Find curl v for v given with respect to right-handed Cartesian coordinates. Show the details of...

 

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Find curl v for v given with respect to right-handed Cartesian coordinates. Show the details of your work. 4. v = [2y²,5x, 0] 5. v = xyz(x, y, z) 6. v = (x²+² +22²)-3/2[x. y. 2] 7. v=0.0. e sin yl 8. v = [e=²²₁ Let v be the velocity vector of a steady fluid flow. Is the flow irrotational? Incompressible? Find the streamlines (the paths of the particles). Hint. See the answers to Probs. 9 and 11 for a determination of a path. 9. v = [0, 32²,0] 10. V== [sec x, csc x, 0] 11. v= [y. -2x, 0] 12. v= [-y, x, TT] 13. v= [x. y. -2] Useful Formulas for the Curl. Assuming sufficient differentiability, show that (a) curl (u + v) = curl u + curl v (b) div (curl v) = 0 (c) curl (fv) = (grad f) x v + f curl v (d) curl (grad /) = 0 (e) div (u x v) = v. curl u- u. curl v With respect to right-handed coordinates, let u = [y. z. x]. v = [yz, zx, xy], f = xyz, and g = x + y + z. Find the given expressions. Check your result by a formula in Proj. 14 if applicable. 15. curl (u + v), curl v 16. curl (gv) 17. v. curl u, u curl v, u. curl u . 18. div (u x V) 19. curl (gu+ v), curl (gu) 20. div (grad (fg)) Find curl v for v given with respect to right-handed Cartesian coordinates. Show the details of your work. 4. v = [2y²,5x, 0] 5. v = xyz(x, y, z) 6. v = (x²+² +22²)-3/2[x. y. 2] 7. v=0.0. e sin yl 8. v = [e=²²₁ Let v be the velocity vector of a steady fluid flow. Is the flow irrotational? Incompressible? Find the streamlines (the paths of the particles). Hint. See the answers to Probs. 9 and 11 for a determination of a path. 9. v = [0, 32²,0] 10. V== [sec x, csc x, 0] 11. v= [y. -2x, 0] 12. v= [-y, x, TT] 13. v= [x. y. -2] Useful Formulas for the Curl. Assuming sufficient differentiability, show that (a) curl (u + v) = curl u + curl v (b) div (curl v) = 0 (c) curl (fv) = (grad f) x v + f curl v (d) curl (grad /) = 0 (e) div (u x v) = v. curl u- u. curl v With respect to right-handed coordinates, let u = [y. z. x]. v = [yz, zx, xy], f = xyz, and g = x + y + z. Find the given expressions. Check your result by a formula in Proj. 14 if applicable. 15. curl (u + v), curl v 16. curl (gv) 17. v. curl u, u curl v, u. curl u . 18. div (u x V) 19. curl (gu+ v), curl (gu) 20. div (grad (fg))

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To find the curl of a vector field with respect to righthanded Cartesian coordinates we use the foll... View the full answer