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

question 12

Short Answer

Determine whether $\mathbf { F }$ is conservative. If so, find a function $f$ such that $\mathbf { F } = \nabla f . .$
$$\mathbf { F } ( x , y , z ) = ( 2 \sinh 2 z ) \mathbf { i } + \left( 4 e ^ { 4 z } \cos 4 y \right) \mathbf { j } + ( 4 x \cosh 2 z ) \mathbf { k }$$

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Definitions:

Owners' Equity

The residual interest in the assets of a company after deducting its liabilities, representing the ownership stake held by shareholders.

Earnings Per Share

A financial ratio that calculates the portion of a company's profit allocated to each outstanding share of common stock, serving as an indicator of the company's profitability.

Financial Performance

An assessment of how well a company can use assets from its primary mode of business and generate revenues.

Capital Required

The total amount of funds needed to operate a business or undertake a project, including costs for assets, resources, and day-to-day operations.