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CalculusMultivariable CalculusSuppose S and E satisfy the conditions of the Divergence Theorem and f is a scalar function with continuous partial derivatives. Prove that ∬ S f n d S = ∭ E ∇ f d V These surface and triple integrals of vector functions are vectors defined by integrating each component function. [ Hint: Start by applying the Divergence Theorem to F = f c, where c is an arbitrary constant vector.]BuyFind*arrow_forward*

8th Edition

James Stewart

Publisher: Cengage Learning

ISBN: 9781305266643

Chapter 16.9, Problem 31E

Textbook Problem

Suppose *S* and *E* satisfy the conditions of the Divergence Theorem and *f* is a scalar function with continuous partial derivatives. Prove that
*Hint:* Start by applying the Divergence Theorem to F = *f*c, where c is an arbitrary constant vector.]

Multivariable Calculus

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Ch. 16.1 - Sketch the vector field F by drawing a diagram...Ch. 16.1 - Sketch the vector field F by drawing a diagram...Ch. 16.1 - Sketch the vector field F by drawing a diagram...Ch. 16.1 - Sketch the vector field F by drawing a diagram...Ch. 16.1 - Sketch the vector field F by drawing a diagram...Ch. 16.1 - Sketch the vector field F by drawing a diagram...Ch. 16.1 - Sketch the vector field F by drawing a diagram...Ch. 16.1 - Sketch the vector field F by drawing a diagram...Ch. 16.1 - Sketch the vector field F by drawing a diagram...Ch. 16.1 - Sketch the vector field F by drawing a diagram...

