CALCULUS EARLY TRANSCENDENTALS W/ WILE
11th Edition
ISBN: 9781119228509
Author: Anton
Publisher: WILEY
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Textbook Question
Chapter 15.3, Problem 38ES
Prove that (b) implies (c) in Theorem 15.3.2.
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Chapter 15 Solutions
CALCULUS EARLY TRANSCENDENTALS W/ WILE
Ch. 15.1 - The function (x,y,z)=xy+yz+xz is a potential for...Ch. 15.1 - The vector field F(x,y,z)=, defined for...Ch. 15.1 - An inverse-square field is one that can be written...Ch. 15.1 - The vector field has divergence and curl
Ch. 15.1 - Match the vector field F(x,y) with one of the...Ch. 15.1 - Match the vector field F(x,y) with one of the...Ch. 15.1 - Determine whether the statement about the vector...Ch. 15.1 - Determine whether the statement about the vector...Ch. 15.1 - Sketch the vector field by drawing some...Ch. 15.1 - Sketch the vector field by drawing some...
Ch. 15.1 - Sketch the vector field by drawing some...Ch. 15.1 - Sketch the vector field by drawing some...Ch. 15.1 - Use a graphing utility to generate a plot of the...Ch. 15.1 - Use a graphing utility to generate a plot of the...Ch. 15.1 - Determine whether the statement is true or false....Ch. 15.1 - Determine whether the statement is true or false....Ch. 15.1 - Determine whether the statement is true or false....Ch. 15.1 - Determine whether the statement is true or false....Ch. 15.1 - Confirm that is a potential function for F(r) on...Ch. 15.1 - Confirm that is a potential function for F(r) on...Ch. 15.1 - Find div F and curl F . F(x,y,z)=x2i2j+yzkCh. 15.1 - Find div F and curl F . F(x,y,z)=xz3i+2y4x2j+5z2ykCh. 15.1 - Find div and curl .
Ch. 15.1 - Find div and curl .
Ch. 15.1 - Find div F and curl F ....Ch. 15.1 - Find div F and curl F ....Ch. 15.1 - Find(FG).F(x,y,z)=2xi+j+4ykG(x,y,z)=xi+yjzkCh. 15.1 - Find(FG).F(x,y,z)=yzi+xzj+xykG(x,y,z)=xyj+xyzkCh. 15.1 - Find(F).F(x,y,z)=sinxi+cos(xy)j+zkCh. 15.1 - Find(F).F(x,y,z)=exzi+3xeyjeyzkCh. 15.1 - Find(F).F(x,y,z)=xyj+xyzkCh. 15.1 - Find(F).F(x,y,z)=y2xi3yzj+xykCh. 15.1 - Use a CAS to check the calculations in Exercises...Ch. 15.1 - Use a CAS to check the calculations in Exercises...Ch. 15.1 - Let k be a constant, F=F(x,y,z),G=G(x,y,z), and...Ch. 15.1 - Let k be a constant, F=F(x,y,z),G=G(x,y,z), and...Ch. 15.1 - Let k be a constant, F=F(x,y,z),G=G(x,y,z), and...Ch. 15.1 - Let k be a constant, F=F(x,y,z),G=G(x,y,z), and...Ch. 15.1 - Let k be a constant, F=F(x,y,z),G=G(x,y,z), and...Ch. 15.1 - Let k be a constant, F=F(x,y,z),G=G(x,y,z), and...Ch. 15.1 - Let k be a constant, F=F(x,y,z),G=G(x,y,z), and...Ch. 15.1 - Let k be a constant, F=F(x,y,z),G=G(x,y,z), and...Ch. 15.1 - Rewrite the identities in Exercises 31, 33, 35,...Ch. 15.1 - Rewrite the identities in Exercises 32, 34, 36,...Ch. 15.1 - Verify that the radius vector r=xi+yj+zk has the...Ch. 15.1 - Verify that the radius vector r=xi+yj+zk has the...Ch. 15.1 - Letr=xi+yj+zk,letr=r,letf be a differentiable...Ch. 15.1 - (a) Use part (a) of Exercise 43, Exercise 36, and...Ch. 15.1 - Use the result in Exercise 43(b) to show that the...Ch. 15.1 - Use the result of Exercise 43(b) to show that if F...Ch. 15.1 - A curve C is called a flow line of a vector field...Ch. 15.1 - Find a differential equation satisfied by the flow...Ch. 15.1 - Find a differential equation satisfied by the flow...Ch. 15.1 - Find a differential equation satisfied by the flow...Ch. 15.2 - The area of the surface extending upward from the...Ch. 15.2 - Suppose that a wire has equation y=1x(0x1) and...Ch. 15.2 - If C is the curve represented by the equations...Ch. 15.2 - Prob. 4QCECh. 