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

question 70

Multiple Choice

Show that 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 $B \left( x _ { 1 } , y _ { 1 } \right)$ . $\mathbf { F } ( x , y ) = \left( 12 x ^ { 2 } y ^ { 2 } - 16 x y ^ { 3 } \right) \mathbf { i } + \left( 8 x ^ { 3 } y - 24 x ^ { 2 } y ^ { 2 } \right) \mathbf { j }$ ; $A ( 3,1 )$ and $B ( 3,2 )$

• A) \(\int _ { C } \mathbf { F } \cdot d \mathbf { r } = - 414\)
• B) \(\int _ { C } \mathbf { F } \cdot d \mathbf { r } = - 152\)
• C) \(\int _ { C } \mathbf { F } \cdot d \mathbf { r } = - 236\)
• D) \(\int _ { C } \mathbf { F } \cdot d \mathbf { r } = - 180\)

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