Ch. 16.1 - Match the vector fields F with the plots labeled...Ch. 16.1 - Match the vector fields F with the plots labeled...Ch. 16.1 - Match the vector fields F with the plots labeled...Ch. 16.1 - Match the vector fields F with the plots labeled...Ch. 16.1 - Match the vector fields F on 3 with the plots...Ch. 16.1 - Match the vector fields F on 3 with the plots...Ch. 16.1 - Match the vector fields F on 3 with the plots...Ch. 16.1 - Match the vector fields F on 3 with the plots...Ch. 16.1 - Find the gradient vector field of f. 21. f(x, y) =...Ch. 16.1 - Find the gradient vector field of f. 22. f(s, t) =...Ch. 16.1 - Find the gradient vector field of f. 23. f(x, y,...Ch. 16.1 - Find the gradient vector field of f. 24. f(x, y,...Ch. 16.1 - Find the gradient vector field f of f and sketch...Ch. 16.1 - Find the gradient vector field f of f and sketch...Ch. 16.1 - Match the functions f with the plots of their...Ch. 16.1 - Match the functions f with the plots of their...Ch. 16.1 - Match the functions f with the plots of their...Ch. 16.1 - Match the functions f with the plots of their...Ch. 16.1 - A particle moves in a velocity field V(x, y) = x2,...Ch. 16.1 - At time t = 1, a particle is located at position...Ch. 16.1 - The flow lines (or streamlines) of a vector field...Ch. 16.1 - (a) Sketch the vector field F(x, y) = i + x j and...Ch. 16.2 - Evaluate the line integral, where C is the given...Ch. 16.2 - Evaluate the line integral, where C is the given...Ch. 16.2 - Evaluate the line integral, where C is the given...Ch. 16.2 - Evaluate the line integral, where C is the given...Ch. 16.2 - Evaluate the line integral, where C is the given...Ch. 16.2 - Evaluate the line integral, where C is the given...Ch. 16.2 - Evaluate the line integral, where C is the given...Ch. 16.2 - Evaluate the line integral, where C is the given...Ch. 16.2 - Evaluate the line integral, where C is the given...Ch. 16.2 - Evaluate the line integral, where C is the given...Ch. 16.2 - Evaluate the line integral, where C is the given...Ch. 16.2 - Evaluate the line integral, where C is the given...Ch. 16.2 - Evaluate the line integral, where C is the given...Ch. 16.2 - Evaluate the line integral, where C is the given...Ch. 16.2 - Evaluate the line integral, where C is the given...Ch. 16.2 - Evaluate the line integral, where C is the given...Ch. 16.2 - Let F be the vector field shown in the figure. (a)...Ch. 16.2 - The figure shows a vector field F and two curves...Ch. 16.2 - Evaluate the line integral C F dr, where C is...Ch. 16.2 - Evaluate the line integral C F dr, where C is...Ch. 16.2 - Evaluate the line integral C F dr, where C is...Ch. 16.2 - Evaluate the line integral C F dr, where C is...Ch. 16.2 - Use a calculator to evaluate the line integral...Ch. 16.2 - Use a calculator to evaluate the line integral...Ch. 16.2 - Use a calculator to evaluate the line integral...Ch. 16.2 - Use a calculator to evaluate the line integral...Ch. 16.2 - Find the exact value of C x3y2 z ds, where C is...Ch. 16.2 - (a) Find the work done by the force field F(x, y)...Ch. 16.2 - A thin wire is bent into the shape of a semicircle...Ch. 16.2 - A thin wire has the shape of the first-quadrant...Ch. 16.2 - (a) Write the formulas similar to Equations 4 for...Ch. 16.2 - Find the mass and center of mass of a wire in the...Ch. 16.2 - If a wire with linear density (x, y) lies along a...Ch. 16.2 - If a wire with linear density (x, y, z) lies along...Ch. 16.2 - Find the work done by the force field F(x, y) = x...Ch. 16.2 - Find the work done by the force field F(x, y) = x2...Ch. 16.2 - Find the work done by the force field F(x, y, z) =...Ch. 16.2 - The force exerted by an electric charge at the...Ch. 16.2 - The position of an object with mass m at time t is...Ch. 16.2 - An object with mass m moves with position function...Ch. 16.2 - A 160-lb man carries a 25-lb can of paint