15.2 - Let C be the line segment from (0.0)to(0,1). In...Ch. 15.2 - Let C be the line segment from (0,2)to(0,4). ln...Ch. 15.2 - Evaluate CFdr by inspection for the force field...Ch. 15.2 - Evaluate CFdr by inspection for the force field...Ch. 15.2 - Use (30) to explain why the line integral in part...Ch. 15.2 - (a) Use (30) to explain why the line integral in...Ch. 15.2 - Evaluate CFdr along the line segment C from PtoQ....Ch. 15.2 - Evaluate CFdr along the line segment C from PtoQ....Ch. 15.2 - Evaluate CFdr along the line segment C from PtoQ....Ch. 15.2 - Evaluate CFdr along the line segment C from PtoQ....Ch. 15.2 - Let C be the curve represented by the equations...Ch. 15.2 - Let C be the curve represented by the equations...Ch. 15.2 - In each part, evaluate the integral...Ch. 15.2 - In each part, evaluate the integral Cydx+zdyxdz...Ch. 15.2 - Prob. 15ESCh. 15.2 - Determine whether the statement is true or false....Ch. 15.2 - Determine whether the statement is true or false....Ch. 15.2 - If a smooth oriented curve C in the xy-plane is a...Ch. 15.2 - Evaluate the line integral with respect to s along...Ch. 15.2 - Evaluate the line integral with respect to s along...Ch. 15.2 - Evaluate the line integral with respect to s along...Ch. 15.2 - Evaluate the line integral with respect to s along...Ch. 15.2 - Evaluate the line integral along the curve C....Ch. 15.2 - Evaluate the line integral along the curve C....Ch. 15.2 - Evaluate the line integral along the curve C....Ch. 15.2 - Evaluate the line integral along the curve C....Ch. 15.2 - Evaluate the line integral along the curve C....Ch. 15.2 - Evaluate the line integral along the curve C....Ch. 15.2 - Evaluate the line integral along the curve C....Ch. 15.2 - Evaluate the line integral along the curve C....Ch. 15.2 - Use a CAS to evaluate the line integrals along the...Ch. 15.2 - Use a CAS to evaluate the line integrals along the...Ch. 15.2 - Evaluate Cydxxdy along the curve C shown in the...Ch. 15.2 - Evaluate Cydxxdy along the curve C shown in the...Ch. 15.2 - Evaluate Cx2zdxyx2dy+3dz along the curve C shown...Ch. 15.2 - Evaluate Cx2zdxyx2dy+3dz along the curve C shown...Ch. 15.2 - Evaluate CFdr along the curve C....Ch. 15.2 - Evaluate CFdr along the curves C....Ch. 15.2 - Evaluate CFdr along the curves C....Ch. 15.2 - Prob. 40ESCh. 15.2 - Find the mass of a thin wire shaped in the form of...Ch. 15.2 - Find the mass of a thin wire shaped in the form of...Ch. 15.2 - Find the mass of a thin wire shaped in the form of...Ch. 15.2 - Find the mass of a thin wire shaped in the form of...Ch. 15.2 - Find the work done by the force field F on a...Ch. 15.2 - Find the work done by the force F on a particle...Ch. 15.2 - Find the work done by the force field F on a...Ch. 15.2 - Find the work done by the force field F on a...Ch. 15.2 - Find the work done by the force field...Ch. 15.2 - Find the work done by the force field...Ch. 15.2 - Use a line integral to find the area of the...Ch. 15.2 - Use a line integral to find the area of the...Ch. 15.2 - As illustrated in the accompanying figure, a...Ch. 15.2 - Evaluate the integral Cxdyydxx2+y2, where C is the...Ch. 15.2 - Suppose that a particle moves through the force...Ch. 15.2 - A farmer weighting 150 lb carries a sack of gain...Ch. 15.2 - Suppose that a curve C in the xy-plane is smoothly...Ch. 15.3 - If C is a piecewise smooth curve from...Ch. 15.3 - Prob. 2QCECh. 15.3 - A potential function for the vector field...Ch. 15.3 - If a, b. and c are nonzero real numbers such that...Ch. 15.3 - Determine whether F is a conservative vector...Ch. 15.3 - Determine whether F is a conservative vector...Ch. 15.3 - Determine whether F is a conservative vector...Ch. 15.3 - Determine whether F is a conservative vector...Ch. 15.3 - Determine whether F is a conservative vector...Ch. 15.3 - Determine whether F is a conservative vector...Ch. 15.3 - In each part, evaluate C2xy3dx+1+3x2y2dy over the...Ch. 15.3 - Prob. 8ESCh. 15.3 - Show that the integral is independent of the path,...Ch. 15.3 - Show that the integral is independent of the path,...Ch. 15.3 - Show that the integral is independent of the path,...Ch. 15.3 - Show that the integral is independent of the path,...Ch. 15.3 - Show that the integral is independent of the path,...Ch. 15.3 - Show that the integral is independent of the path,...Ch. 15.3 - Confirm that the force field F is conservative in...Ch. 15.3 - Confirm that the force field F is conservative in...Ch. 15.3 - Confirm that the force field F is conservative in...Ch. 15.3 - Confirm that the force field F is conservative in...Ch. 15.3 - Determine whether the statement is true or false....Ch. 15.3 - Determine whether the statement is true or false....Ch. 15.3 - Determine whether the statement is true or false....Ch. 15.3 - Prob. 22ESCh. 15.3 - Find the exact value of CFdr using any method....Ch. 15.3 - Find the exact value of CFdr using any method....Ch. 15.3 - Use the numerical integration capability of a CAS...Ch. 15.3 - Use the numerical integration capability of a CAS...Ch. 15.3 - Is the vector field conservative? Explain.Ch. 15.3 - Is the vector field conservative? Explain.Ch. 15.3 - Prove: If Fx,y,z=fx,y,zi+gx,y,zj+hx,y,zk is a...Ch. 15.3 - Use the result in Exercise 31 to show that the...Ch. 15.3 - Find a nonzero function h for which...Ch. 15.3 - (a) In Example 3 of Section 15.1 we showed that...Ch. 15.3 - Use the result in Exercise 34(b). In each part,...Ch. 15.3 - Use the result in Exercise 34(b). Let...Ch. 15.3 - Prove Theorem 15.3.1 if C is a piecewise smooth...Ch. 15.3 - Prove that (b) implies (c) in Theorem 15.3.2.Ch. 15.3 - Complete the proof of Theorem 15.3.2 by showing...Ch. 15.4 - If C is the square with vertices 1,1 oriented...Ch. 15.4 - If C is the triangle with vertices 0,0,1,0,and1,1...Ch. 15.4 - Prob. 3QCECh. 