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A Course in Mathematics for Students of Physics 1

Paul Bamberg, Shlomo Sternberg

Chapter 5

Calculus in the plane - all with Video Answers

Educators

Chapter Questions

Problem 1

Show that if $f: V \rightarrow W$ is differentiable at $\alpha$ and if $T: W \rightarrow Z$ is linear, then $T \circ f$ is differentiable at $\alpha$ and

$$
\mathrm{d}(T \circ f)_\alpha=T \circ \mathrm{~d} f_{\alpha^*}
$$

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01:19

Problem 2

Let $F: V \rightarrow \mathbb{R}$ be differentiable at $\alpha$ and let $f: \mathbb{R} \rightarrow \mathbb{R}$ be a function whose derivative exists at $a=F(\boldsymbol{\alpha})$. Prove that $f \circ F$ is differentiable at $\alpha$ and that

$$
\mathrm{d}(f \circ F)_\alpha=f^{\prime}(a) \mathrm{d} F_\alpha
$$

Carson Merrill
Carson Merrill
Numerade Educator
01:26

Problem 3

Let $F: V \rightarrow W$ and $G: W \rightarrow V$ be continuous maps with $G \circ F(\mathbf{v})=\mathbf{v}$ and $F \circ G(\mathbf{w})=\mathbf{w}$ for all $\mathbf{v}$ in $V$ and $\mathbf{w}$ in $W$. Suppose that $F$ is differentiable at $\boldsymbol{\alpha}$ and $G$ is differentiable at $\boldsymbol{\beta}=F(\boldsymbol{\alpha})$. Prove that

$$
\mathrm{d} G_\beta=\left(\mathrm{d} F_\alpha\right)^{-1}
$$

Doruk Isik
Doruk Isik
Numerade Educator
01:20

Problem 4

Let $f: V \rightarrow \mathbb{R}$ be differentiable at $\boldsymbol{\alpha}$. Show that $g=f^n$ is differentiable at $\boldsymbol{\alpha}$ and that

$$
\mathrm{d} g_\alpha=n f^{n-1} \mathrm{~d} f_\alpha
$$

Ekaveera Kumar
Ekaveera Kumar
Numerade Educator
04:06

Problem 5

Let $\gamma: \mathbb{R} \rightarrow \mathbb{R}^2$ denote the curve $\gamma(t)=\binom{\mathrm{e}^t}{\sin t}$, and let $F: \mathbb{R}^2 \rightarrow \mathbb{R}^2$ be the mapping

$$
F\left(\binom{x}{y}\right)=\binom{3 x^2 y}{x^2 y^3}
$$

(a) Compute the tangent vector for $\gamma$ at $t=0$ and $t=\pi / 2$
(b) Find the directional derivative of $F$ with respect to each of these tangent vectors.

Eric Mockensturm
Eric Mockensturm
Numerade Educator

Problem 6

Let $f: \mathbb{R}^2 \rightarrow \mathbb{R}^2$ and $g: \mathbb{R}^2 \rightarrow \mathbb{R}^2$ be given by

$$
f\left(\binom{x}{y}\right)=\binom{x^2 y}{y^3}, g\left(\binom{x}{y}\right)=\binom{\cos x y}{\sin x y} .
$$

Verify the chain rule for the mapping $g \circ f: \mathbb{R}^2 \rightarrow \mathbb{R}^2$.

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04:42

Problem 7

Let $g: V \rightarrow W$ be the mapping $g\left(\binom{x}{y}\right)=\binom{x^2 \mathrm{e}^y}{\cos x y}$, and let $\lambda: \mathbb{R} \rightarrow V$ be the straight line

$$
\lambda(t)=\binom{3}{-1}+t\binom{1}{2}
$$
(a) Find the tangent vector at $\binom{3}{-1}$ to the curve $g \circ \lambda$.
(b) Compute the directional derivative $\mathrm{D}_{\left(-\frac{3}{1}\right),\left(\frac{1}{2}\right)}(g)$ in two ways.

