Skip to main content

Bartleby Sitemap - Textbook Solutions

All Textbook Solutions for Multivariable Calculus

41RE 1. Let S be a smooth parametric surface and let P be a point such that each line that starts at P intersects S at most once. The solid angle (S) subtended by S at P is the set of lines starting at P and passing through S. Let S(a) be the intersection of (S) with the surface of the sphere with center P and radius a. Then the measure of the solid angle (in steradians) is defined to be |(S)| = areaofS(a)a2 Apply the Divergence Theorem to the part of (S) between S(a) and S to show that |(S)| = Srnr3 dS where r is the radius vector from P to any point on S, r = | r |, and the unit normal vector n is directed away from P. This shows that the definition of the measure of a solid angle is independent of the radius a of the sphere. Thus the measure of the solid angle is equal to the area subtended on a unit sphere. (Note the analogy with the definition of radian measure.) The total solid angle subtended by a sphere at its center is thus 4 steradians.Find the positively oriented simple closed curve C for which the value of the line integral C (y3 y) dx 2x3dy is a maximum.Let C be a simple closed piecewise-smooth space curve that lies in a plane with unit normal vector n = a, b, c and has positive orientation with respect to n. Show that the plane area enclosed by C is 12 C (bz - cy) dx + (cx az) dy + (ay bx) dz Prove the following identity: (F G) = (F )G + (G )F + F curl G + G curl F The figure depicts the sequence of events in each cylinder of a four-cylinder internal combustion engine. Each piston moves up and down and is connected by a pivoted arm to a rotating crankshaft. Let P(t) and V(t) be the pressure and volume within a cylinder at time t, where a t b gives the time required for a complete cycle. The graph shows how P and V vary through one cycle of a four-stroke engine. During the intake stroke (from to ) a mixture of air and gasoline at atmospheric pressure is drawn into a cylinder through the intake valve as the piston moves downward. Then the piston rapidly compresses the mix with the valves closed in the compression stroke (from to ) during which the pressure rises and the volume decreases. At the sparkplug ignites the fuel, raising the temperature and pressure at almost constant volume to . Then, with valves closed, the rapid expansion forces the piston downward during the power stroke (from to ). The exhaust valve opens, temperature and pressure drop, and mechanical energy stored in a rotating flywheel pushes the piston upward, forcing the waste products out of the exhaust valve in the exhaust stroke. The exhaust valve closes and the intake valve opens. Were now back at and the cycle starts again. (a) Show that the work done on the piston during one cycle of a four-stroke engine is W = C P dV, where C is the curve in the PV-plane shown in the figure. [Hint: Let x(t) be the distance from the piston to the top of the cylinder and note that the force on the piston is F = AP(t) i, where A is the area of the top of the piston. Then W = C1F dr, where C1 is given by r(t)= x(t) i, a t b. An alternative approach is to work directly with Riemann sums.] (b) Use Formula 16.4.5 to show that the work is the difference of the areas enclosed by the two loops of C.Solve the differential equation. 1. y" y' 6y = 0 2E 3E Solve the differential equation. 4. y" + y' 12y = 0 5E 6E 7E 8E Solve the differential equation. 9. y" 4y' + 13y = 0 10E 11E 12E 13E 14E 15E 16E 17E 18E Solve the initial-value problem. 19. 9y" + 12y' + 4y = 0, y(0) = 1, y'(0) = 0 20E 21E 22E Solve the initial-value problem. 23. y" y' 12y = 0, y(1) = 0, y'(1) = 1 Solve the initial-value problem. 24. 4y" + 4y' + 3y = 0, y(0) = 0, y'(0) = 1 25E 26E 27E 28E 29E 30E 31E Solve the boundary-value problem, if possible. 32. y" + 4y' + 20y = 0, y(0) = 1, y() = e-2 33E If a, b, and c are all positive constants and y(x) is a solution of the differential equation ay" + by' + cy = 0, show that limx y(x) = 0.1E 2E 3E 4E 5E 6E 7E 8E 9E 10E 11E 12E 13E 14E 15E 16E 17E 18E 19E 20E Solve the differential equation using (a) undetermined coefficients and (b) variation of parameters. 21. y" 2y' + y = e2x 22E 23E Solve the differential equation using the method of variation of parameters. 24. y" + y = sec3x, 0 x /2 Solve the differential equation using the method of variation of parameters. 25. y3y+2y=11+ex Solve the differential equation using the method of variation of parameters. 