Q7 Part I and II needed to be solved in 30 minutes Question 7A spherical vector field is given byF(r, 0, 0) 2r cos(0) ê, + 3r sin(0) eo + 0(T0) êø.=i) Compute the divergence of F using the spherical definition. That is, for a gen-eral spherical vector field F = F₁ êr + F₂ eo + F3 ê1 ər² ƏrV.F= (r²F₁) +$=ii) Compute the total flux through the solid hemisphere (z ≥ 0) of radius 2 (Figure7) using21 Әr sin(0) 20$=-2(sin(0) F2) +•///.V.Fiii) Confirm your total flux calculation by computing the surface integral2· ElleΣi=1#fr. ñ ds-2V.FdV.1 OF3r sin(0) doYou should compute both surface integrals in spherical coordinates.F. ds.0Figure 7: The solid hemisphere (z ≥ 0) of radius 2 with a sample of unit normal vectorson the two surfaces S₁ and S₂.

It is given that Fr,θ,ϕ=2rcosθe^r+3rsinθe^θ+θπ-θe^ϕ. Thus, F1=2rcosθ, F2=3rsinθ, and F3=θπ-θ.

Answered: A spherical vector field is given by…

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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1) Consider the conservative vector field given by: F(x, y) = (exy3 + 2e2xy, e2x + 3exy2) A potential function that generates the vector field F corresponds to: A) f(x, y) = exy + exy3 B) f(x, y) = 3exy2 +(e2x/2)+(exy4)/4 C) f(x, y) = e2xy + exy3 D) f(x, y) = exy + e2xy3 2) Consider the vector field F(x, y, z) = (y - z sinx, x, 2z + cosx). The work that performs the F field to displace a body, from point A (3π, −1, 1) to point B (π, 2, 0) corresponds approximately to: A) 28, 45 JB) 32, 42 JC) 15, 71 JD) 13, 72 J
Which of the following statements are true for all vector fields, and which are true only for conservative vector fields? (a) The line integral along a path from P to Q does not depend on which path is chosen. (b) The line integral over an oriented curve C does not depend on how C is parametrized. (c) The line integral around a closed curve is zero. (d) The line integral changes sign if the orientation is reversed. (e ) The line integral is equal to the difference of a potential function at the two endpoints. (f) The line integral is equal to the integral of the tangential component along the curve. (g) The cross partial derivatives of the components are equal. 3

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