Concept explainers
Evaluating a Line Integral In Exercises 23-32, evaluate
along each path. (Hint: If F is conservative, the
(a)
(b) C2: The closed path consisting of line segments from (0, 3) to (0. 0), from (0, 0) to (3, 0), and then from (3, 0) to (0. 3)
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Chapter 15 Solutions
Calculus: Early Transcendental Functions (MindTap Course List)
- Showing Linear Independence In Exercises 27-30, show that the set of solutions of a second-order linear homogeneous differential equation is linearly independent. {eax,xeax}arrow_forwardShowing Linear Independence In Exercises 27-30, show that the set of solutions of a second-order linear homogeneous differential equation is linearly independent. {eax,ebx}, abarrow_forwardFind the potential function f for the field F.F = (y - z) i + (x + 2y - z) j - (x + y) k a) f(x, y, z) = x(y + y2) - xz - yz + C b) f(x, y, z) = xy + y2 - x - y + C c) f(x, y, z) = xy + y2 - xz - yz + C d) f(x, y, z) = x + y2 - xz - yz + Carrow_forward
- Evaluate the line integral using Green's Theorem and check the answer by evaluating it directly. ∮C4 y2dx+6 x2dy∮C4 y2dx+6 x2dy, where CC is the square with vertices (0,0)(0,0), (3,0)(3,0), (3,3)(3,3), and (0,3)(0,3) oriented counterclockwise. ∮C4 y2dx+6 x2dy=arrow_forwardThe figure shows a vector field F and two curves C_1 and C_2. Are the line integrals of F over C_1 and C_2 positive, negative, or zero? Explain.arrow_forwardEvaluate C F · dr using the Fundamental Theorem of Line Integrals. F(x, y) = e2xi + e2yj C: line segment from (−1, −1) to (0, 0)arrow_forward
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- 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_forwardUsing Stoke’s theorem, evaluate the circulation of the field F( x, y, z )=x ^2i +2xj+ z ^2k around the ellipse 4x^2+y^2=4 in the xy plane, counterclockwise when viewed from abovearrow_forwardIn Exercises 1-6, evaluate the integral using the Integration by Parts formula with the given choice of u and d v. f xe2x dx; u = x dv = e2x dxarrow_forward
- Elementary Linear Algebra (MindTap Course List)AlgebraISBN:9781305658004Author:Ron LarsonPublisher:Cengage Learning