Concept explainers
Evaluating a Line
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Calculus
- Application of Green's theorem Assume that u and v are continuously differentiable functions. Using Green's theorem, prove that SS'S D Ux Vx |u₁|dA= udv, C Wy Vy where D is some domain enclosed by a simple closed curve C with positive orientation.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_forwardEvaluate the line integral using Green's Theorem and check the answer by evaluating it directly. $ 5 y dx + 5 x²dy, where Cis the square with vertices (0, 0), (2, 0), (2, 2), and (0, 2) oriented counterclockwise. + iarrow_forward
- Evaluating Polar Integrals In Exercises 9-22, change the Cartesian integral into an equivalent polar integral. Then evaluate the polar integral. μl pV²-3² 11 12. Jo Jo ra I √a²-x² тугилау dy dx JOJOarrow_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) = x + 3y; C: r(t) = ⟨2 - t, t⟩ , for 0 ≤ t ≤ 2arrow_forwardScalar line integrals Evaluate the following line integral along the curve C.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, z) = xy + xz + yz; C: r(t) = ⟨t, 2t, 3t⟩ , for 0 ≤ t ≤ 4arrow_forwardLine integrals Use Green’s Theorem to evaluate the following line integral. Assume all curves are oriented counterclockwise.A sketch is helpful. The flux line integral of F = ⟨ex - y, ey - x⟩, where C is theboundary of {(x, y): 0 ≤ y ≤ x, 0 ≤ x ≤ 1}arrow_forward人工知能を使用せず、 すべてを段階的にデジタル形式で解決してください。 ありがとう SOLVE STEP BY STEP IN DIGITAL FORMAT DON'T USE CHATGPT For Exercises 1-4, use Green's Theorem to evaluate the given line integral around the curve C, traversed counterclockwise. 1. f(x² - y²) dx + 2xydy; C is the boundary of R = {(x,y): 0≤x≤ 1, 2x² ≤ y ≤ 2x) x³y dx + 2xydy; C is the boundary of R = {(x, y): 0 ≤x≤1, x² ≤ y ≤ x} $² 2ydx-3xd y; C is the circle x² + y² = 1 2. 3. 4. ·f (ex² + y²) dx + (e¹² + x³)dy; C is the boundary of the triangle with vertices (0,0), (4,0) and (0,4)arrow_forward
- ulus III |Uni Use Green's Theorem to evaluate the line integral cos (y) dx + x²sin (y) dy along CoS the positively oriented curve C, where C is the rectangle with vertices(0,0), (4, 0), (4, 2) and (0, 2).arrow_forwarda) Find the value(s) of (1+ i)2/3. b) Show that cos² z + sin? z = 1, for all z = x + iy, , y E R, V-I= i. c) Find a complex valued analytic function f(x, y) = u(x, y) + iv(x, y), whose real part u(x, y) = 213 – 3r?y – 6xy? + y°.arrow_forwardUsing the Fundamental Theorem of Line Integrals In Exercises 47-50, evaluate F· dr using the Fundamental Theorem of Line Integrals. F (x, y) = e²*i + e²»j C : line segment from (-1, –1) to (0,0)arrow_forward
- Elementary Linear Algebra (MindTap Course List)AlgebraISBN:9781305658004Author:Ron LarsonPublisher:Cengage Learning