15.4 - What region R and choice of functions fx,yandgx,y...Ch. 15.4 - Evaluate the line integral using Green’s Theorem...Ch. 15.4 - Prob. 2ESCh. 15.4 - Use Green’s Theorem to evaluate the integral. In...Ch. 15.4 - Use Green’s Theorem to evaluate the integral. In...Ch. 15.4 - Use Green’s Theorem to evaluate the integral. In...Ch. 15.4 - Use Green’s Theorem to evaluate the integral. In...Ch. 15.4 - Use Green’s Theorem to evaluate the integral. In...Ch. 15.4 - Use Green’s Theorem to evaluate the integral. In...Ch. 15.4 - Use Green’s Theorem to evaluate the integral. In...Ch. 15.4 - Use Green’s Theorem to evaluate the integral. In...Ch. 15.4 - Use Green’s Theorem to evaluate the integral. In...Ch. 15.4 - Use Green’s Theorem to evaluate the integral. In...Ch. 15.4 - Use Green’s Theorem to evaluate the integral. In...Ch. 15.4 - Let C be the boundary of the region enclosed...Ch. 15.4 - Determine whether the statement is true or false....Ch. 15.4 - Determine whether the statement is true or false....Ch. 15.4 - Determine whether the statement is true or false....Ch. 15.4 - Determine whether the statement is true or false....Ch. 15.4 - Use a CAS to check Green’s Theorem by evaluating...Ch. 15.4 - In Example 3, we used Green’s Theorem to obtain...Ch. 15.4 - Use a line integral to find the area of the region...Ch. 15.4 - Use a line integral to find the area of the...Ch. 15.4 - Use the formula A=12Cydx+xdy to find the area of...Ch. 15.4 - Use the formula A=12Cydx+xdy to find the area of...Ch. 15.4 - Suppose that Fx,y=fx,yi+gx,yj is a vector field...Ch. 15.4 - Suppose that Fx,y,=fx,yi+gx,yj is a vector field...Ch. 15.4 - Prob. 27ESCh. 15.4 - In the accompanying figure, C is a smooth oriented...Ch. 15.4 - Use Green’s Theorem to find the work done by the...Ch. 15.4 - Use Green’s Theorem to find the work done by the...Ch. 15.4 - Evaluate Cydxxdy, where C is cardioid r=a1+cos02Ch. 15.4 - Let R be a plane region with area A whose boundary...Ch. 15.4 - Use the result in Exercise 32 to find the centroid...Ch. 15.4 - Use the result in Exercise 32 to find the centroid...Ch. 15.4 - Use the result in Exercise 32 to find the centroid...Ch. 15.4 - Use the result in Exercise 32 to find the centroid...Ch. 15.4 - Find a simple closed curve C with counterclockwise...Ch. 15.4 - (a) Let C be the line segment from a point a,b to...Ch. 15.4 - Evaluate the integral CFdr, where C is the...Ch. 15.4 - Evaluate the integral CFdr, where C is the...Ch. 15.4 - Discuss the role of the Fundamental Theorem of...Ch. 15.5 - Consider the surface integral fx,y,zdS. (a) If is...Ch. 15.5 - If is the triangular region with vertices...Ch. 15.5 - If is the sphere of radius 2 centered at the...Ch. 15.5 - If fx,y,z is the mass density function of a curved...Ch. 15.5 - Evaluate the surface integral fx,y,zdS fx,y,z=x2;...Ch. 15.5 - Evaluate the surface integral fx,y,zdS fx,y,z=xy;...Ch. 15.5 - Evaluate the surface integral fx,y,zdS fx,y,z=x2y;...Ch. 15.5 - Evaluate the surface integral fx,y,zdS...Ch. 15.5 - Evaluate the surface integral fx,y,zdS fx,y,z=xyz;...Ch. 15.5 - Evaluate the surface integral fx,y,zdS fx,y,z=x+y;...Ch. 15.5 - Evaluate the surface integral fx,y,zdS...Ch. 15.5 - Evaluate the surface integral fx,y,zdS...Ch. 15.5 - Determine whether the statement is true or false....Ch. 15.5 - Determine whether the statement is true or false....Ch. 15.5 - Determine whether the statement is true or false....Ch. 15.5 - Determine whether the statement is true or false....Ch. 15.5 - Sometimes evaluating a surface integral results in...Ch. 15.5 - Sometimes evaluating a surface integral results in...Ch. 15.5 - In some cases it is possible to use