Show That $\mathbf { F }$ Is Conservative and Find a Function

question 14

Short Answer

Show that $\mathbf { F }$ is conservative and find a function $f$ such that $\mathbf { F } = \nabla f$ , and use this result to evaluate $\int _ { C } \mathbf { F } \cdot d \mathbf { r }$ , where $C$ is any path from $A \left( x _ { 0 } , y _ { 0 } \right)$ to $B \left( x _ { 1 } , y _ { 1 } \right)$ .
$\mathbf { F } ( x , y ) = \left( 16 x y ^ { 2 } - 16 x y ^ { 3 } \right) \mathbf { i } + \left( 16 x ^ { 2 } y - 24 x ^ { 2 } y ^ { 2 } \right) \mathbf { j } ; A ( 1,0 )$ and $B ( 1,0 )$

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Definitions:

Incurred Costs

Expenses that have been realized in the course of business operations.

Alternatives

Different options or choices available in a decision-making process.

Traceable

Costs or items that can be directly linked or attributed to a specific product, job, or department, facilitating accurate costing and financial analysis.

Avoidable Cost

A cost that can be eliminated by choosing one alternative over another in a decision. This term is synonymous with differential cost and relevant cost.