Questions and Answers of Mathematical Methods For Physicists

For each problem, locate the critical points and classify each one using the second derivative test.a. \(f(x, y)=(x+y)^{2}\).b. \(f(x, y)=x^{2} y+x y^{2}\).c. \(f(x, y)=x^{4} y+x y^{4}-x y\).d.
For each problem, locate the critical points and evaluate the Hessian matrix at each critical point.a. \(f(x, y)=(x+y)^{2}\).b. \(f(x, y)=x^{2} y+x y^{2}\).c. \(f(x, y)=x^{4} y+x y^{4}-x y\).d.
Find the absolute maxima and minima of the function \(f(x, y)=x^{2}+\) \(x y+y^{2}\) on the unit circle.
A thin plate has a temperature distribution of \(T(x, y)=x^{2}-y^{3}-x^{2} y+\) \(y+20\) for \(0 \leq x, y_{,} \leq 2\). Find the coldest and hottest points on the plate.
Find the extrema of the given function subject to the given constraint.a. \(f(x, y)=(x+y)^{2}, x^{2}+y=1\).b. \(f(x, y)=x^{2} y+x y^{2}, x^{2}+y^{2}=2\).c. \(f(x, y)=2 x+3 y, 3 x^{2}+2 y^{2}=3\).d.
A particle moves under the force field \(\mathbf{F}=-abla V\), where the potential function is given by \(V(x, y)=x^{3}+y^{3}-3 x y+5\). Find the equilibrium points of \(\mathbf{F}\) and determine if
For each of the following, find a path that extremizes the given integral.a. \(f_{1}^{2}\left(y^{\prime 2}+2 y y^{\prime}+y^{2}\right) d y, y(1)=0, y(2)=1\).b. \(f_{0}^{2} y^{2}\left(1-y^{\prime
In 1929 E. Hubble confirmed the linear dependence of the velocity on the distance by using the observed values of these quantities. He proposed that the observed radial velocity \(v\), and the
For each of the following, find a path that extremizes the given integral.a. \(\int_{1}^{2}\left(y^{\prime 2}+2 y y^{\prime}+y^{2}\right) d y, y(1)=0, y(2)=1\).b. \(\int_{0}^{3}
A bead slides frictionlessly down a wire from the point \((1,0)\) to \((0,0)\).a. Determine the equation of the path (shape of the wire) such that the bead takes the leat time between the given
A light ray travels from point \(\mathrm{A}\) in a medium with index of refraction \(n_{1}\) toward point \(B\) in a medium with index of refraction \(n_{2}\). Assume that the rays travel in straight
Given the cylinder defined by \(x^{2}+y^{2}=4\), find the path of shortest length connecting the given points.a. \((2,0,0)\) and \((0,2,5)\).b. \((2,0,0)\) and \((2,0,5)\).
The shape of a hanging chain between the points \((-a, b)\) and \((a, b)\) is such that the gravitational potential energy\[V[y]=ho g \int_{-a}^{a} y \sqrt{1+y^{\prime 2}} d x\]is minimized subject
In Example 10. 34, the geodesic equations for geodesics in the plane in polar coordinates were found as\[\begin{align*} r\left(\frac{d \phi}{d s}\right)^{2}-\frac{d^{2} r}{d s^{2}} & =0 \\
Use a Lagrange multiplier to find the curve \(x(y)\) of length \(L=\pi\) on the interval \([0,1]\) which maximizes the integral \(I=\int_{0}^{1} y(x) d x\) and pass through the points \((0,0)\) and
A mass \(m\) lies on a table and is connected to a string of length \(\ell\) as shown in Figure 10. 45. The string passes through a hole in the table and is connected to another mass \(M\) that is
An inclined plane of mass \(M\) and inclination \(\theta\) lies on a frictionless horizontal surface. A small block of mass \(m\) is placed carefully on the plane. Find the acceleration of the
Two similar masses are connected by a string of fixed length and hung over two pulleys that are the same height as depicted in Figure 10. 47. One mass is set into pendular motion and the other only
For those sequences that converge, find the \(\operatorname{limit}_{\lim _{n \rightarrow \infty}} a_{n}\).a. \(a_{n}=\frac{n^{2}+1}{n^{3}+1}\).b. \(a_{n}=\frac{3 n+1}{n+2}\).c.
Find the sum for each of the series:a. \(\sum_{n=0}^{\infty} \frac{(-1)^{n} 3}{4^{n}}\).b. \(\sum_{n=2}^{\infty} \frac{2}{5^{n}}\).c.