26. y" + 3y' +2y = sin(ex)27E 28E A spring has natural length 0.75 m and a 5-kg mass. A force of 25 N is needed to keep the spring stretched to a length of1 m. If the spring is stretched to a length of 1.1 m and thenreleased with velocity 0, find the position of the mass aftert seconds.2E A spring with a mass of 2 kg has damping constant 14, and a force of 6 N is required to keep the spring stretched 0.5 mbeyond its natural length. The spring is stretched 1 mbeyond its natural length and then released with zero velocity. Find the position of the mass at any time t.4E 5E 6E 7E 8E Suppose a spring has mass m and spring constant k and let =k/m. Suppose that the damping constant is so smallthat the damping force is negligible. If an external forceF(t) = F0 cos 0t is applied, where 0 , use the methodof undetermined coefficients to show that the motion of themass is described by Equation 6.10E 11E 12E 13E 14E 15E The battery in Exercise 14 is replaced by a generator producing a voltage of E(t) = 12 sin 10t. (a) Find the charge at time t. (b) Graph the charge function.17E 18E 1E 2E 3E 4E 5E 6E 7E 8E 9E 10E 11E 12E (a) Write the general form of a second-order homogeneous linear differential equation with constant coefficients. (b) Write the auxiliary equation. (c) How do you use the roots of the auxiliary equation to solve the differential equation? Write the form of the solution for each of the three cases that can occur.2RCC (a) Write the general form of a second-order nonhomogeneous linear differential equation with constant coefficients. (b) What is the complementary equation? How does it help solve the original differential equation? (c) Explain how the method of undetermined coefficients works. (d) Explain how the method of variation of parameters works.4RCC 5RCC 1RQ 2RQ 3RQ 4RQ Solve the differential equation. 1. 4y" y =0 2RE 3RE Solve the differential equation. 4. y" + 8y' + 16y =0 5RE 6RE 7RE 8RE 9RE Solve the differential equation. 10. d2ydx2+y=cscx,0x/2 11RE Solve the initial-value problem. 12. y" 6y' + 25y = 0, y(0) = 2, y'(0) = 1 Solve the initial-value problem. 13. y" 5y' + 4y = 0, y(0) = 0, y'(0) = 1 14RE Solve the boundary-value problem, if possible. 15. y" + 4y' + 29y = 0, y(0) = 1, y() = 1 16RE 17RE 18RE A series circuit contains a resistor with R = 40 , an inductor with L = 2 H, a capacitor with C = 0.0025 F, and a 12-V battery. The initial charge is Q = 0.01 C and the initial currentis 0. Find the charge at time t.20RE Assume that the earth is a solid sphere of uniform density with mass M and radius R = 3960 mi. For a particle of mass mwithin the earth at a distance r from the earths center, thegravitational force attracting the particle to the center is Fr=GMrmr2 where G is the gravitational constant and Mr is the mass of theearth within the sphere of radius r. (a) Show that Fr=GMmR3r. (b) Suppose a hole is drilled through the earth along a diameter. Show that if a particle of mass m is dropped from rest at the surface, into the hole, then the distance y = y(t) ofthe particle from the center of the earth at time t is given by y"(t) = -k2y(t) where k2 = GM/R3 = g/R. (c) Conclude from part (b) that the particle undergoes simple harmonic motion. Find the period T. (d) With what speed does the particle pass through the center of the earth?Evaluate the expression and write your answer in the form a + bi. 1. (5 6i) + (3 + 2i)Evaluate the expression and write your answer in the form a + bi. 2. (412i)(9+52i)3E 4E 5E Evaluate the expression and write your answer in the form a + bi. 6. 2i(12i)Evaluate the expression and write your answer in the form a + bi. 7. 1+4i3+2i 8E 9E 10E 11E Evaluate the expression and write your answer in the form a + bi. 12. i100 Evaluate the expression and write your answer in the form a + bi. 13. 25 14E 15E 16E 17E 18E 19E 20E 21E 22E 23E 24E 25E 26E 27E 28E 29E Find polar forms for zw, z/w, and 1/z by first putting z and w into polar form. 30. z=434i,w=8i Find polar forms for zw, z/w, and 1/z by first putting z and w into polar form. 31. z=232i, w = 1 + i 32E 33E 34E 35E 36E 37E 38E 39E 40E Write the number in the form a + bi. 41. ei/2 42E 43E 44E 45E 46E 47E 48E If u(x) = f(x) + ig(x) is a complex-valued function of a real variable x and the real and imaginary parts f(x) and g(x) are differentiable functions of x, then the derivative of u is defined to be u(x) = f(x) + ig(x). Use this together with Equation 7 to prove that if F(x) = erx, then F(x) = rerx when r = a + bi is a complex number.(a) If u is a complex-valued function of a real variable, its indefinite integral u(x)dx is an antiderivative of u. Evaluate e(1+i)xdx (b) By considering the real and imaginary parts of the integral in part (a), evaluate the real integrals excosxdxandexsinxdx (c) Compare with the method used in Example 7.1.4.

Page: [1][2][3][4][5][6][7][8][9][10][11][12]