Determine Whether $\mathbf { F }$ Is Conservative $f$ Such That $\mathbf { F } = \nabla f$

question 111

Short Answer

Determine whether $\mathbf { F }$ is conservative. If so, find a function $f$ such that $\mathbf { F } = \nabla f$ .
$$\mathbf { F } ( x , y ) = \left( 6 x ^ { 2 } - 4 x ^ { 3 } y \right) e ^ { - 2 x y } \mathbf { i } - 4 x ^ { 4 } e ^ { - 2 x y } \mathbf { j }$$

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Definitions:

Tax System

The organized method by which a government collects taxes from its citizens and businesses to fund public services and infrastructure.

Revenue

The total income generated by a business or organization from its activities, before any expenses are deducted.

Proportional Tax

A tax system where the tax rate is fixed and applied equally to all income levels, resulting in taxpayers paying the same percentage of their income regardless of earnings.

Progressive Tax

A tax system in which the tax rate increases as the taxable income or amount increases.