Determine if the following converge, or diverge, using one of the convergence tests. If the series converges, is it absolute or conditional?a. \(\sum_{n=1}^{\infty} \frac{n+4}{2 n^{3}+1}\).b.
Do the following:a. Compute: \(\lim _{n \rightarrow \infty} n \ln \left(1-\frac{3}{n}\right)\).b. Use L'Hopital's Rule to evaluate \(L=\lim _{x \rightarrow \infty}\left(1-\frac{4}{x}\right)^{x}\).
Consider the sum \(\sum_{n=1}^{\infty} \frac{1}{(n+2)(n+1)}\).a. Use an appropriate convergence test to show that this series converges.b. Verify that\[\sum_{n=1}^{\infty}
Recall that the alternating harmonic series converges conditionally.a. From the Taylor series expansion for \(f(x)=\ln (1+x)\), inserting \(x=1\) gives the alternating harmonic series. What is the
Find the Taylor series centered at \(x=a\) and its corresponding radius of convergence for the given function. In most cases, you need not employ the direct method of computation of the Taylor
Test for pointwise and uniform convergence on the given set. [The Weierstraß M-Test might be helpful.]a. \(f(x)=\sum_{n=1}^{\infty} \frac{\ln n x}{n^{2}}, x \in[1,2]\).b. \(f(x)=\sum_{n=1}^{\infty}
Use deMoivre's Theorem to write \(\sin ^{3} \theta\) in terms of \(\sin \theta\) and \(\sin 3 \theta\). [Focus on the imaginary part of \(e^{3 i \theta}\).]
Evaluate the following expressions at the given point. Use your calculator and your computer (such as Maple). Then use series expansions to find an approximation to the value of the expression to as
Determine the order, \(O\left(x^{p}\right)\), of the following functions. You may need to use series expansions in powers of \(x\) when \(x \rightarrow 0\), or series expansions in powers of \(1 /
Compute \(\mathbf{u} \times \mathbf{v}\) using the permutation symbol. Verify your answer by computing these products using traditional methods.a. \(\mathbf{u}=2 \mathbf{i}-3 \mathbf{k}, \mathbf{v}=3
Compute the following determinants using the permutation symbol. Verify your answer.a. \(\left|\begin{array}{ccc}3 & 2 & 0 \\ 1 & 4 & -2 \\ -1 & 4 & 3\end{array}\right|\)b.
For the given expressions, write out all values for \(i, j=1,2,3\).a. \(\epsilon_{i 2 j}\).b. \(\epsilon_{i 13}\).c. \(\epsilon_{i j 1} \epsilon_{i 32}\).
Show thata. \(\delta_{i i}=3\).b. \(\delta_{i j} \epsilon_{i j k}=0\)c. \(\epsilon_{i m n} \epsilon_{j m n}=2 \delta_{i j}\).d. \(\epsilon_{i j k} \epsilon_{i j k}=6\).
Show that the vector \((\mathbf{a} \times \mathbf{b}) \times(\mathbf{c} \times \mathbf{d})\) lies on the line of intersection of the two planes: (1) the plane containing \(\mathbf{a}\) and
Prove the following vector identities:a. \((\mathbf{a} \times \mathbf{b}) \cdot(\mathbf{c} \times \mathbf{d})=(\mathbf{a} \cdot \mathbf{c})(\mathbf{b} \cdot \mathbf{d})-(\mathbf{a} \cdot
Use Problem 6a to prove that \(|\mathbf{a} \times \mathbf{b}|=a b \sin \theta\).Data from Problem 6Prove the following vector identities:a. \((\mathbf{a} \times \mathbf{b}) \cdot(\mathbf{c} \times
A particle moves on a straight line, \(\mathbf{r}=t \mathbf{u}\), from the center of a disk. If the disk is rotating with angular velocity \(\omega\), then \(\mathbf{u}\) rotates. Let \(\mathbf{u}=\)
Compute the gradient of the following:a. \(f(x, y)=x^{2}-y^{2}\).b. \(f(x, y, z)=y z+x y+x z\).c. \(f(x, y)=\tan ^{-1}\left(\frac{y}{x}\right)\).d. \(f(x, y, z)=2 y^{x} \cos z-5 \sin z \cos y\).
