb R}\) is differentiable. (HINT: Use the coordinate projection function and the chain rule for one implication and consider \begin{eqnarray*} && \Big[\frac{\| h({\bf x})\,-\,h({\bf x}_0) -{\bf D}h({\bf x}_0) ({\bf x}\,-\,{\bf x}_0) \| }{ \| {\bf x}\,-\,{\bf x}_0 \|}\Big]^2\\[4pt] &&=\frac{\sum^m_{i=1}[h_i({\bf x})-h_i({\bf x}_0){\bf D} h_i({\bf x}_0)({\bf x}\,-\,{\bf x}_0)]^2}{\|{\bf x}\,-\,{\bf x}_0 \|^2} \end{eqnarray*} to obtain the other.)

Question 2.159

Use the chain rule and differentiation under the integral sign, namely, \[ \frac{d}{dx}\int^b_a f(x,y)\,dy= \int^b_a\frac{\partial f}{\partial x}(x,y)\,dy, \] to show that \[ \frac{d}{d x}\int ^x_0f(x,y)\,dy=f(x,x)+\int^x_0\frac{\partial f}{\partial x}(x,y)\,dy. \]

Question 2.160

For what integers \(p>0\) is \[ f(x)=\Big\{ \begin{array}{lc} x^p\sin\, (1/x) & x\not=0\\[1pt] 0& x=0 \end{array} \] differentiable? For what \(p\) is the derivative continuous?

Question 2.161

Suppose \(f\colon\, {\mathbb R}^n\to {\mathbb R}\) and \(g\colon\, {\mathbb R}^n\to {\mathbb R}^m\) are differentiable. Show that the product function \(h({\bf x})=f({\bf x})g({\bf x})\) from \({\mathbb R}^n\) to \({\mathbb R}^m\) is differentiable and that if \({\bf x}_0\) and \({\bf y}\) are in \({\mathbb R}^n\), then \([{\bf D}h({\bf x}_0)]{\bf y}= f({\bf x}_0)\{[{\bf D}g({\bf x}_0)]{\bf y}\}+\{[{\bf D}\! f({\bf x}_0)]{\bf y}\}g({\bf x}_0)\).

Question 2.162

Let \(g(u, v)=(e^u, u+\sin v)\) and \(f(x, y, z)=(xy, yz)\). Compute \(\textbf{D}(g \circ f)\) at (0, 1, 0) using the chain rule.

Question 2.163

Let \(f\colon\mathbb{R}^4 \to \mathbb{R}\) and \(\textbf{c}(t)\colon \mathbb{R} \to \mathbb{R}^4\). Suppose \(\nabla f(1, 1, \pi, e^6)=(0, 1, 3, -7), \ \textbf{c}(\pi)= (1, 1, \pi, e^6),\) and \(\textbf{c}'(\pi)= (19, 11, 0, 1)\). Find \(\displaystyle \frac{d(f \circ \textbf{c})}{dt}\) when \(t=\pi\).

Question 2.164

Suppose \(f\colon \mathbb{R}^n \to \mathbb{R}^m\) and \(g\colon \mathbb{R}^p \to \mathbb{R}^q\).

  • (a) What must be true about the numbers \(n, m, p,\) and \(q\) for \(f \circ g\) to make sense?
  • (b) What must be true about the numbers \(n, m, p,\) and \(q\) for \(g \circ f\) to make sense?
  • (c) When does \(f \circ f\) make sense?

Question 2.165

If \(z=f(x-y)\), use the chain rule to show that \(\displaystyle \frac{\partial z}{\partial x}+\frac{\partial z}{\partial y}=0\).

Question 2.166

Let \(w=x^2+y^2+z^2, \ x=uv, \ y=u \cos v, z=u \sin v\). Use the chain rule to find \(\displaystyle \frac{\partial w}{\partial u}\) when\((u, v)=(1, 0)\).