Mathematics of Physics and Engineering II: Homework problems

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1 Mathematics of Physics and Engineering II: Homework problems Homework. Problem. Consider four points in R 3 : P (,, ), Q(,, 2), R(,, ), S( + a,, 2a), where a is a real number. () Compute the coordinates of P Q, P R, P S (2) Compute the value of a so that the angle RP S is the right angle. (3) Compute the area of the triangle P QR. (4) Compute the equation of the plane containing points P, Q, R. (5) Compute the volume of the parallelepiped spanned by the vectors P Q, P R, P S (it will be a function of a). (6) For what value of a will the point S lie in the same plane as the points P, Q, R? (7) Write the vector parametric equation of the line that passes through the point (,, ) and is perpendicular to the plane through points P, Q, R, and compute the coordinates of the point of intersection of the line and the plane. Problem 2. A particle moves so that its position at time t is given by the vector function r(t) = t 2, t 3, + t 2, t. Compute: () Coordinates of the particle at time t = : (2) Velocity of the particle for t : (3) Speed of the particle for t : (4) Acceleration of the particle for t : (5) Vector parametric equation of the tangent line to the trajectory at (,, 2). (6) The coordinates of the point of intersection of the trajectory with the plane z x = 2. (7) Distance traveled (arc length) from t = to t =. Homework 2. Solve each problem. Then make and solve -2 similar problems that you Problem. Consider a function f(x, y) = 2x 2 xy + y 2 x + y. () Compute the gradient of f. (2) Compute the rate of change of f at (, ) in the direction toward the origin. Is the function increasing or decreasing in that direction? (3) Determine the direction of most rapid decrease of f at (, ) and compute the rate of change of the function in that direction. (4) (Warning: this one can be time-consuming) Write the equation of the path of steepest ascent on the surface z = f(x, y) starting from point (,, ). What path will you get on the topographic map? Problem 2. Evaluate the following integrals: () (2y 2 + 2xz)dx + 4xydy + x 2 dz, where C is the path C x(t) = cos t, y(t) = sin t, z(t) = t, t 2π. (2) y 2 dx + x 2 dy, where C is the boundary of the rectangle with vertices (, ), (3, ), C (3, 2), (, 2), oriented counterclockwise. (3) ydx zdy + ydz, where C is the ellipse x 2 + y 2 = ; 3x + 4y + z = 2 oriented C counterclockwise as seen from the point (,, ). Problem 3. Compute the following quantities using a suitable integral:

2 2 () The mass of the curve shaped as a helix r(t) = cos t, sin t, t, t [, 2π], if the density at every point is the square of the distance of the point to the origin. (2) The area between x-axis and the curve r(t) = t sin t, cos t, t [, 2π]. (3) The average distance to the (x, y) plane of the points on the hemisphere z = x 2 y 2. (4) The flux of the vector field F = x, y, z through the lateral surface of the cylinder x 2 + y 2 =, z [, 2]. Problem 4. Let G be the parallelogram with vertices (, ), (3, ), (4, ), (, ), and f, a continuous function. Write the limits in the iterated integrals below: f da = fdx dy = fdy dx = frdr dθ = fdθ rdr. G Homework 3. Solve each problem. Then make and solve -2 similar problems that you Problem. (a) Write equation of the unit sphere in cylindrical coordinates. (b) Given a twice continuously differentiable function f, write the expression for the Laplacian 2 f of f in the spherical coordinates (r, θ, φ), where x = r cos θ sin φ, y = r sin θ sin φ, z = r cos φ. Problem 2. () Write the number 2 i in the form x + iy. 3i 4 (2) Write i in the polar form (3) Solve the following equation: z 4 2z =. Write the answer in the polar form. (4) Compute the anti-derivative e 2x sin(3x)dx without integrating by parts. (5) Find all the values of 6 i. (6) Find all the values of + i 3 + i 3. (7) Write cos(5x) as a polynomial in cos x. Problem 3. Let α = e 2iπ/5. (a) Let u = α + α 4. Verify that u 2 + u = and u = α + ᾱ. (b) Find an algebraic expression for Reα = cos(2π/5). (c) As the final prize, use the results to construct a regular pentagon using compass and straight edge. Homework 4. Solve each problem. Then make and solve -2 similar problems that you Problem. () For f(z) = z 3 2z + find Re(f) and Im(f). Is this function analytic? (2) For f(z) = ze z find Re(f) and Im(f). Is this function analytic? (3) For f(z) = cos z find Re(f) and Im(f). Is this function analytic? (4) Suppose that f is analytic everywhere and Re(f) =. What can you say about Im(f)? (5) Find a and b so that u(x, y) = ax 3 + bxy is harmonic, and then find a conjugate harmonic of u. Problem 2. Find the image of each of the following sets under the map f(z) = /z : () {z : z < }. (2) {z : Re(z) > }. (3) {z : < Im(z) < }. Homework 5. Solve each problem. Then make and solve -2 similar problems that you

