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All Textbook Solutions for Multivariable Calculus

23E 24E 25E 26E 27E 28E 29E 30E 31E 32E 33E 34E 35E 36E 37E 38E 39E 40E 41E 42E 43E 1E 2E 3E 4E 5E 6E 7E Verifying the Divergence Theorem In Exercises 38, verify the Divergence Theorem by evaluating sFNdS as a surface integral and as a triple integral. F(x,y,z)=xy2i+yx2j+ek S: surface bounded by z=x2+y2 and z=4 9E 10E 11E 12E 13E 14E 15E Using the Divergence Theorem In Exercises 9-18, use the Divergence Theorem to evaluate SFNdS and find the outward flux of F through the surface of the solid S bounded by the graphs of the equations. Use a computer algebra system to verify your results. F(x,y,z)=(xy2+cosz)i+(x2y+sinz)j+ezkS:z=12x2+y2,z=8 17E 18E 19E 20E Classifying a Point In Exercises 19-22, a vector field and a point in the vector field are given. Determine whether the point is a source, a sink, or incompressible. F(x,y,z)=sinxi+cosyj+z3sinyk,(2,,4)22E 23E 24E 25E 26E Volume (a) Use the Divergence Theorem to verify that the volume of the solid bounded by a surface S is Sxdydz=Sydzdx=Szdxdy. (b) Verify the result of part (a) for the cube bounded by x=0,x=a,y=0,y=a,z=0,andz=a.Constant Vector Field For the constant vector field F(x,y,z)=a1i+a2j+a3k, verify the following integral for any closed surface S. SFNdS=0 29E 30E 31E 32E CONCEPT CHECK Stokess Theorem Explain the benefit of Stokess Theorem when the boundary of the surface is a piecewise curve.2E Verifying Stokess Theorem In Exercises 3-6, verify Stokess Theorem by evaluating CFdr as a line integral and as a double integral. F(x,y,z)=(y+z)i+(xz)j+(xy)kS:z=9x2y2,z0 Verifying Stokess Theorem In Exercises 3-6, verify Stokess Theorem by evaluating CFdr as a line integral and as a double integral. F(x,y,z)=(y+z)i+(xz)j+(xy)kS:z=1x2y2 5E 6E 7E 8E 9E 10E 11E Using Stokess TheoremIn Exercises 716, use Stokess Theorem to evaluate CFdr. In each case, C is oriented counterclockwise as viewed from above. F(x,y,z)=x2i+z2jxyzkS:z=4x2y2 Using Stokess Theorem In Exercises 7-16, use Stokess Theorem to evaluate CFdr . In each case, C is oriented counterclockwise as viewed from above. F(x,y,z)=lnx2+y2i+arctanxyj+k S:z=92x3y over r=2sin2 in the first octant Using Stokess Theorem In Exercises 7-16, use Stokess Theorem to evaluate CFdr . In each case, C is oriented counterclockwise as viewed from above. F(x,y,z)=yzi+(23y)j+(x2+y2)k,x2+y216S:thefirst-octantprotionofx2+z2=16overx2+y2=16 15E 16E 17E 18E 19E 20E 21E 1RE 2RE 3RE 4RE 5RE Solving an Exact Differential Equation In Exercises 3-6, verify that the differential equation is exact. Then find the general solution. ysin(xy)dx+[xsin(xy)+y]dy=0 7RE 8RE 9RE 10RE 11RE 12RE 13RE 14RE 15RE 16RE 17RE 18RE 19RE 20RE 21RE 22RE 23RE 24RE 25RE 26RE 27RE 28RE 29RE 30RE 31RE 32RE 33RE 34RE 35RE Motion of a SpringIn Exercise 35-36, a 64-pound weight stretched a spring 43 foot from its natural length. Use the given information to find a formula for the position of the weight as a function of time. The weight is pulled 34 foot below equilibrium and released. The motion takes place in a medium that furnishes a damping force of magnitude 18|v| at all times.37RE 38RE 39RE 40RE 41RE 42RE 43RE 44RE 45RE Using Initial Conditions In Exercises 45-50, solve the differential equation by the method of undetermined coefficients subject to the initial conditions. y+25y=ex y(0)=0,y(0)=0 47RE 48RE 49RE 50RE Method of Variation of Parameters In Exercises 51-54, solve the differential equation by the method of variation of parameters. y+9y=csc3x 52RE 53RE 54RE 55RE 56RE 57RE 58RE 59RE 60RE 61RE 62RE 1PS 2PS 3PS 4PS 5PS 6PS 7PS 8PS Pendulum Consider a pendulum of length L that swings by the force of gravity only. For small values, the motion of the pendulum can be approximated by the differential equation d2dt2+gL=0 where g is the acceleration due to gravity. Find the general solution of the differential equation and show that it can be written in the form (t)=Acos[gL(t+)] Find the particular solution for a pendulum of length 0.25 meter when the initial conditions are (0)=0.1 radian and (0)=0.5 radian per second. (Use g=9.8 meters per second per second.) Determine the period of the pendulum. Determine the maximum value of . How much time from t = 0 does it take for to be 0 the first time? The second time? What is the angular velocity when =0 the first time? the second time?10PS 11PS 12PS 13PS 14PS 15PS ChebyshevsEquation ConsiderChebyshevs equation (1x2)yxy+k2y=0 Polynomial solutions of this differential equation are called Chebyshev polynomials and are denoted by Tk(x). They satisfy the recursion equation Tn+1(x)=2xTn(x)Tn1(x). Given that T0(x)=1andT1(x)=x, determine the Chebyshevpolynomials T2(x), T3(x) and T4(x). Verify that T0(x),T1(x),T2(x),T3(x),andT4(x) are solutions of the given differential equation. Verify the following Chebyshev polynomials. T5(x)=16x520x3+5x T6(x)=32x648x4+18x21 T7(x)=64x7112x5+56x37x 17PS 18PS 19PS Laguerres Equation Consider Laguerres Equation xy+(1x)y+ky=0. Polynomial solutions of Laguerres equation are called Laguerre polynomials and are denoted by Lk(x). Use a power series of the form y=n=0anxn to show that Lk(x)=n=0k(1)nk!xn(kn)!