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Vector Fields : Divergence, Scalar Curl and Gradient

1) Given a vector field P(x, y) i + Q(x, y) j in R2, its scalar curl is the k - component
of the vector curl (Pi + Qj + 0k) in R3 (the i and j components of that curl are 0).
In other words, the scalar curl of the vector field Pi + Qj equals ∂Q/∂x - ∂P/∂y.
Find the scalar curl of F(x, y) = sin (xy) i + cos (xy) j.
2) Find such a function that f(x, y) that ∇f = e2r?(r/k) for all r not equal to 0 (here r
= (x, y), r = ⎢⎢r ⎢⎢ = sqrt (x2 + y2). Hint: the answer is a function depending only
on r.
3) Let f(x, y, z) = x2y2 + x2z2. Find curl (grad f), or ∇× (∇f). Writing just the
answer is not enough.

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Solution Summary

Vector Fields, Divergence, Scalar Curl and Gradient are investigated. The solution is detailed and well presented.

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