up a...Ch. 16.2 - Suppose there is a hole in the can of paint in...Ch. 16.2 - (a) Show that a constant force field does zero...Ch. 16.2 - The base of a circular fence with radius 10 m is...Ch. 16.2 - If C is a smooth curve given by a vector function...Ch. 16.2 - If C is a smooth curve given by a vector function...Ch. 16.2 - An object moves along the curve C shown in the...Ch. 16.2 - Experiments show that a steady current I in a long...Ch. 16.3 - The figure shows a curve C and a contour map of a...Ch. 16.3 - A table of values of a function f with continuous...Ch. 16.3 - Determine whether or not F is a conservative...Ch. 16.3 - Determine whether or not F is a conservative...Ch. 16.3 - Determine whether or not F is a conservative...Ch. 16.3 - Determine whether or not F is a conservative...Ch. 16.3 - Determine whether or not F is a conservative...Ch. 16.3 - Determine whether or not F is a conservative...Ch. 16.3 - Determine whether or not F is a conservative...Ch. 16.3 - Determine whether or not F is a conservative...Ch. 16.3 - The figure shows the vector field F(x, y) = 2xy,...Ch. 16.3 - (a) Find a function f such that F = f and (b) use...Ch. 16.3 - (a) Find a function f such that F = f and (b) use...Ch. 16.3 - (a) Find a function f such that F = f and (b) use...Ch. 16.3 - (a) Find a function f such that F = f and (b) use...Ch. 16.3 - (a) Find a function f such that F = f and (b) use...Ch. 16.3 - (a) Find a function f such that F = f and (b) use...Ch. 16.3 - (a) Find a function f such that F = f and (b) use...Ch. 16.3 - Show that the line integral is independent of path...Ch. 16.3 - Show that the line integral is independent of path...Ch. 16.3 - Suppose youre asked to determine the curve that...Ch. 16.3 - Suppose an experiment determines that the amount...Ch. 16.3 - Find the work done by the force field F in moving...Ch. 16.3 - Find the work done by the force field F in moving...Ch. 16.3 - Is the vector field shown in the figure...Ch. 16.3 - Is the vector field shown in the figure...Ch. 16.3 - Let F = f, where f(x, y) = sin(x 2y). Find...Ch. 16.3 - Show that if the vector field F = P i + Q j + R k...Ch. 16.3 - Use Exercise 29 to show that the line integral C y...Ch. 16.3 - Determine whether or not the given set is (a)...Ch. 16.3 - Determine whether or not the given set is (a)...Ch. 16.3 - Determine whether or not the given set is (a)...Ch. 16.3 - Determine whether or not the given set is (a)...Ch. 16.3 - Let F(x, y) = yi+xjx2+y2 (a) Show that P/y=Q/x....Ch. 16.4 - Evaluate the line integral by two methods: (a)...Ch. 16.4 - Evaluate the line integral by two methods: (a)...Ch. 16.4 - Evaluate the line integral by two methods: (a)...Ch. 16.4 - Evaluate the line integral by two methods: (a)...Ch. 16.4 - Use Greens Theorem to evaluate the line integral...Ch. 16.4 - Use Greens Theorem to evaluate the line integral...Ch. 16.4 - Use Greens Theorem to evaluate the line integral...Ch. 16.4 - Use Greens Theorem to evaluate the line integral...Ch. 16.4 - Use Greens Theorem to evaluate the line integral...Ch. 16.4 - Use Greens Theorem to evaluate the line integral...Ch. 16.4 - Use Greens Theorem to evaluate C F dr. (Check the...Ch. 16.4 - Use Greens Theorem to evaluate C F dr. (Check the...Ch. 16.4 - Use Greens Theorem to evaluate C F dr. (Check the...Ch. 16.4 - Use Greens Theorem to evaluate C F dr. (Check the...Ch. 16.4 - Use Greens Theorem to find the work done by the...Ch. 16.4 - A particle starts at the origin, moves along the...Ch. 16.4 - Use one of the formulas in (5) to find the area...Ch. 16.4 - If a circle C with radius 1 rolls along the...Ch. 16.4 - (a) If C is the line segment connecting the point...Ch. 16.4 - Let D be a region bounded by a simple closed path...Ch. 16.4 - Use Exercise 22 to find the centroid of a...Ch. 16.4 - Use Exercise 22 to find the centroid of the...Ch. 16.4 - A plane lamina with