Solve all and explain the pass to pass Gauss's Divergence TheoremConsider the vector fields F(x, y, z)(x,y,z), and the surface of the= ||(x,y,z)||³ellispsoid + + 4 = 1. Considering the outward orientation, calculate:x² y² z²8² 2² 1²i) the flux through this surface without using Gauss's theorem;ii) the flux through this surface using Gauss's theorem;Sketch the surface and the vector field."

Given the vector field .The surface is an ellipsoid (i)Let us paremetrize the surface using the…

Answered: Gauss's Divergence Theorem Consider the…

Algebra & Trigonometry with Analytic Geometry

13th Edition

ISBN:9781133382119

Author:Swokowski

Publisher:Swokowski

Chapter8: Applications Of Trigonometry

Section8.3: Vectors

Problem 60E

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Similar questions

Streamlines and equipotential lines Assume that on ℝ2, the vectorfield F = ⟨ƒ, g⟩ has a potential function φ such that ƒ = φxand g = φy, and it has a stream function ψ such that ƒ = ψy andg = -ψx. Show that the equipotential curves (level curves of φ)and the streamlines (level curves of ψ) are everywhere orthogonal.
Flux of a vector field? Let S be a closed surface consisting of a paraboloid (z = x²+y²), with (0≤z≤1), and capped by the disc (x²+y² ≤1) on the plane (z=1). Determine the flow of the vector field F (x,y,z) = zj − yk, in the direction that points out across the surface S.
Rain 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.
Divergence Theorem for more general regions Use the DivergenceTheorem to compute the net outward flux of the following vectorfields across the boundary of the given regions D.
Find the flux of the vector field F(x, y, z) =< x^3, y^3, z^3 > across the surface σ,where σ is the surface of the solid G bounded below by z = 1 + x^2 + y^2, andabove by z = 2 and is oriented by outward normals. Sketch the surface σaccurately. Simplify your answer
Circulation on a half-annulus Consider the vector field F = ⟨y2, x2⟩on the half-annulus R = {(x, y): 1 ≤ x2 + y2 ≤ 9, y ≥ 0}, whose boundary is C. Find the circulation on C, assuming it has the orientation shown.
Using Green's Theorem on this vector field problem, compute a) the circulation on the boundary of R in terms of a and b, and b) the outward flux across the boundary of R in terms of a and b.
Conservative fields Use Stokes’ Theorem to find the circulationof the vector field F = ∇(10 - x2 + y2 + z2) around anysmooth closed curve C with counterclockwise orientation.
Divergence Theorem for more general regions Use the DivergenceTheorem to compute the net outward flux of the following vectorfields across the boundary of the given regions D. F = ⟨x2, -y2, z2⟩; D is the region in the first octant between theplanes z = 4 - x - y and z = 2 - x - y.
a. 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.
Using Gauss' theorem to calculate the flow of the vector field 3x3 F: F (x, y, z) = (x^2z, 2x^2, 3z^2) exiting the cylinder defined from the relations x ^2+y ^2<=1, 1<= z <= 2.
Line integrals of vector fields in the plane Given the followingvector fields and oriented curves C, evaluate ∫C F ⋅ T ds.

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