Prove that the divergence of a curl is always zero. Check it for function va = x2x-hat + 3xz2y-hat − 2xzz-hat.
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Prove that the divergence of a curl is always zero. Check it for function va = x2x-hat + 3xz2y-hat − 2xzz-hat.
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- Verify the divergence theoremGive an example of a vector field F (x, y, z) that has value 0 on precisely one line and such that curl F is nonzero every-where. Be sure to identify the line and compute the curl.A vector field is given by F~ = (xyz, xyz, xyz). Show by explicit calculation that ∇ · [∇ × F ] = 0.
- Give an example of a vector field F (x, y, z) that has value 0 at only one point and such that curl F is nonzero everywhere. Be sure to identify the point and compute the curl.Check if the field is conservative, if so, calculate by the Fundamental Theorem of Calculus for Line IntegralsFor any vector field F, show that div(curl F) = 0.