constant density (x, y) = ...Ch. 16.4 - Use Exercise 25 to find the moment of inertia of a...Ch. 16.4 - Use the method of Example 5 to calculate C F dr,...Ch. 16.4 - Calculate C F dr, where F(x, y) = x2 + y, 3x y2...Ch. 16.4 - If F is the vector field of Example 5, show that C...Ch. 16.4 - Complete the proof of the special case of Greens...Ch. 16.4 - Use Greens Theorem to prove the change of...Ch. 16.5 - Find (a) the curl and (b) the divergence of the...Ch. 16.5 - Find (a) the curl and (b) the divergence of the...Ch. 16.5 - Find (a) the curl and (b) the divergence of the...Ch. 16.5 - Find (a) the curl and (b) the divergence of the...Ch. 16.5 - Find (a) the curl and (b) the divergence of the...Ch. 16.5 - Find (a) the curl and (b) the divergence of the...Ch. 16.5 - Find (a) the curl and (b) the divergence of the...Ch. 16.5 - Find (a) the curl and (b) the divergence of the...Ch. 16.5 - The vector field F is shown in the xy-plane and...Ch. 16.5 - The vector field F is shown in the xy-plane and...Ch. 16.5 - The vector field F is shown in the xy-plane and...Ch. 16.5 - Let f be a scalar field and F a vector field....Ch. 16.5 - Determine whether or not the vector field is...Ch. 16.5 - Determine whether or not the vector field is...Ch. 16.5 - Determine whether or not the vector field is...Ch. 16.5 - Determine whether or not the vector field is...Ch. 16.5 - Determine whether or not the vector field is...Ch. 16.5 - Determine whether or not the vector field is...Ch. 16.5 - Is there a vector field G on 3 such that curl G =...Ch. 16.5 - Is there a vector field G on 3 such that curl G =...Ch. 16.5 - Show that any vector field of the form F(x, y, z)...Ch. 16.5 - Show that any vector field of the form F(x, y, z)...Ch. 16.5 - Prove the identity, assuming that the appropriate...Ch. 16.5 - Prove the identity, assuming that the appropriate...Ch. 16.5 - Prove the identity, assuming that the appropriate...Ch. 16.5 - Prove the identity, assuming that the appropriate...Ch. 16.5 - Prove the identity, assuming that the appropriate...Ch. 16.5 - Prove the identity, assuming that the appropriate...Ch. 16.5 - Prove the identity, assuming that the appropriate...Ch. 16.5 - Let r = x i + y j + z k and r = |r|. 30. Verify...Ch. 16.5 - Let r = x i + y j + z k and r = |r|. 31. Verify...Ch. 16.5 - Let r = x i + y j + z k and r = |r|. 32. If F =...Ch. 16.5 - Use Greens Theorem in the form of Equation 13 to...Ch. 16.5 - Use Greens first identity (Exercise 33) to prove...Ch. 16.5 - Recall from Section 14.3 that a function g is...Ch. 16.5 - Use Greens first identity to show that if f is...Ch. 16.5 - This exercise demonstrates a connection between...Ch. 16.5 - Maxwells equations relating the electric field E...Ch. 16.5 - We have seen that all vector fields of the form F...Ch. 16.6 - Determine whether the points P and Q lie on the...Ch. 16.6 - Determine whether the points P and Q lie on the...Ch. 16.6 - Identify the surface with the given vector...Ch. 16.6 - Identify the surface with the given vector...Ch. 16.6 - Identify the surface with the given vector...Ch. 16.6 - Identify the surface with the given vector...Ch. 16.6 - Match the equations with the graphs labeled IVI...Ch. 16.6 - Match the equations with the graphs labeled IVI...Ch. 16.6 - Match the equations with the graphs labeled IVI...Ch. 16.6 - Match the equations with the graphs labeled IVI...Ch. 16.6 - Match the equations with the graphs labeled IVI...Ch. 16.6 - Match the equations with the graphs labeled IVI...Ch. 16.6 - Find a parametric representation for the surface....Ch. 16.6 - Find a parametric representation for the surface....Ch. 16.6 - Find a parametric representation for the surface....Ch. 16.6 - Find a parametric representation for the surface....Ch. 16.6 - Find a parametric representation for the surface....Ch. 16.6 - Find a parametric representation for the surface....Ch. 