3 3 Problem. Determine the radius of convergence of the following power series: () ( ) n 3 i n z 2n. n 3 + 3i (2) (5 + i + n ( )n ) n z 2n. (3) (n!) 2 n (2n)! z3n+. (4) n! n n n z2n+. (5) z 2n+ (4n)! n (3n) 4n (7 + ( ) n ). n Problem 2. Write the Taylor series of the given function f = f(z) at the given point z and determine the radius of convergence of the series. () f(z) = 2z + 3, z =. (2) f(z) = z2 + z +, z =. (3) f(z) = (2z ), z 2 =. (4) f(z) = z 2 + 2z + 2, z =. Problem 3. Write the Laurent series of the function f(z) = following regions: () < z < 2 (2) z 2 > 2 (3) < z 2 < 2 (4) < z + < 3 2z z 2 in each of the Problem 4. Determine the type of singularity at the origin (that is, at the point z = ) of the following functions: () f(z) = (e z ) 2 /( cos z) (2) f(z) = (z cos z sin z)/(z sin z) (3) f(z) = ( cos z)/(e z ) 4 (4) f(z) = sin(/z) (5) f(z) = / sin(/z) Homework 6. Solve each problem. Then make and solve -2 similar problems that you Problem. Compute the residue Res z=z f(z) for the following functions f and points z : () f(z) = z + 5 z 3 z, z = ; (2) f(z) = z6 + 3z 4 + 2z (z ) 3, z = ; (3) f(z) = z e z +, z = πi; (4) f(z) = z e z +, z = 3πi; (5) f(z) = z 2 sin(/z), z =. Problem 2. Evaluate the following integrals (residue method is strongly recommended):

4 4 () (2) (3) z: z 2πi =2π 2π z e z + dz + 2 sin t cos t dt dx (4 + 2x + x 2 ) 3 Problem 3. Compute the power series expansion of the solutions of the following equations: () w (z) zw(z) =, w() =, w () = ; (2) z 2 w (z) + zw (z) + z 2 w(z) =, w() =, w () =. (3) w (z) zw (z) + 2w(z) =, w() =, w () =. Problem 4. Determine the values of the parameter λ for which the following equations have polynomial solutions: () w (z) 2zw (z) + λw(z) = (2) ( z 2 )w (z) zw (z) + λw(z) = (3) ( z 2 )w (z) 2zw (z) + λw(z) = Homework 7. Solve each problem. Then make and solve -2 similar problems that you Problem. Give an example of a power series that converges at one point on the boundary of the disk of convergence and diverges at another point. Explain why the convergence in this example is necessarily conditional (in other words, explain why the absolute convergence at one point on the boundary implies absolute convergence at all points of the boundary). Problem 2. For each sequence of functions below, identify the limit and determine whether the convergence is uniform: () x n, x (, ) (2) sin(x/n), x ( π, π) (3) nx/( + nx), x (, ) (4) nx 2 /(n + x), x (, ) Problem 3. Determine whether each of the following series converges (a) absolutely for each x in the indicated interval (b) uniformly over the indicated interval. () n nxn, x <. cos(nx) (2) n, < x < +. 2 (3) x n n, x <. n 3/2 (4) n, < x < +. x n n! Problem 4. For functions f, g below, find a, b, c, d so that g(x) = a + bf(cx + d). () f(x) = x, x π, f is even and 2π-periodic; g(x) = 2x, x /2; g(x) = 2 2x, /2 x, g is odd and is periodic with period 2. (2) f(x) =, x < π; f(x) =, π x < 2π, f is 2π-periodic; g(x) =, x < /2; g(x) =, /2 x <, g is periodic with period. (3) f(x) = x, π x < π, f is 2π-periodic; g(x) = x, x <, g is periodic with period. Problem 5. Compute the Fourier series expansion of each of the six functions in problem 4. (You will have to compute some integrals, but not for all six functions). Problem 6. The function f(x) = x 2, < x < is to be expanded in a Fourier series. You have three options: (a) take the periodic extension of the function with period.

