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

question 118

Multiple Choice

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

• A) \(\int _ { C } \mathbf { F } \cdot d \mathbf { r } = 34\)
• B) \(\int _ { C } \mathbf { F } \cdot d \mathbf { r } = 90\)
• C) \(\int _ { C } \mathbf { F } \cdot d \mathbf { r } = - 144\)
• D) \(\int _ { C } \mathbf { F } \cdot d \mathbf { r } = 118\)

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