Line
is the same for each parametric representation of C.
(i)
(ii)
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Calculus (MindTap Course List)
- Proof Let V and W be two subspaces of vector space U. (a) Prove that the set V+W={u:u=v+w,vVandwW} is a subspace of U. (b) Describe V+W when V and W are subspaces of U=R2: V={(x,0):xisarealnumber} and W={(0,y):yisarealnumber}.arrow_forwardProof Use the concept of a fixed point of a linear transformation T:VV. A vector u is a fixed point when T(u)=u. (a) Prove that 0 is a fixed point of a liner transformation T:VV. (b) Prove that the set of fixed points of a linear transformation T:VV is a subspace of V. (c) Determine all fixed points of the linear transformation T:R2R2 represented by T(x,y)=(x,2y). (d) Determine all fixed points of the linear transformation T:R2R2 represented by T(x,y)=(y,x).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 line segment from (1, 1) to (5, 10)arrow_forward
- A. 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_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. 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_forward
- Splitting a vector field Express the vector field F = ⟨xy, 0, 0⟩in the form V + W, where ∇ ⋅ V = 0 and ∇ x W = 0.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 = ⟨x, y⟩ on ℝ2arrow_forwardFlux 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_forward
- Line integrals of vector fields in the plane Given the followingvector fields and oriented curves C, evaluate ∫C F ⋅ T ds. F = ⟨x, y⟩ on the parabola r(t) = ⟨4t, t2⟩ , for 0 ≤ t ≤ 1arrow_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 = ⟨y + z, x + z, x + y⟩ on ℝ3arrow_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