5 (b) take the periodic extension of the even extension of the function with period 2. (c) take the periodic extension of the odd extension of the function with period 2. Draw the pictures of the sum of the resulting Fourier series in each case. Which option would you choose and why? (Note: to answer these questions you do not need to compute the Fourier coefficients of the function.) Problem 7. Use the results of Problem 5 to evaluate the following infinite sums: () k (2) k (3) k (4) k ( ) k 2k+ (2k+) 4 (2k+) 2 k 2 Problem 8. Let f(x) = 2x, x <. Denote by S f (x) the sum of the Fourier series of f. Draw the graph of S f and evaluate (a) S f (3) (b) S f (5/2). Homework 8. Solve each problem. Then make and solve -2 similar problems that you Problem. Compute the Fourier transform of the function f in each of the following cases: () f(x) = e 2x, x >, f(x) = otherwise. (2) f(x) = x, a < x < b, f(x) = otherwise. (3) f (x) f(x) = u(x), where u(x) =, x <, u(x) = otherwise. 5 Problem 2. Compute the Fourier transform of the function f(x) = e x and use the result cos(wx) to evaluate the integrals dw and + w2 ( + w 2 ) dw. 2 Problem 3. The Fourier transform of the function f(x) = e x2 /2 is f(ω) = e ω2 /2. Compute the Fourier transform of the function g(x) = e x2. Problem 4. Confirm that if then ĥ(w) = 2π ˆf(w)ĝ(w). h(x) = + f(x y)g(y)dy, Problem 5. Let f = f(x) be a bounded continuous function such that + f(x) dx < and + f 2 (x)dx =. For t > define Verify that, for all t >, u(t, x) = 4πit e i(x y)2 /(4t) f(y)dy, i =. u(t, x) 2 dx =. Food for thought. Is it possible that f is bounded, continuous, + f(x) dx <, but + f 2 (x)dx =? Is it possible that f is bounded, continuous, + f 2 (x)dx <, but f(x) dx =? What if we drop the assumption that f is bounded? + Homework 9. Solve each problem. Then make and solve -2 similar problems that you

6 6 Problem. Find the general solution of the equation u t + sin(t) u x =. Problem 2. Find the general solution of the equation x 2 u x y 2 u y =. Problem 3. Find the general solution of the boundary value problems () u t = u xx, t >, < x <, u x= =, u x x= =. (2) u t = u xx, t >, < x <, u x= =, (u u x ) x= =. (3) u t = u xx, t >, < x <, u x= =, u x= =. (4) u t = u xx + t 2 u, t >, < x <, u x= =, u x= =. Problem 4. Consider the initial value problem u t =.5u xx, x (, + ), t >, u t= = f(x). () Find u if f(x) = 2π e x2 2. (2) Suppose that f(x) and + f(x)dx =. Show that, for all t >, u(x, t) and + u(x, t)dx =. Problem 5. Let f = f(t), t be a continuous function. Show that the function u(t, x) = t x f(s) 4πc2 (t s) 3 e satisfies u t = c 2 u xx, t >, x >, u t= =, u x= = f(t). x 2 4c 2 (t s) ds Homework. Solve each problem. Then make and solve -2 similar problems that you Problem. Find a solution of the following equations using separation of variables. () u xx + u yy =. (2) y 2 u x x 2 u y =. (3) u x + u y = (x + y)u. (4) xu xy + 2yu =. (5) u t + uu x = Problem 2. Solve the following initial-boundary value problems. () u tt = 4u xx, t >, < x <, u x x= = u x x= =, u t= = f(x), u t t= = f (x), where f(x) = x, x /2, f(x) = x, /2 x ; f (x) = cos(2πx). (2) u tt = u xx + sin x, t >, < x < π, u t= = sin(2x), u t t= = sin x, u x= = u x=π =. (3) u tt = 4u xx, t >, < x <, u x x= = u x x= =, u t= = u t t= = f(x), where f(x) = x, x /2, f(x) = x, /2 x. Problem 3. Suppose that q = q(x), x is a continuous non-negative function and v, v 2 are nontrivial solutions of v i q(x)v i = a i v i, v i() v i () = v i() v i () =, i =, 2, a a 2. What is the value of v (x)v 2 (x)dx? Homework. [Optional] Problem. Solve the following boundary value problems () u xx + 4u yy =, (x, y) G = (, π) (, π); u(, y) = u(x, ) = u(π, y) =, u(x, π) = sin x (2) u xx + u yy =, (x, y) G = (, π) (, π); u G = (3) u xx + u yy =, x 2 + y 2 < ; u(x, y) =, x 2 + y 2 = (4) 2 u(r, θ) =, r < ; u r r= =. Problem 2. Solve the following initial-boundary value problems in polar coordinates (with u = u(t, r, θ)) () u t = 2 u, r < 2; u(t, r, θ) =, r = 2, u(, r, θ) = f(r).

7 7 (2) u tt = 2 u, r < ; u(t, r, θ) =, r =, u(, r, θ) = f(r) cos θ, u t (, r, θ) =. Problem 3. Use the power series representation of the Bessel functions ( ) k ( z ) 2k+N J N (z) =, N =,, 2, 3,..., k!(k + N)! 2 k= to verify the following properties of the Bessel functions: () J N ( z) = ( ) N J N (z), (2) (z N J N (z)) = z N J N (z), (3) (z N J N (z)) = z N J N+ (z), (4) zj N (z) = NJ N(z) zj N+ (z) = NJ N (z) + zj N (z), (5) C(,z) ζn J N (ζ)dζ = z N J N (z), C(, z) is a path from to z. Problem 4. For each of the following names, recall at least one item from this class that is associated with the name (for example, divergence theorem for Gauss). Feel free to add to the list. () Bessel (2) Cauchy (3) Dirichlet (4) Euler (5) Fourier (6) Gauss (7) Gibbs (8) Green (9) Laplace () Poisson () Riemann (2) Stokes