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

question 12

Multiple Choice

Show that F is conservative, and find a function f such that $\mathbf { F } = \nabla f$ , and use the result to evaluate $\int _ { C } \mathbf { F } \cdot d \mathbf { r }$ , where C is any curve from $A \left( x _ { 0 } , y _ { 0 } , z _ { 0 } \right)$ to $B \left( x _ { 1 } , y _ { 1 } , z _ { 1 } \right)$ . $\mathbf { F } ( x , y , z ) = 12 x ^ { 2 } y \mathbf { i } + \left( 4 x ^ { 3 } + 10 y z ^ { 2 } \right) \mathbf { j } + 10 y ^ { 2 } z \mathbf { k }$ ; $A ( 0,0,0 )$ and $B ( 0,3,1 )$

• A) -109
• B) 45
• C) -62
• D) -44

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