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Determining if a Vector Field is a Curl

Multivariable calculus core

Is it possible for \( \mathbf{G}(x, y, z) = \langle \sin x, \cos y, \sin(xyz) \rangle \) to be the curl of a vector field?

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What you need first

divergence of a vector field
curl of a vector field
partial derivatives
the chain rule

What this problem tests

This tests the student's knowledge of the identity div(curl F) = 0 and their ability to compute the divergence of a vector field.

This problem typically appears in a Calculus III or Vector Calculus course when discussing the properties of the curl and divergence operators.

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