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

question 61

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 $B \left( x _ { 1 } , y _ { 1 } \right)$ .
$$\mathbf { F } ( x , y ) = \left( 15 x ^ { 2 } y ^ { 2 } - 14 x y ^ { 4 } \right) \mathbf { i } + \left( 10 x ^ { 3 } y - 28 x ^ { 2 } y ^ { 3 } \right) \mathbf { j } ; A ( 1 , - 2 ) \text { and } B ( 1 , - 1 )$$

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Definitions:

National Defense

The governmental and military activities aimed at defending a country's territorial integrity and protecting its citizens from external threats.

Misallocated

A situation where resources are used inefficiently, resulting in lost potential output or wasteful expenditures.

Market Failure

A situation in which the allocation of goods and services by a free market is not efficient, often leading to a social welfare loss.

Public Good

An offering of goods or services to society's members at no cost, made available by public institutions or private bodies or individuals, not aiming to generate profits.