Definition...Ch. 15.5 - In some cases it is possible to use Definition...Ch. 15.5 - In some cases it is possible to use Definition...Ch. 15.5 - Set up, but do not evaluate, an iterated integral...Ch. 15.5 - Set up, but do not evaluate, an iterated integral...Ch. 15.5 - Use a CAS to confirm that the three integrals you...Ch. 15.5 - Try to confirm with a CAS that the three integrals...Ch. 15.5 - Set up, but do not evaluate, two different...Ch. 15.5 - Set up, but do not evaluate, two different...Ch. 15.5 - Use a CAS to confirm that the two integrals you...Ch. 15.5 - Use a CAS to find the value of the surface...Ch. 15.5 - Find the mass of the lamina with constant density...Ch. 15.5 - Find the mass of the lamina with constant density...Ch. 15.5 - Find the mass of the lamina that is the portion of...Ch. 15.5 - Find the mass of the lamina that is the portion of...Ch. 15.5 - If a curved lamina has constant density 0, what...Ch. 15.5 - Show that if the density of the lamina x2+y2+z2=a2...Ch. 15.5 - The centroid of a surface is defined by...Ch. 15.5 - The centroid of a surface is defined by...Ch. 15.5 - Evaluate the integral fx,y,zdS over the surface ...Ch. 15.5 - Prob. 36ESCh. 15.5 - Prob. 37ESCh. 15.5 - Prob. 38ESCh. 15.5 - Use a CAS to approximate the mass of the curved...Ch. 15.5 - The surface shown in the accompanying figure on...Ch. 15.5 - Discuss the similarities and differences between...Ch. 15.6 - In these exercises, F(x,y,z) denotes a vector...Ch. 15.6 - In these exercises, F(x,y,z) denotes a vector...Ch. 15.6 - In these exercises, F(x,y,z) denotes a vector...Ch. 15.6 - In these exercises, F(x,y,z) denotes a vector...Ch. 15.6 - Suppose that the surface of the unit cube in the...Ch. 15.6 - Find the flux of the constant vector field...Ch. 15.6 - Find the flux of F(x,y,z)=xi through a square of...Ch. 15.6 - Find the flux of F(x,y,z)=(y+1)j through a square...Ch. 15.6 - Find the flux of Fx,y,z=xi+yj+z2+4k through a 23...Ch. 15.6 - Find the flux of F(x,y,z)=2i+3j through a disk of...Ch. 15.6 - Find the flux of F(x,y,z)=9j+8k through a disk of...Ch. 15.6 - Prob. 8ESCh. 15.6 - Find the flux of the vector field F across ....Ch. 15.6 - Find the flux of the vector field F across ....Ch. 15.6 - Find the flux of the vector field F across ....Ch. 15.6 - Find the flux of the vector field F across ....Ch. 15.6 - Find the flux of the vector field F across ....Ch. 15.6 - Find the flux of the vector field F across ....Ch. 15.6 - Find the flux of the vector field F across ....Ch. 15.6 - Find the flux of the vector field F across ....Ch. 15.6 - Find the flux of the vector field F across in the...Ch. 15.6 - Find the flux of the vector field F across in the...Ch. 15.6 - Find the flux of the vector field F across in the...Ch. 15.6 - Find the flux of the vector field F across in the...Ch. 15.6 - Let be the surface of the cube bounded by the...Ch. 15.6 - Let be the closed surface consisting of the...Ch. 15.6 - Determine whether the statement is true or false....Ch. 15.6 - Determine whether the statement is true or false....Ch. 15.6 - Determine whether the statement is true or false....Ch. 15.6 - Determine whether the statement is true or false....Ch. 15.6 - Find the flux of F across the surface by...Ch. 15.6 - Find the flux of F across the surface by...Ch. 