Verify whether a given scalar function is a potential function for a specific vector field by computing its gradient.

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Checking if a Function is a Potential Function

Multivariable calculus warmup

Is \( f(x, y, z) = x^2 \cos(yz) + y^2 z^2 \) a potential function for \( \mathbf{F}(x, y, z) = \langle 2x \cos(yz), -x^2 z \sin(yz) + 2yz^2, y^2 \rangle \)?

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What you need first

how to compute partial derivatives
the definition of the gradient of a scalar function
the chain rule for differentiation

What this problem tests

Tests the ability to compute partial derivatives and understand the relationship between a potential function and its gradient vector field.

This type of problem is common in multivariable calculus when introducing conservative vector fields and the fundamental theorem for line integrals.

Checking if a Function is a Potential Function

What you need first

What this problem tests

More on Multivariable calculus