Find the directional derivative of the given function at the indicated point in the given direction.a. \(f(x, y)=x^{2}-y^{2},(3,2), \mathbf{u}=\mathbf{i}+\mathbf{j}\).b. \(f(x, y)=\frac{y}{x},(2,1),
Zaphod Beeblebrox was in trouble after the infinite improbability drive caused the Heart of Gold, the spaceship Zaphod had stolen when he was President of the Galaxy, to appear between a small
A particle moves under the force field \(\mathbf{F}=-abla V\), where the potential function is given by \(V(x, y, z)=x^{3}+y^{3}-3 x y+5\). Find the equilibrium points of \(\mathbf{F}\) and determine
For the given vector field, find the divergence and curl of the field.a. \(\mathbf{F}=x \mathbf{i}+y \mathbf{j}\)b. \(\mathbf{F}=\frac{y}{r} \mathbf{i}-\frac{x}{r} \mathbf{j}\), for
Write the following using \(\epsilon_{i j k}\) notation and simplify if possible.a. \(\mathbf{C} \times(\mathbf{A} \times(\mathbf{A} \times \mathbf{C}))\).b. \(abla \times(abla \times
Prove the identities:a. \(abla \cdot(abla \times \mathbf{A})=0\).b. \(abla \cdot(f abla g-g abla f)=f abla^{2} g-g abla^{2} f\).c. \(abla r^{n}=n r^{n-2} \mathbf{r}, \quad n \geq 2\).
For \(\mathbf{r}=x \mathbf{i}+y \mathbf{j}+z \mathbf{k}\) and \(r=|\mathbf{r}|\), simplify the following.a. \(abla \times(\mathbf{k} \times \mathbf{r})\).b. \(abla
Newton's Law of Gravitation gives the gravitational force between two masses as\[\mathbf{F}=-\frac{G m M}{r^{3}} \mathbf{r}\]a. Prove that \(\mathbf{F}\) is irrotational.b. Find a scalar potential
Consider a constant electric dipole moment \(\mathbf{p}\) at the origin. It produces an electric potential of \(\phi=\frac{\boldsymbol{p} \cdot \mathbf{r}}{4 \pi \epsilon_{0} r^{3}}\) outside the
In fluid dynamics, the Euler equations govern inviscid fluid flow and provide quantitative statements on the conservation of mass, momentum, and energy. The continuity equation is given
Find the lengths of the following curves:a. \(y(x)=x\) for \(x \in[0,2]\).b. \((x, y, z)=(t, \ln t, 2 \sqrt{2} t)\) for \(1 \leq t \leq 2\).c. \(y(x)=\cosh x, x \in[-2,2]\). (Recall the hanging chain
Consider the integral \(\int_{C} y^{2} d x-2 x^{2} d y\). Evaluate this integral for the following curves:a. \(C\) is a straight line from \((0,2)\) to \((1,1)\).b. \(C\) is the parabolic curve
Evaluate \(\int_{C}\left(x^{2}-2 x y+y^{2}\right) d s\) for the curve \(x(t)=2 \cos t, y(t)=2 \sin t\), \(0 \leq t \leq \pi\)
Prove that the magnetic flux density, B, satisfies the wave equation.
Let \(C\) be a closed curve and \(D\) the enclosed region. Prove the identity\[\int_{C} \phi abla \phi \cdot \mathbf{n} d s=\int_{D}\left(\phi abla^{2} \phi+abla \phi \cdot abla \phi\right) d A\]
Let \(S\) be a closed surface and \(V\) the enclosed volume. Prove Green's first and second identities, respectively.a. \(\int_{S} \phi abla \psi \cdot \mathbf{n} d S=\int_{V}\left(\phi abla^{2}
Let \(C\) be a closed curve and \(D\) the enclosed region. Prove Green's identities in two dimensions.a. First prove\[\int_{D}(v abla \cdot \mathbf{F}+\mathbf{F} \cdot abla v) d A=\int_{C}(v
Compute the work done by the force \(\mathbf{F}=\left(x^{2}-y^{2}\right) \mathbf{i}+2 x y \mathbf{j}\) in moving a particle counterclockwise around the boundary of the rectangle \(R=[0,3] \times\)
Compute the following integrals:a. \(\int_{C}\left(x^{2}+y\right) d x+\left(3 x+y^{3}\right) d y\) for \(C\) the ellipse \(x^{2}+4 y^{2}=4\).b. \(\int_{S}(x-y) d y d z+\left(y^{2}+z^{2}\right) d z d