Brian Sipko
Brian Sipko
Numerade Educator
01:02

Problem 8

Let $\phi: V \rightarrow W$ be the mapping $\phi:\binom{r}{\theta} \rightarrow\binom{r \cos \theta}{r \sin \theta}$ and let $f: W \rightarrow \mathbb{R}$ be given by $f:\binom{x}{y} \rightarrow x^3 y^4$. Verify that

$$
\mathrm{D}_{\xi}\left(\phi^* f\right)=\mathrm{D}_{\mathrm{d} \phi_d(\xi)} f
$$

for all tangent vectors $\boldsymbol{\xi}=(\boldsymbol{\alpha}, \mathbf{v})$.

Hoan Nguyen
Hoan Nguyen
Numerade Educator

Problem 9

Define mappings $F: \mathbb{R}^2 \rightarrow \mathbb{R}^2, G: \mathbb{R}^2 \rightarrow \mathbb{R}^2, f: \mathbb{R}^1 \rightarrow \mathbb{R}^2$, and $g: \mathbb{R}^2 \rightarrow \mathbb{R}^1$ by

$$
\begin{gathered}
F\left(\binom{x}{y}\right)=\binom{x^3+y^2}{x y}, \quad G\left(\binom{x}{y}\right)=\binom{2 x^2 y}{y^2}, \\
f(t)=\binom{t^2+1+\cos t}{3 t+2}, \quad g\left(\binom{x}{y}\right)=\left(x^3 y\right)
\end{gathered}
$$

Verify that
(a) $\mathrm{d}(F \circ G)_{\binom{x}{y}}=\mathrm{d} F_{G\left(\binom{x}{y}\right)} \circ \mathrm{d} G_{\binom{x}{y}}$
(b) $\mathrm{d}(G \circ F)_{\binom{x}{y}}=\mathrm{d} G_{\left.F\binom{x}{y}\right)} \mathrm{d}_{\binom{(x}{y}}$
(c) $\mathrm{d}(g \circ f)_{(t)}=\mathrm{d} g_{f(t)} \circ \mathrm{d} F_{(t)}$
(d) $\mathrm{d}(f \circ g)_{\binom{x}{y}}=\mathrm{d} f_{g\left(\binom{x}{y}\right.} \circ{ }^{\circ} g_{\binom{x}{y}}$

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Problem 10

Let $f: \mathbb{R}^2 \rightarrow \mathbb{R}$ be differentiable in some neighborhood of $\binom{x_0}{y_0}$ and satisfy $f\left(\binom{x_0}{y_0}\right) \neq 0$. Show that in some neighborhood of $\binom{x_0}{y_0}$ the mapping $g$, given by $\mathrm{g}\left(\binom{x}{y}\right)=1 / f\left(\binom{x}{y}\right)$ is differentiable and that

$$
\mathrm{d} g_{\binom{x_0}{y_0}}=-\mathrm{d} f_{\binom{x_0}{y_0}} /\left(f\left(\binom{x_0}{y_0}\right)\right)^2 .
$$

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02:06

Problem 11

A function $f$ on the plane is defined in terms of affine coordinates $x$ and $y$ by

$$
f(P)=\sqrt{ }(|x(P) y(P)|)
$$

(a) Is $f$ continuous at the origin $P_0(x=0, y=0)$ ? Justify your answer carefully in terms of the definition of continuity.
(b) Is $f$ differentiable at the origin? Justify your answer carefully in terms of the definition of differentiability.

Tanishq Gupta
Tanishq Gupta
Numerade Educator

Problem 12

So-called parabolic coordinates on the plane are defined in terms of Cartesian coordinates $x$ and $y$ by

$$
\binom{u}{v}=\binom{\sqrt{ }\left(x^2+y^2\right)-x}{\sqrt{ }\left(x^2+y^2\right)+x}
$$
(a) Express $\binom{\mathrm{d} u}{\mathrm{~d} v}$ in terms of $\binom{\mathrm{d} x}{\mathrm{~d} y}$ by means of a $2 \times 2$ matrix, then invert this matrix to express $\binom{\mathrm{d} x}{\mathrm{~d} y}$ in terms of $\binom{\mathrm{d} u}{\mathrm{~d} v}$.
(b) Invert the coordinate transformation by solving for $\binom{x}{y}$ in terms of $u$ and $v$. Differentiate to express $\binom{\mathrm{d} x}{\mathrm{~d} y}$ in terms of $\binom{\mathrm{d} u}{\mathrm{~d} v}$.
(c) Show that the curves $u=$ constant and $v=$ constant are parabolas which are perpendicular where they cross. Sketch these families of curves.
(d) Consider the function $f$ on the plane defined by $f(p)=1 /(u(p)+v(p))$. Express $\mathrm{d}_{\mathbf{q}} f$, where $\mathbf{q}$ is the point with coordinates $u(\mathbf{q})=4, v(\mathbf{q})=16$, in terms of $\mathrm{d} u$ and $\mathrm{d} v$, then in terms of $\mathrm{d} x$ and $\mathrm{d} y$.
(e) Suppose that a particle moves along the path defined by the function $\alpha: \mathbb{R} \rightarrow \mathbb{R}^2$ such that

