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A spherical vector field is given by F(r,0,0)=2r cos(0) êr + 3r sin(0) ê +0(π- 0) ê. 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 ê V.F = 10 r² Ər (r²F₁) + 1 ə r sin(0) 00 (sin(0) F₂) + $= ii) Compute the total flux through the solid hemisphere (z > 0) of radius 2 (Figure 7) using = /[[, V. Edv. V.FdV. 1 аF3 r sin(0) do

A spherical vector field is given by F(r,0,0)=2r cos(0) êr + 3r sin(0) ê +0(π- 0) ê. 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 ê V.F = 10 r² Ər (r²F₁) + 1 ə r sin(0) 00 (sin(0) F₂) + $= ii) Compute the total flux through the solid hemisphere (z > 0) of radius 2 (Figure 7) using = /[[, V. Edv. V.FdV. 1 аF3 r sin(0) do

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Q7 Part I and II needed to be solved in 30 minutes
Question 7 A spherical vector field is given by F(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 ê 1 ə r² Ər V.F= (r²F₁) + $= ii) Compute the total flux through the solid hemisphere (z ≥ 0) of radius 2 (Figure 7) using 2 1 Ә r sin(0) 20 $= -2 (sin(0) F2) + •///.V.F iii) Confirm your total flux calculation by computing the surface integral 2 · Elle Σ i=1 #fr. ñ ds -2 V.FdV. 1 OF3 r sin(0) do You should compute both surface integrals in spherical coordinates. F. ds. 0 Figure 7: The solid hemisphere (z ≥ 0) of radius 2 with a sample of unit normal vectors on the two surfaces S₁ and S₂.
Transcribed Image Text:Question 7 A spherical vector field is given by F(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 ê 1 ə r² Ər V.F= (r²F₁) + $= ii) Compute the total flux through the solid hemisphere (z ≥ 0) of radius 2 (Figure 7) using 2 1 Ә r sin(0) 20 $= -2 (sin(0) F2) + •///.V.F iii) Confirm your total flux calculation by computing the surface integral 2 · Elle Σ i=1 #fr. ñ ds -2 V.FdV. 1 OF3 r sin(0) do You should compute both surface integrals in spherical coordinates. F. ds. 0 Figure 7: The solid hemisphere (z ≥ 0) of radius 2 with a sample of unit normal vectors on the two surfaces S₁ and S₂.
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