16.6 - Find a parametric representation for the surface....Ch. 16.6 - Find a parametric representation for the surface....Ch. 16.6 - Find parametric equations for the surface obtained...Ch. 16.6 - Find parametric equations for the surface obtained...Ch. 16.6 - Find an equation of the tangent plane to the given...Ch. 16.6 - Find an equation of the tangent plane to the given...Ch. 16.6 - Find an equation of the tangent plane to the given...Ch. 16.6 - Find an equation of the tangent plane to the given...Ch. 16.6 - Find an equation of the tangent plane to the given...Ch. 16.6 - Find an equation of the tangent plane to the given...Ch. 16.6 - Find the area of the surface. 39. The part of the...Ch. 16.6 - Find the area of the surface. 40.The part of the...Ch. 16.6 - Find the area of the surface. 41. The part of the...Ch. 16.6 - Find the area of the surface. 42. The part of the...Ch. 16.6 - Find the area of the surface. 43.The surface z =...Ch. 16.6 - Find the area of the surface. 44. The part of the...Ch. 16.6 - Find the area of the surface. 45. The part of the...Ch. 16.6 - Find the area of the surface. 46. The part of the...Ch. 16.6 - Find the area of the surface. 47. The part of the...Ch. 16.6 - Find the area of the surface. 48.The helicoid (or...Ch. 16.6 - Find the area of the surface. 49. The surface with...Ch. 16.6 - Find the area of the surface. 50.The part of the...Ch. 16.6 - If the equation of a surfaceSis z =f(x,y),...Ch. 16.6 - Find the area of the surface correct to four...Ch. 16.6 - Find the area of the surface correct to four...Ch. 16.6 - Find, to four decimal places, the area of the part...Ch. 16.6 - Find the area of the surface with vector equation...Ch. 16.6 - (a) Show that the parametric equations x...Ch. 16.6 - (a) Show that the parametric equationsx = acosh u...Ch. 16.6 - Find the area of the part of the spherex2+y2+ z2=...Ch. 16.6 - The figure shows the surface created when the...Ch. 16.6 - Find the area of the part of the spherex2+y2+ z2 =...Ch. 16.7 - LetSbe the surface of the box enclosed by the...Ch. 16.7 - A surface S consists of the cylinderx2+ y2=1, 1 z...Ch. 16.7 - LetHbe the hemispherex2+y2+ z2= 50,z 0, and...Ch. 16.7 - Suppose thatf(x, y,z)=g(), where g is a function...Ch. 16.7 - Evaluate the surface integral. 5. s (x + y + z)...Ch. 16.7 - Evaluate the surface integral. 6. s xyz dS, Sis...Ch. 16.7 - Evaluate the surface integral. 7. s y dS,Sis the...Ch. 16.7 - Evaluate the surface integral. 8.s (x2+ y2)dS, Sis...Ch. 16.7 - Evaluate the surface integral. 9. s x2yz dS, Sis...Ch. 16.7 - Evaluate the surface integral. 10. s xz dS, S is...Ch. 16.7 - Evaluate the surface integral. 11. s x dS, S is...Ch. 16.7 - Evaluate the surface integral. 12. s y dS, S is...Ch. 16.7 - Evaluate the surface integral. 13. s z2dS, S is...Ch. 16.7 - Evaluate the surface integral. 14. s y2z2 dS, S is...Ch. 16.7 - Evaluate the surface integral. 15. s x dS, S is...Ch. 16.7 - Evaluate the surface integral. 16 s y2 dS, S is...Ch. 16.7 - Evaluate the surface integral. 17. s (x2z +...Ch. 16.7 - Evaluate the surface integral. 18. s (x + y + z)...Ch. 16.7 - Evaluate the surface integral. 19. s xz dS, S is...Ch. 16.7 - Evaluate the surface integral. 20. s (x2 + y2 +...Ch. 16.7 - Evaluate the surface integral s F dS for the...Ch. 16.7 - Evaluate the surface integral s F dS for the...Ch. 16.7 - Evaluate the surface integral s F dS for the...Ch. 16.7 - Evaluate the surface integral s F dS for the...Ch. 16.7 - Evaluate the surface integral s F dS for the...Ch. 16.7 - Evaluate the surface integral s F dS for the...Ch. 16.7 - Evaluate the surface integral s F dS for the...Ch. 16.7 - Find a formula for s F dS similar to Formula 10...Ch. 16.7 - Find a formula for s F dS similar to Formula 10...Ch. 16.7 - Find the center of mass of the hemisphere x2 + y2...Ch. 16.7 - Find the mass of a thin funnel in the shape of a...Ch. 16.7 - (a) Give an integral expression for the moment of...Ch. 16.7 - Let S be the part of the sphere x2 + y2 + z2 = 25...Ch. 16.7 - A fluid has density 870 kg/m3 and flows with...Ch. 16.7 - Seawater has density 1025 kg/m3 and flows in a...Ch. 16.7 - Use Gausss Law to find the charge contained in the...Ch. 16.7 - Use Gausss Law to find the charge enclosed by the...Ch. 16.7 - The temperature at the point (x, y, z) in a...Ch. 16.7 - The temperature at a point in a ball with...Ch. 16.7 - Let F be an inverse square field, that is, |F(r) =...Ch. 16.8 - 1. A hemisphere H and a portion P of a paraboloid...Ch. 16.8 - Use Stokes Theorem to evaluate s curl F dS. 2....Ch. 16.8 - Use Stokes Theorem to evaluate s curl F dS. 3....Ch. 16.8 - Use Stokes Theorem to evaluate s curl F dS. 4....Ch. 16.8 - (x, y, z) = xyz i + xy j + x2yz k. S consists of...Ch. 16.8 - Use Stokes Theorem to evaluate s curl F dS. 6....Ch. 16.8 - Use Stokes Theorem to evaluate c F dr. In each...Ch. 16.8 - Use Stokes Theorem to evaluate c F dr. In each...Ch. 16.8 - Use Stokes Theorem to evaluate c F dr. In each...Ch. 16.8 - Use Stokes Theorem to evaluate c F dr. In each...Ch. 16.8 - (a) Use Stokes Theorem to evaluate c F dr, where...Ch. 16.8 - (a) Use Stokes Theorem to evaluate c F dr, where...Ch. 16.8 - Verify that Stokes Theorem is true for the given...Ch. 16.8 - Verify that Stokes Theorem is true for the given...Ch. 16.8 - Verify that Stokes Theorem is true for the given...Ch. 16.8 - A particle moves along line segments from the...Ch. 16.8 - Evaluate c (y + sin x) dx + (z2 + cos y) dy + x3...Ch. 16.8 - If S is a sphere and F satisfies the hypotheses of...Ch. 16.8 - Suppose S and C satisfy the hypotheses of Stokes...Ch. 16.9 - Verify that the Divergence Theorem is true for the...Ch. 16.9 - Verify that the Divergence Theorem is true for the...Ch. 16.9 - Verify that the Divergence Theorem is true for the...Ch. 16.9 - Verify that the Divergence Theorem is true for the...Ch. 16.9 - Use the Divergence Theorem to calculate the...Ch. 16.9 - Use the Divergence Theorem to calculate the...Ch. 16.9 - Use the Divergence Theorem to calculate the...Ch. 16.9 - Use the Divergence Theorem to calculate the...Ch. 16.9 - Use the Divergence Theorem to calculate the...Ch. 16.9 - Use the Divergence Theorem to calculate the...Ch. 16.9 - Use the Divergence Theorem to calculate the...Ch. 16.9 - Use the Divergence Theorem to calculate the...Ch. 16.9 - Use the Divergence Theorem to calculate the...Ch. 16.9 - Use the Divergence Theorem to calculate the...Ch. 16.9 - Use the Divergence Theorem to evaluate s F dS,...Ch. 16.9 - Let F(x, y, z) = z tan-1(y2) i + z3 ln(x2 + 1) j +...Ch. 16.9 - A vector field F is shown. Use the interpretation...Ch. 16.9 - (a) Are the points P1 and P2 sources or sinks for...Ch. 16.9 - Verify that div E = 0 for the electric field...Ch. 16.9 - Use the Divergence Theorem to evaluate...Ch. 16.9 - Prove each identity, assuming that S and E satisfy...Ch. 16.9 - Prove each identity, assuming that S and E satisfy...Ch. 16.9 - Prove each identity, assuming that S and E satisfy...Ch. 16.9 - Prove each identity, assuming that S and E satisfy...Ch. 16.9 - Prove each identity, assuming that S and E satisfy...Ch. 16.9 - Prove each identity, assuming that S and E satisfy...Ch. 16.9 - Suppose S and E satisfy the conditions of the...Ch. 16.9 - A solid occupies a region E with surface S and is...Ch. 16 - What is a vector field? Give three examples that...Ch. 16 - (a) What is a conservative vector field? (b) What...Ch. 16 - (a) Write the definition of the line integral of a...Ch. 16 - (a) Define the line integral of a vector field F...Ch. 16 - State the Fundamental Theorem for Line Integrals.Ch. 16 - (a) What does it mean to say that C F dris...Ch. 16 - State Greens Theorem.Ch. 16 - Write expressions for the area enclosed by a curve...Ch. 16 - Suppose F is a vector