(n!)2 Assume that a0=1. Determine the Laguerre polynomials L0(x),L1(x),L2(x),L3(x),andL4(x).Exactness What does it mean for the differentialequation M(x,y)dx+N(x,y)dy=0 to be exact? Explainhow to determine whether this differential equation is exact.Integrating Factor When is it beneficial to use an integrating factor to find the solution of the differential equation M(x,y)dx+N(x,y)dy=0.Testing for Exactness In Exercises 3-6, determine whether the differential equation is exact. (2x+xy2)dx+(3+x2y)dy=0 Testing for Exactness In Exercises 3-6, determine whether the differential equation is exact. (2xyy)dx+(x2xy)dy=0 5E 6E 7E Solving an Exact Differential Equation In Exercises 714, verify that the differential equation is exact. Then find the general solution. yexdx+exdy=0 9E 10E 11E 12E 13E 14E 15E Graphical and Analytic AnalysisIn Exercises 15 and 16, (a) sketch an approximate solution of the differential equation satisfying the initialcondition on the slope field, (b) find the particular solution that satisfies the initial condition, and (c) use a graphing utility to graph the particular solution. Compare the graph with the sketch in part (a). Differential Equation Initial Condition 1x2+y2(xdx+ydy)=0 y(4)=3 Figure for 16.17E 18E 19E Finding a Particular SolutionIn Exercises 17-22, find the particular solution of the differential equation that satisfies the initial condition. (x2+y2)dx+2xydy=0, y(3)=1 21E 22E 23E 24E 25E 26E 27E Finding an Integrating Factor In Exercises 23-32, find the integrating factor that is a function of x or y alone and use it to find the general solution of the differential equation. (2x2y1)dx+x3dy=0 29E 30E 31E 32E 33E Using an Integrating Factor In Exercises 33-36, use the integrating factor to find the general solution of the differential equation. Integrating Factor Differential Equation u(x,y)=x2y (3y2+5x2y)dx+(3xy+2x3)dy=0 35E 36E 37E 38E Tangent Curves In Exercises 39-42, use agraphing utility to graph the family of curvestangent to the force field. F(x,y)=yx2+y2ixx2+y2j 40E 41E 42E 43E Finding an Equation of a Curve In Exercises 43 and 44, find an equation of the curve with the specified slope passing through the given point. Slope Point dydx=2xyx2+y2 (-1, 1)Cost In a manufacturing process where y=C(x) represents the cost of producing x units, the elasticity of cost is defined as E(x)=marginalcostaveragecost=C(x)C(x)/x=xydydx. Find the cost function when the elasticity function is E(x)=20xy2y10x where C(100)=500 and x100.HOW DO YOU SEE? The graph shows several representative curves from the family of curves tangent to a force field F. Which is the equation of the force field? Explain your reasoning. a) F(x,y)=i+2j b) F(x,y)=3xi+yj c) F(x,y)=exij d) F(x,y)=2i+eyj 47E 48E 49E 50E 51E 52E 53E 54E 55E 56E 57E 58E 1E 2E 3E 4E 5E 6E 7E 8E 9E 10E 11E 12E 13E 14E 15E 16E 17E 18E 19E 20E 21E 22E 23E 24E 25E 26E 27E 28E 29E 30E 31E Finding a General Solution In exercises 9-36, find the general solution of the linear differential equation. yyy+y=0 33E 34E 35E 36E 37E Finding a Particular Solution Determine C and such that y=Csin3t is a particular solution of the differential equation y+y=0, where y(0)=5 39E 40E 41E Find a Particular Solution: Initial ConditionsIn Exercises 39-44, find the particular solution of the linear differential equation that satisfies the initial conditions. 9y6y+y=0 y(0)=2,y(0)=1 43E 44E 45E 46E 47E Finding a Particular Solution: Boundary ConditionsIn Exercises 45-50, find the particular solution of the linear differential equation that satisfies the boundary conditions. 4y+20y+21y=0 y(0)=3andy(2)=0 49E 50E 51E 52E Several shock absorbers are shown at the right. Do you think the motion of the spring in a shock absorber is undamped or damped.54E 55E 56E Motion of a Spring In Exercise 55-58, match the differential equation with the graph of a particular solution. [The graphs are labeled (a), (b), (c) and (d).] The correct match can be made by comparing the frequency of the oscillations or the rate at which the oscillations are being damped with the appropriate coefficient in the differential equation. y+2y+10y=0 58E 59E 60E 61E 62E 63E 64E 65E 66E 67E True or False? In exercises 67-70, determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false. y1=xandy2=x2 are linearly independent.69E 70E Wronskian The Wronskian of two differentiable functions f and g, denoted by W (f, g), is defined as the function given W(f,g)=|fgfg|. The function f and g are linearly independent when there exists at least one value of x for which W(f,g)0. In exercise 71-74, use the Wronskian to verify that the two functions are linearly independent. y1=eaxy2=ebx,ab 72E 73E 74E 1E Choosing a MethodDetermine whether you woulduse the method of undetermined coefficients or the method of variation of parameters to find the general solution ofeach differential equation. Explain your reasoning. (Donot solve the equations.) (a) y+3y+y=x2 (b) y+y=cscx (c) y6y=3xe2x 3E 4E 5E 6E 7E 8E 9E 10E 11E 12E

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