Use Stokes' Theorem to evaluate the integral\[\int_{C}-y^{3} d x+x^{3} d y-z^{3} d z\]for \(C\) the (positively oriented) curve of intersection between the cylinder \(x^{2}+y^{2}=1\) and the plane
Use Stokes' Theorem to derive the integral form of Faraday's law,\[\int_{C} \mathbf{E} \cdot d \mathbf{s}=-\frac{\partial}{\partial t} \iint_{D} \mathbf{B} \cdot d \mathbf{S}\]from the differential
For cylindrical coordinates,\[\begin{align*} & x=r \cos \theta \\ & y=r \sin \theta \\ & z=z \tag{9.128} \end{align*}\]find the scale factors and derive the following
For spherical coordinates,\[\begin{align*} x & =ho \sin \theta \cos \phi \\ y & =ho \sin \theta \sin \phi \\ z & =ho \cos \theta \tag{9.133} \end{align*}\]find the scale factors and
The moments of inertia for a system of point masses are given by sums instead of integrals. For example, \(I_{x x}=\sum_{i} m_{i}\left(y_{i}^{2}+z_{i}^{2}\right)\) and \(I_{x y}=\) \(-\sum_{i} m_{i}
Consider the octant of a uniform sphere of density 5 grams per cubic centimeter and radius \(a\) lying in the first octant.a. Find the inertia tensor about the origin.b. What are the principal
Let \(T^{\alpha}\) be a contravariant vector and \(S_{\alpha}\) be a covariant vector.a. Show that \(R_{\beta}=g_{\alpha \beta} T^{\alpha}\) is a covariant vector.b. Show that \(R^{\beta}=g^{\alpha
Show that \(T^{\alpha \beta \gamma \delta ho} S_{\beta ho}\) is a tensor. What is its rank?
The line element in terms of the metric tensor, \(q_{\alpha \beta}\) is given by\[d s^{2}=g_{\alpha \beta} d x^{\alpha} d x^{\beta}\]Show that the transformed metric for the transformation
In this problem you will show that the sequence of functions\[f_{n}(x)=\frac{n}{\pi}\left(\frac{1}{1+n^{2} x^{2}}\right)\]approaches \(\delta(x)\) as \(n \rightarrow \infty\). Use the following to
Verify that the sequence of functions \(\left\{f_{n}(x)\right\}_{n=1}^{\infty}\), defined by \(f_{n}(x)=\) \(\frac{n}{2} e^{-n|x|}\), approaches a delta function.
Evaluate the following integrals:a. \(\int_{0}^{\pi} \sin x \delta\left(x-\frac{\pi}{2}\right) d x\)b. \(\int_{-\infty}^{\infty} \delta\left(\frac{x-5}{3} e^{2 x}\right)\left(3 x^{2}-7 x+2\right) d
For the case that a function has multiple roots, \(f\left(x_{i}\right)=0, i=1,2, \ldots\), it can be shown that\[\delta(f(x))=\sum_{i=1}^{n}
Find a Fourier series representation of the Dirac delta function, \(\delta(x)\), on \([-L, L]\).
For \(a>0\), find the Fourier transform, \(\hat{f}(k)\), of \(f(x)=e^{-a|x|}\).
Use the result from Problem 6 plus properties of the Fourier transform to find the Fourier transform, of \(f(x)=x^{2} e^{-a|x|}\) for \(a>0\).Data from Problem 6For \(a>0\), find the Fourier
Find the Fourier transform, \(\hat{f}(k)\), of \(f(x)=e^{-2 x^{2}+x}\).
Prove the Second Shift Property in the form\[F\left[e^{i \beta x} f(x)\right]=\hat{f}(k+\beta)\]
A damped harmonic oscillator is given by\[f(t)=\left\{\begin{array}{cc} A e^{-\alpha t} e^{i \omega_{0} t}, & t \geq 0 \\ 0, & t
Show that the convolution operation is associative: \((f *(g * h))(t)=\) \(((f * g) * h)(t)\).
In this problem, you will directly compute the convolution of two Gaussian functions in two steps.a. Use completing the square to evaluate\[\int_{-\infty}^{\infty} e^{-\alpha t^{2}+\beta t} d t\]b.