15.6 - Let x,y,andz be measured in meters, and suppose...Ch. 15.6 - Let x,y,andz be measured in meters, and suppose...Ch. 15.6 - (a) Derive the analogs of Formulas (12) and (13)...Ch. 15.6 - (a) Derive the analogs of Formulas (12) and (13)...Ch. 15.6 - Let F=rkr,wherer=xi+yj+zkandkisaconstant. (Note...Ch. 15.6 - Let F(x,y,z)=a2xi+(y/a)j+az2k and let be the...Ch. 15.6 - Let F(x,y,z)=6a+1xi4ayj+a2zk and let be the...Ch. 15.6 - Discuss the similarities and differences between...Ch. 15.6 - Write a paragraph explaining the concept of flux...Ch. 15.7 - Let G be a solid whose surface is oriented outward...Ch. 15.7 - The outward flux of Fx,y,z=xi+yj+zk across any...Ch. 15.7 - If Fx,y,z is the velocity vector field for a...Ch. 15.7 - If F(r)=cr3r is an inverse-square field, and if ...Ch. 15.7 - Verify Formula (1) in the Divergence Theorem by...Ch. 15.7 - Verify Formula (1) in the Divergence Theorem by...Ch. 15.7 - Verify Formula (1) in the Divergence Theorem by...Ch. 15.7 - Verify Formula (1) in the Divergence Theorem by...Ch. 15.7 - Determine whether the statement is true or false....Ch. 15.7 - Determine whether the statement is true or false....Ch. 15.7 - Determine whether the statement is true or false....Ch. 15.7 - Determine whether the statement is true or false....Ch. 15.7 - Use the Divergence Theorem to find the flux of F...Ch. 15.7 - Use the Divergence Theorem to find the flux of F...Ch. 15.7 - Use the Divergence Theorem to find the flux of F...Ch. 15.7 - Use the Divergence Theorem to find the flux of F...Ch. 15.7 - Use the Divergence Theorem to find the flux of F...Ch. 15.7 - Use the Divergence Theorem to find the flux of F...Ch. 15.7 - Use the Divergence Theorem to find the flux of F...Ch. 15.7 - Use the Divergence Theorem to find the flux of F...Ch. 15.7 - Use the Divergence Theorem to find the flux of F...Ch. 15.7 - Use the Divergence Theorem to find the flux of F...Ch. 15.7 - Use the Divergence Theorem to find the flux of F...Ch. 15.7 - Prove that if r=xi+yj+zkand is the surface of a...Ch. 15.7 - Use the result in Exercise 20 to find the outward...Ch. 15.7 - Let Fx,y,z=ai+bj+ck be a constant vector field and...Ch. 15.7 - Find a vector field Fx,y,z that has (a) positive...Ch. 15.7 - Let Fx,y,z be a nonzero vector field in 3-space...Ch. 15.7 - Does the result in Exercise 25 remain true if the...Ch. 15.7 - Prove the identity, assuming that F, , and G...Ch. 15.7 - Prove the identity, assuming that F, , and G...Ch. 15.7 - Prove the identity, assuming that F, , and G...Ch. 15.7 - Prove the identity, assuming that F, , and G...Ch. 15.7 - Prove the identity, assuming that F, , and G...Ch. 15.7 - Use the Divergence Theorem to find all positive...Ch. 15.7 - Determine whether the vector field Fx,y,z is free...Ch. 15.7 - Determine whether the vector field Fx,y,z is free...Ch. 15.7 - Determine whether the vector field Fx,y,z is free...Ch. 15.7 - Determine whether the vector field Fx,y,z is free...Ch. 15.7 - Let be the surface of the solid G that is...Ch. 15.8 - Let be a piecewise smooth oriented surface that...Ch. 15.8 - We showed in Example 2 that the vector field...Ch. 15.8 - (a) If 1and2 are two oriented surface that have...Ch. 15.8 - For steady-state flow, the maximum circulation...Ch. 15.8 - Verify Formula (2) in stokes’ Theorem by...Ch. 15.8 - Verify Formula (2) in stokes’ Theorem by...Ch. 15.8 - Verify Formula (2) in stokes’ Theorem by...Ch. 15.8 - Verify Formula (2) in stokes’ Theorem by...Ch. 15.8 - Use Stokes’s Theorem to evaluate CF.dr....Ch. 15.8 - Use Stokes’s Theorem to evaluate CF.dr....Ch. 15.8 - Use Stokes’s Theorem to evaluate CF.dr....Ch. 15.8 - Use Stokes’s Theorem to evaluate CF.dr....Ch. 15.8 - Use Stokes’s Theorem to evaluate CF.dr....Ch. 15.8 - Use Stokes’s Theorem to evaluate CF.dr....Ch. 15.8 - Use Stokes’s Theorem to evaluate CF.dr....Ch. 15.8 - Use Stokes’s Theorem to evaluate CF.dr....Ch. 15.8 - Determine whether the statement is true or false....Ch. 15.8 - Determine whether the statement is true or false....Ch. 15.8 - Determine whether the statement is true or false....Ch. 15.8 - Determine whether the statement is true or false....Ch. 15.8 - Consider the vector field given by the formula...Ch. 15.8 - (a) Let denote the surface of a solid G with n...Ch. 15.8 - The figures in these exercises show a horizontal...Ch. 15.8 - The figures in these exercises show a horizontal...Ch. 15.8 - Let F(x,y,z) be a conservation vector field in...Ch. 15.8 - In 1831 the physicist Michael Faraday discovered...Ch. 15.8 - Let be the portion of the paraboloid z=1x2y2 for...Ch. 15.8 - Discuss what it mean to say that the curl of a...Ch. 15.8 - Compare and contrast the Fundamental Theorem of...Ch. 15 - In words, what is a vector field? Give some...Ch. 15 - (a) Give a physical example of an inverse-square...Ch. 15 - Find an explicit coordinate expression for the...Ch. 15 - Find x+yxy.Ch. 15 - Find curl zi+xj+yk.Ch. 15 - Let Fx,y,z=xx2+y2i+yx2+y2j+zx2+y2k Sketch the...Ch. 15 - Assume that C is the parametric curve x=xt,y=yt,...Ch. 15 - (a) Express the mass M of a thin wire in 3-space...Ch. 15 - Give a physical interpretation of CFTds.Ch. 15 - State some alternative notations for CFTds.Ch. 15 - Evaluate the line integral. Cxyds;C:x2+y2=1Ch. 15 - Evaluate the line integral....Ch. 15 - Evaluate the line integral....Ch. 15 - Find the work done by the force field Fx,y=y2i+xyj...Ch. 15 - State the Fundamental Theorem of Line Integrals,...Ch. 15 - Evaluate Cfdr where fx,y,z=xy2z3 and...Ch. 15 - Let Fx,y=yi2xj. (a) Find a nonzero function hx...Ch. 15 - Let Fx,y=yexy1i+xexyj. (a) Show that F is a...Ch. 15 - State Green's Theorem, including all of the...Ch. 15 - Express the area of a plane region as a line...Ch. 15 - Let and denote angles that satisfy 02 and assume...Ch. 15 - (a) Use Green's Theorem to prove that Cfxdx+gydy=0...Ch. 15 - Assume that is the parametric surface...Ch. 15 - Evaluate zdS;:x2+y2=10z1.Ch. 15 - Do you think that the surface in the accompanying...Ch. 15 - Give a physical interpretation of FndS.Ch. 15 - Find the flux of Fx,y,z=xi+yj+2zk through the...Ch. 15 - Find the flux of Fx,y,z=xi+2yj+3zk through the...Ch. 15 - State the Divergence Theorem and Stokes' Theorem,...Ch. 15 - Let G be a solid with the surface oriented by...Ch. 15 - Let be the sphere x2+y2+z2=1, let n be an inward...Ch. 15 - Use Stokes' Theorem to evaluate curlFndS where...Ch. 15 - Prob. 33RECh. 15 - With the aid of Exercise 33, determine whether F...Ch. 15 - With the aid of Exercise 33, determine whether F...Ch. 15 - As discussed in Example 1 of Section 15.1,...