field on 3. (a) Define curl...Ch. 16 - If F = P i + Q j, how do you determine whether F...Ch. 16 - (a) What is a parametric surface? What arc its...Ch. 16 - (a) Write the definition of the surface integral...Ch. 16 - (a) What is an oriented surface? Give an example...Ch. 16 - State Stokes Theorem.Ch. 16 - State the Divergence Theorem.Ch. 16 - In what ways are the Fundamental Theorem for Line...Ch. 16 - Determine whether the statement is true or false....Ch. 16 - Determine whether the statement is true or false....Ch. 16 - Determine whether the statement is true or false....Ch. 16 - Determine whether the statement is true or false....Ch. 16 - Determine whether the statement is true or false....Ch. 16 - Determine whether the statement is true or false....Ch. 16 - Determine whether the statement is true or false....Ch. 16 - Determine whether the statement is true or false....Ch. 16 - Determine whether the statement is true or false....Ch. 16 - Determine whether the statement is true or false....Ch. 16 - Determine whether the statement is true or false....Ch. 16 - Determine whether the statement is true or false....Ch. 16 - Determine whether the statement is true or false....Ch. 16 - A vector field F, a curve C, and a point P are...Ch. 16 - Evaluate the line integral. 2. C x ds, C is the...Ch. 16 - Evaluate the line integral. 3. C yz cos x ds, C: x...Ch. 16 - Evaluate the line integral. 4. C y dx + (x + y2)...Ch. 16 - Evaluate the line integral. 5. C y3 dx + x2 dy, C...Ch. 16 - Evaluate the line integral. 6. C xy dx + ey dy +...Ch. 16 - Evaluate the line integral. 7. C xy dx + y2 dy +...Ch. 16 - Evaluate the line integral. 8. C F dr, where F(x,...Ch. 16 - Evaluate the line integral. 9. C F dr, where...Ch. 16 - Find the work done by the force field F(x, y, z) =...Ch. 16 - Show that F is a conservative vector field. Then...Ch. 16 - Show that F is a conservative vector field. Then...Ch. 16 - Show that F is a conservative and use this fact to...Ch. 16 - Show that F is a conservative and use this fact to...Ch. 16 - Verify that Greens Theorem is true for the line...Ch. 16 - Use Greens Theorem to evaluate C 1+x3dx + 2xydy...Ch. 16 - Use Greens Theorem to evaluate C x2y dx xy2dy,...Ch. 16 - Find curl F and div F if F(x, y, z) = e-x sin y i...Ch. 16 - Show that there is no vector field G such that...Ch. 16 - If F and G are vector fields whose component...Ch. 16 - If C is any piecewise-smooth simple closed plane...Ch. 16 - If f and g are twice differentiable functions,...Ch. 16 - If f is a harmonic function, that is, 2f = 0, show...Ch. 16 - (a) Sketch the curve C with parametric equations x...Ch. 16 - Find the area of the part of the surface z = x2 +...Ch. 16 - Evaluate the surface integral. 27. S z dS, where S...Ch. 16 - Evaluate the surface integral. 28. s (x2z +...Ch. 16 - Evaluate the surface integral. 29. S F dS, where...Ch. 16 - Evaluate the surface integral. 30. S F dS, where...Ch. 16 - Verify that Stokes Theorem is true for the vector...Ch. 16 - Use Stokes Theorem to evaluate s curl F dS, where...Ch. 16 - Use Stokes Theorem to evaluate C F dr, where F(x,...Ch. 16 - Use the Divergence Theorem to calculate the...Ch. 16 - Verify that the Divergence Theorem is true for the...Ch. 16 - Compute the outward flux of F(x, y, z) =...Ch. 16 - Let F(x, y, z) = (3x2 yz 3y) i + (x3z 3x) j +...Ch. 16 - Let F(x, y) = (2x3+2xy22y)i+(2y3+2x2y+2x)jx2+y2...Ch. 16 - Find S F n dS, where F(x, y, z) = x i + y j + z k...Ch. 16 - If the components of F have continuous second...Ch. 16 - If a is a constant vector, r = x i + y j + z k,...Ch. 16 - 1. Let S be a smooth parametric surface and let P...Ch. 16 - Find the positively oriented simple closed curve C...Ch. 16 - Let C be a simple closed piecewise-smooth space...Ch. 16 - Prove the following identity: (F G) = (F )G + (G...Ch. 16 - The figure depicts the sequence of events in each...