You will compute the (Fourier) convolution of two box functions of the same width. Recall that the box function is given by\[f_{a}(x)= \begin{cases}1, & |x| \leq a \\ 0, &
Define the integrals \(I_{n}=\int_{-\infty}^{\infty} x^{2 n} e^{-x^{2}} d x\). Noting that \(I_{0}=\sqrt{\pi}\),a. Find a recursive relation between \(I_{n}\) and \(I_{n-1}\).b. Use this relation to
Find the Laplace transform of the following functions:a. \(f(t)=9 t^{2}-7\).b. \(f(t)=e^{5 t-3}\).c. \(f(t)=\cos 7 t\)d. \(f(t)=e^{4 t} \sin 2 t\).e. \(f(t)=e^{2 t}(t+\cosh t)\).f. \(f(t)=t^{2}
Find the inverse Laplace transform of the following functions using the properties of Laplace transforms and the table of Laplace transform pairs.a. \(F(s)=\frac{18}{s^{3}}+\frac{7}{s}\).b.
Compute the convolution \((f * g)(t)\) (in the Laplace transform sense) and its corresponding Laplace transform \(\mathcal{L}[f * g]\) for the following functions:a. \(f(t)=t^{2}, g(t)=t^{3}\).b.
For the following problems, draw the given function and find the Laplace transform in closed form.a. \(f(t)=1+\sum_{n=1}^{\infty}(-1)^{n} H(t-n)\).b. \(f(t)=\sum_{n=0}^{\infty}[H(t-2 n+1)-H(t-2
Use the Convolution Theorem to compute the inverse transform of the following:a. \(F(s)=\frac{2}{s^{2}\left(s^{2}+1\right)}\).b. \(F(s)=\frac{e^{-3 s}}{s^{2}}\).c. \(F(s)=\frac{1}{s\left(s^{2}+2
Find the inverse Laplace transform in two different ways: (i) Use tables.(ii) Use the Bromwich Integral.a. \(F(s)=\frac{1}{s^{3}(s+4)^{2}}\).b. \(F(s)=\frac{1}{s^{2}-4 s-5}\).c.
Use Laplace transforms to solve the following initial value problems. Where possible, describe the solution behavior in terms of oscillation and decay.a. \(y^{\prime \prime}-5 y^{\prime}+6 y=0,
Use Laplace transforms to convert the following system of differential equations into an algebraic system, and find the solution of the differential equations.\[\begin{aligned} & x^{\prime \prime}=3
Use Laplace transforms to convert the following nonhomogeneous systems of differential equations into an algebraic system, and find the solutions of the differential equations.a.\[\begin{aligned}
Consider the series circuit in Problem 2. 20 and in Figure 2. 7 with \(L=\) \(1.00 \mathrm{H}, R=1.00 \times 10^{2} \Omega, C=1.00 \times 10^{-4} \mathrm{~F}\), and \(V_{0}=1.00 \times 10^{3}
Use Laplace transforms to sum the following series or write as a single integral.a. \(\sum_{n=0}^{\infty} \frac{(-1)^{n}}{1+2 n}\).b. \(\sum_{n=1}^{\infty} \frac{1}{n(n+3)}\).c. \(\sum_{n=1}^{\infty}
Use Laplace transforms to prove\[\sum_{n=1}^{\infty} \frac{1}{(n+a)(n+b)}=\frac{1}{b-a} \int_{0}^{1} \frac{u^{a}-u^{b}}{1-u} d u\]Use this result to evaluate the following sums:a.
Do the following:a. Find the first four nonvanishing terms of the Maclaurin series expansion of \(f(x)=\frac{x}{e^{x}-1}\).b. Use the result in parta. to determine the first four nonvanishing
Given the following Laplace transforms \(F(s)\), find the function \(f(t)\). Note that in each case there are an infinite number of poles, resulting in an infinite series representation.a.
Consider the initial boundary value problem for the heat equation:\[\begin{array}{cc} u_{t}=2 u_{x x}, & 0
Write the following in standard form.a. \((4+5 i)(2-3 i)\).b. \((1+i)^{3}\).c. \(\frac{5+3 i}{1-i}\).
Write the following in polar form, \(z=r e^{i \theta}\).a. \(i-1\).b. \(-2 i\).c. \(\sqrt{3}+3 i\).
Write the following in rectangular form, \(z=a+i b\).a. \(4 e^{i \pi / 6}\).b. \(\sqrt{2} e^{5 i \pi / 4}\).c. \((1-i)^{100}\).
Find all \(z\) such that \(z^{4}=16 i\). Write the solutions in rectangular form, \(z=a+i b\), with no decimal approximation or trig functions.
Show that \(\sin (x+i y)=\sin x \cosh y+i \cos x \sinh y\) using trigonometric identities and the exponential forms of these functions.

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