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- 3. Consider the sequences of functions f₁: [-π, π] → R, sin(n²x) An(2) n f pointwise as (i) Find a function ƒ : [-T,π] → R such that fn n∞. Further, show that fn →f uniformly on [-π,π] as n → ∞. [20 Marks] (ii) Does the sequence of derivatives f(x) has a pointwise limit on [-7, 7]? Justify your answer. [10 Marks]arrow_forward1. (i) Give the definition of a metric on a set X. [5 Marks] (ii) Let X = {a, b, c} and let a function d : XxX → [0, ∞) be defined as d(a, a) = d(b,b) = d(c, c) 0, d(a, c) = d(c, a) 1, d(a, b) = d(b, a) = 4, d(b, c) = d(c,b) = 2. Decide whether d is a metric on X. Justify your answer. = (iii) Consider a metric space (R, d.), where = [10 Marks] 0 if x = y, d* (x, y) 5 if xy. In the metric space (R, d*), describe: (a) open ball B2(0) of radius 2 centred at 0; (b) closed ball B5(0) of radius 5 centred at 0; (c) sphere S10 (0) of radius 10 centred at 0. [5 Marks] [5 Marks] [5 Marks]arrow_forward(c) sphere S10 (0) of radius 10 centred at 0. [5 Marks] 2. Let C([a, b]) be the metric space of continuous functions on the interval [a, b] with the metric doo (f,g) = max f(x)g(x)|. xЄ[a,b] = 1x. Find: Let f(x) = 1 - x² and g(x): (i) do(f, g) in C'([0, 1]); (ii) do(f,g) in C([−1, 1]). [20 Marks] [20 Marks]arrow_forward
- Given lim x-4 f (x) = 1,limx-49 (x) = 10, and lim→-4 h (x) = -7 use the limit properties to find lim→-4 1 [2h (x) — h(x) + 7 f(x)] : - h(x)+7f(x) 3 O DNEarrow_forward17. Suppose we know that the graph below is the graph of a solution to dy/dt = f(t). (a) How much of the slope field can you sketch from this information? [Hint: Note that the differential equation depends only on t.] (b) What can you say about the solu- tion with y(0) = 2? (For example, can you sketch the graph of this so- lution?) y(0) = 1 y ANarrow_forward(b) Find the (instantaneous) rate of change of y at x = 5. In the previous part, we found the average rate of change for several intervals of decreasing size starting at x = 5. The instantaneous rate of change of fat x = 5 is the limit of the average rate of change over the interval [x, x + h] as h approaches 0. This is given by the derivative in the following limit. lim h→0 - f(x + h) − f(x) h The first step to find this limit is to compute f(x + h). Recall that this means replacing the input variable x with the expression x + h in the rule defining f. f(x + h) = (x + h)² - 5(x+ h) = 2xh+h2_ x² + 2xh + h² 5✔ - 5 )x - 5h Step 4 - The second step for finding the derivative of fat x is to find the difference f(x + h) − f(x). - f(x + h) f(x) = = (x² x² + 2xh + h² - ])- = 2x + h² - 5h ])x-5h) - (x² - 5x) = ]) (2x + h - 5) Macbook Proarrow_forward
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