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Probability and Statistics for Engineering and the Sciences

Write each of the following in terms of sin and cos ; then simplify if possible. sectan

Trigonometry (MindTap Course List)

Find the number of sides for a regular polygon whose exterior angles each measure: a 24 b 18

Elementary Geometry For College Students, 7e

In each of the following parts, a relation is defined on the set of all human beings. Determine whether the rel...

Elements Of Modern Algebra

Round each number to a. the nearest tenth and) b. the nearest thousandth: 0.05731

Elementary Technical Mathematics

Set up and solve equations for each of the following business situations. A regular lightbulb uses 20 watts les...

Contemporary Mathematics for Business & Consumers

Use two disjoint sets to show that 0+2=0.

Mathematical Excursions (MindTap Course List)

Trigonometric Substitution In Exercises 14, state the trigonometric substitution you would use to find the inde...

Calculus: Early Transcendental Functions (MindTap Course List)

Evaluate the integral. 46. (x1)exx2dx

Single Variable Calculus: Early Transcendentals

Review In Exercises 53-70, determine the convergence or divergence of the series using any appropriate test fro...

Calculus of a Single Variable

In Exercises 9 to 28, use the drawings as needed to answer the following questions. Suppose that a point C lies...

Elementary Geometry for College Students

An experiment consists of selecting a card from a standard deck of playing cards and noting whether the card is...

Finite Mathematics for the Managerial, Life, and Social Sciences

Direction Cosines Consider the vector v=v1,v2,v3. What is the meaning of arccosv2v=30?

Multivariable Calculus

ln |2x + 1| + C
2 ln |2x + 1| + C

Study Guide for Stewart's Single Variable Calculus: Early Transcendentals, 8th

Solve y″ − 2y′ + 2 = 0, y(0) = 2, y′(0) = 1.
y = e−x(cos x − 2 sin x)
y = e−x(2 cos x − sin x)
y = ex(cos x − 2...

Study Guide for Stewart's Multivariable Calculus, 8th

The following data are a sample of survival times (days from diagnosis) for patients suffering from chronic leu...

Introduction To Statistics And Data Analysis

A researcher has a sample of 30 rats that are all cloned from the same source. The 30 rats are genetically iden...

Research Methods for the Behavioral Sciences (MindTap Course List)

Solve these examples of combined operations. Round the answers to 2 decimal places where necessary. 31. 18.327....

Mathematics For Machine Technology

A researcher claims that intelligence can be measured by measuring the length of a persons right-hand ring fing...

Research Methods for the Behavioral Sciences (MindTap Course List)

The Food Marketing Institute shows that 17% of households spend more than 100 per week on groceries. Assume the...

STATISTICS F/BUSINESS+ECONOMICS-TEXT

To the Internal Revenue Service, the reasonableness of total itemized deductions depends on the taxpayers adjus...

Statistics for Business & Economics, Revised (MindTap Course List)

Practice Solve each problem. Uniform motion problem By increasing her usual speed by 25 kilometers per hour, a ...

College Algebra (MindTap Course List)

Converting to Polar CoordinatesIn Exercises 25 and 26, evaluate the iterated integral by converting to polar co...

Calculus (MindTap Course List)

Exercises 1-5 contains a while loop and a predicate. In each case show that if the predicate is true before ent...

Discrete Mathematics With Applications

One measure of the risk or volatility of an individual stock is the standard deviation of the total return (cap...

Modern Business Statistics with Microsoft Office Excel (with XLSTAT Education Edition Printed Access Card) (MindTap Course List)

A pair of dice is rolled once. a.Find the probability of rolling a 6. b.Find the probability of rolling a 6 giv...

Mathematics: A Practical Odyssey

Living with Family. Consider the following example survey results of 18- to 34-year-olds in the United States, ...

Essentials Of Statistics For Business & Economics

Getting a Formula In Exercises S-4 through S-13, a verbal description of a function is given. Use a formula to ...

Functions and Change: A Modeling Approach to College Algebra (MindTap Course List)

Eigenvalues and Eigenfunctions In Problems 920 find the eigenvalues and eigenfunctions for the given boundary-v...

A First Course in Differential Equations with Modeling Applications (MindTap Course List)

Using Technology In Exercises 59-68, use a graphing utility to graph the inequality and identify the solution s...

College Algebra

[T] In the following exercises, use a calculator to draw the graph of each piecewise-defined function and study...

Calculus Volume 1