Finding Work in a Conservative Force Field
In exercise 19-22, (a) show the
Trending nowThis is a popular solution!
Chapter 15 Solutions
Calculus: Early Transcendental Functions
- Work integrals Given the force field F, find the work required to move an object on the given oriented curve. F = ⟨x, y⟩ on the line segment from (-1, 0) to (0, 8) followedby the line segment from (0, 8) to (2, 8)arrow_forwardWork integrals in ℝ3 Given the force field F, find the workrequired to move an object on the given curve. F = ⟨ -y, z, x⟩ on the path consisting of the line segment from(0, 0, 0) to (0, 1, 0) followed by the line segment from (0, 1, 0)to (0, 1, 4)arrow_forwardGravitational potential The potential function for the gravitational force field due to a mass M at the origin acting on a mass m is φ = GMm/ | r | , where r = ⟨x, y, z⟩ is the position vector of the mass m, and G is the gravitational constant.a. Compute the gravitational force field F = -∇φ .b. Show that the field is irrotational; that is, show that ∇ x F = 0.arrow_forward
- Evaluating line integrals Use the given potential function φ of the gradient field F and the curve C to evaluate the line integral ∫C F ⋅ dr in two ways.a. Use a parametric description of C and evaluate the integral directly.b. Use the Fundamental Theorem for line integrals. φ(x, y) = x + 3y; C: r(t) = ⟨2 - t, t⟩ , for 0 ≤ t ≤ 2arrow_forwardA particle moves along line segments from the origin to the points (1, 0, 0), (1, 3, 1), (0, 3, 1), and back to the origin under the influence of the force field F(x, y, z) = z2i + 5xyj + 4y2k. Find the work done.arrow_forwardEvaluating line integrals Use the given potential function φ of the gradient field F and the curve C to evaluate the line integral ∫C F ⋅ dr in two ways.a. Use a parametric description of C and evaluate the integral directly.b. Use the Fundamental Theorem for line integrals. φ(x, y, z) = xy + xz + yz; C: r(t) = ⟨t, 2t, 3t⟩ , for 0 ≤ t ≤ 4arrow_forward
- a. Outward flux and area Show that the outward flux of theposition vector field F = xi + yj across any closed curve towhich Green’s Theorem applies is twice the area of the regionenclosed by the curve.b. Let n be the outward unit normal vector to a closed curve towhich Green’s Theorem applies. Show that it is not possiblefor F = x i + y j to be orthogonal to n at every point of C.arrow_forwardCirculation on a half-annulus Consider the vector field F = ⟨y2, x2⟩on the half-annulus R = {(x, y): 1 ≤ x2 + y2 ≤ 9, y ≥ 0}, whose boundary is C. Find the circulation on C, assuming it has the orientation shown.arrow_forwarda) Show that F (x, y) = (yexy + cos(x + y)) i + (xexy + cos(x + y) j is the gradient of some function f. Find f b) Evaluate the line integral ʃC F dr where the vector field is given by F (x, y) = (yexy + cos(x + y)) i + (xexy + cos(x + y) j and C is the curve on the circle x 2 + y 2 = 9 from (3, 0) to (0, 3) in a counterclockwise direction.arrow_forward
- Verifying path independence Consider the potential functionφ(x, y) = (x2 - y2)/2 and its gradient field F = ⟨x, -y⟩.• Let C1 be the quarter-circle r(t) = ⟨cos t, sin t⟩, for 0 ≤ t ≤ π/2, from A(1, 0) to B(0, 1).• Let C2 be the line r(t) = ⟨1 - t, t⟩, for 0 ≤ t ≤ 1, also from A to B.Evaluate the line integrals of F on C1 and C2, and show that both are equal toφ(B) - φ(A).arrow_forwardEvaluating line integrals Use the given potential function φ of the gradient field F and the curve C to evaluate the line integral ∫C F ⋅ dr in two ways.a. Use a parametric description of C and evaluate the integral directly.b. Use the Fundamental Theorem for line integrals. φ(x, y) = xy; C: r(t) = ⟨cos t, sin t⟩ , for 0 ≤ t ≤ πarrow_forwardLine integrals of vector fields in the plane Given the followingvector fields and oriented curves C, evaluate ∫C F ⋅ T ds. F = ⟨-y, x⟩ on the semicircle r(t) = ⟨4 cos t, 4 sin t⟩ , for 0 ≤ t ≤ πarrow_forward
- Calculus: Early TranscendentalsCalculusISBN:9781285741550Author:James StewartPublisher:Cengage LearningThomas' Calculus (14th Edition)CalculusISBN:9780134438986Author:Joel R. Hass, Christopher E. Heil, Maurice D. WeirPublisher:PEARSONCalculus: Early Transcendentals (3rd Edition)CalculusISBN:9780134763644Author:William L. Briggs, Lyle Cochran, Bernard Gillett, Eric SchulzPublisher:PEARSON
- Calculus: Early TranscendentalsCalculusISBN:9781319050740Author:Jon Rogawski, Colin Adams, Robert FranzosaPublisher:W. H. FreemanCalculus: Early Transcendental FunctionsCalculusISBN:9781337552516Author:Ron Larson, Bruce H. EdwardsPublisher:Cengage Learning