$$
\binom{x}{y} \circ \boldsymbol{\alpha}(t)=\binom{t^2+t}{t^3} .
$$

Calculate the derivatives of $x \circ \boldsymbol{\alpha}, y \circ \boldsymbol{\alpha}, u \circ \boldsymbol{\alpha}$, and $v \circ \boldsymbol{\alpha}$ at $t=2$.

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01:34

Problem 13

Consider coordinates $u$ and $v$ in the plane which are related to $x$ and $y$ by the equations

$$
\binom{x}{y}=\binom{2 u v}{u^2-v^2}
$$

(a) Calculate the derivative (the Jacobian matrix) of this transformation at the point $u=2, v=1$ (equivalently, express $\binom{\mathrm{d} x}{\mathrm{~d} y}$ in terms of $\binom{\mathrm{d} u}{\mathrm{~d} v}$ at this point).
(b) Consider the function $f(u, v)=u^2 v^3$. Find the equation, in terms of $x$ and $y$, of the line tangent to the curve $f(u, v)=4$ at the point $u=2, v=1$ (i.e., at $x=4, y=3$ ). (Do not try to solve for $u$ and $v$ as functions of $x$ and $y$; just use the chain rule.)
(c) Suppose that a particle moves along the path

$$
\binom{x}{y}=\binom{\frac{1}{2} t^3}{\frac{3}{4} t^2}
$$

At the instant $t=2$, when the particle is passing through the point $\binom{u}{v}=\binom{2}{1}$, at what rate are its $u$ and $v$ coordinates changing; i.e., what are $\mathrm{d} u / \mathrm{d} t$ and $\mathrm{d} v / \mathrm{d} t$ at this instant?

Vikash Ranjan
Vikash Ranjan
Numerade Educator
06:42

Problem 14

Let $\mathbb{A}$ denote an affine plane, let $P_0$ be a point in this plane. Invent a function $f: \mathbb{A} \rightarrow \mathbb{R}$, satisfying $f\left(P_0\right)=0$, which has the property that for any affine coordinates $s(P), t(P)$ on the plane, $\left(\frac{\partial f}{\partial s}\right)_t$ and $\left(\frac{\partial f}{\partial t}\right)_s$ are defined and equal to zero at $P_0$, yet $f$ is not differentiable at $P_0$. (Hint: replace 'differentiable' by 'continuous', 'partial derivative' by 'limit as a function of one coordinate', and an answer would be

$$
f(P)= \begin{cases}0 & \text { at } P_0 \\ \frac{x^2 y}{x^4+y^2} & \text { otherwise }\end{cases}
$$

where $x\left(P_0\right)$ and $y\left(P_0\right)$ are both zero.)

Abigail Martyr
Abigail Martyr
Numerade Educator
06:23

Problem 15

In Quadratic Crater National Monument, the altitude above sea level is described by the function

$$
z(x, y)=\sqrt{ }\left(x^2+4 y^2\right), \quad(x, y, z \text { in kilometers })
$$

The Fahrenheit temperature is described by

$$
T(x, y)=100+2 x-\frac{1}{4} x^2 y^2
$$

(a) Express $\mathrm{d} z$ and $\mathrm{d} T$ in terms of $\mathrm{d} x$ and $\mathrm{d} y$ at the point $x=3, y=2$.
(b) Find the equation of the tangent plane to the crater at the point $x=3$, $y=2$.
(c) At the point $x=3, y=2$, along what direction is the temperature changing most rapidly? If one follows a path along this direction, what is the rate of change of temperature with respect to altitude (accurate to the nearest degree per kilometer)?

Lucas Finney
Lucas Finney
Numerade Educator