Concept explainers
Line
is the same for each parametric representation of C.
(i)
(ii)
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Calculus
- Line 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 line segment from (1, 1) to (5, 10)arrow_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 parabola y = x2 from (0, 0) to (1, 1)arrow_forwardA. Give the definition of a path-independent, or conservative, vector field. B. Verify that the vector field F(x, y, z) = (3x^2yz − 3y)i + (x^3z − 3x)j + (x^3 y + 2z)k is path-independent by showing that curl F = 0. C. Find a potential f for F, that is, a scalar function f such that F = grad f.arrow_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_forwardSplitting a vector field Express the vector field F = ⟨xy, 0, 0⟩in the form V + W, where ∇ ⋅ V = 0 and ∇ x W = 0.arrow_forwardLine integrals of vector fields on closed curves Evaluate ∮C F ⋅ dr for the following vector fields and closed oriented curves C by parameterizing C. If the integral is not zero, give an explanation. F = ⟨x, y⟩; C is the triangle with vertices (0, ±1) and (1, 0)oriented counterclockwise.arrow_forward
- Flux across the boundary of an annulus Find the outward flux of the vector field F = ⟨xy2, x2y⟩ across the boundary of the annulusR = {(x, y): 1 ≤ x2 + y2 ≤ 4}, which, when expressed in polar coordinates, is the set {(r, θ): 1 ≤ r ≤ 2, 0 ≤ θ ≤ 2π}arrow_forwardA. Let F = P i + Q j be a smooth vector field on R^2 , C a closed simple curve in R^2 , and D the plane simple region enclosed by C. State Green’s Theorem for F, C, and D. B. Evaluate the line integral in Green’s Theorem when F = (x + y)i + xy j and C is the unit circle with equation x^2 + y^2 = 1. C. Evaluate the double integral in Green’s Theorem when F = (x + y)i + xy j, C is the unit circle with equation x 2 + y 2 = 1, and D is the unit disc bounded by C. Then compare your answers in Parts B and C.arrow_forwardLine integrals of vector fields in the plane Given the followingvector fields and oriented curves C, evaluate ∫C F ⋅ T ds.arrow_forward
- Finding potential functions Determine whether the following vector fields are conservative on the specified region. If so, determine a potential function. Let R* and D* be open regions of ℝ2 and ℝ3, respectively, that do not include the origin. F = ⟨x, y⟩ on ℝ2arrow_forwarduse Green’s Theorem to find the counterclock-wise circulation and outward flux for the field F and curve C. F = (y2 - x2 )i + (x2 + y2 )j C: The triangle bounded by y = 0, x = 3, and y =x.arrow_forwardFinding potential functions Determine whether the following vector fields are conservative on the specified region. If so, determine a potential function. Let R* and D* be open regions of ℝ2 and ℝ3, respectively, that do not include the origin. F = ⟨ez, ez, ez (x - y)⟩ on ℝ3arrow_forward
- Elementary Linear Algebra (MindTap Course List)AlgebraISBN:9781305658004Author:Ron LarsonPublisher:Cengage Learning