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Showing a 2D Vector Field is Not Conservative

Multivariable calculus warmup

Show that the vector field \( \mathbf{F}(x, y) = xy\mathbf{i} - x^{2}y\mathbf{j} \) is not conservative.
vector-fieldsconservative-vector-fieldspartial-derivativescross-partial-testmultivariable-calculusclairauts-theoremgradient-fields2d-vector-fieldscalculus-iiievaluate-conservativeprove-non-conservativecomponent-test

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What you need first

What this problem tests

Tests the ability to identify the components of a vector field, compute partial derivatives, and apply the cross-partial test.

This is a standard exercise in a multivariable calculus course when introducing line integrals and conservative vector fields.

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