Find the outward flux for the field F = ⟨3x^2 + y^2, 2xy⟩ and the curve C which outlines the semicircle y = −√(4 − x^2) (including the diameter along the x-axis): a. By finding a parametrization of the curve C and using it to evaluate the line integral for outward flux. b. By using Green’s Theorem (Flux Form) and evaluating the double integral of the divergence.
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Part a: Parametrization and Line Integral ** Show more…
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In the xy-plane, let F = (y - x, x) and let C be a circle of radius 2 centered at the origin. Find the outward flux of F through C. Consider the surface S defined as a cylinder of radius r that extends from 0 ≤ z ≤ h and that is open on both its top and bottom: a. Parameterize S. (15 pts) b. Find the surface area of S using your parameterization in part (a) and the surface integral presented in the video lecture on "Surface Integrals and Gauss' Divergence Theorem". (15 pts)
Adi S.
3. Let the surface (S) be the part of the elliptic paraboloid z = x^2 + 4y^2 lying below the plane z = 1. We define the orientation of (S) by taking the unit normal vector n pointing in the positive direction of z-axis (the inner normal vector to the surface). Further, let C denotes the curve of the intersection of the paraboloid z = x^2 + 4y^2 and the plane z = 1 oriented counterclockwise when viewed from positive z-axis above the plane and let S1 denotes the part of the plane z = 1 inside the paraboloid z = x^2 + 4y^2 oriented upward. a) Parametrize the curve C and use the parametrization to evaluate the line integral integral_C F . dr, where F(x, y, z) = <y, -xz, xz^2>. b) Find G = del x F, where F(x, y, z) is the vector field from Part a), parameterize the surface S1 and use the parametrization to evaluate the flux of the vector field G. HINT: The area enclosed by an ellipse x^2/a^2 + y^2/b^2 = 1 is ab pi. c) What is the flux of the vector field G = del x F, from Part b), across the surface (S)? Explain why the answers in a), b), and c) must be the same.
What's wrong? Consider the radial field $\mathbf{F}=\frac{\langle x, y\rangle}{x^{2}+y^{2}}$ a. Verify that the divergence of $\mathbf{F}$ is zero, which suggests that the double integral in the flux form of Green's Theorem is zero. b. Use a line integral to verify that the outward flux across the unit circle of the vector field is $2 \pi$ c. Explain why the results of parts (a) and (b) do not agree.
Vector Calculus
Green's Theorem
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