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is the same for each parametric representation of C.
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Chapter 15 Solutions
Calculus Loose Leaf Bundle W/webassign
- 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_forwardRain on a roof Consider the vertical vector field F = ⟨0, 0, -1⟩, correspondingto a constant downward flow. Find the flux in the downward direction acrossthe surface S, which is the plane z = 4 - 2x - y in the first octant.arrow_forward
- 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_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_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
- 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 parabola y = x2 from (0, 0) to (1, 1)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_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_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 = ⟨y + z, x + z, x + y⟩ on ℝ3arrow_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_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