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

question 105

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 $B \left( x _ { 1 } , y _ { 1 } , z _ { 1 } \right)$ . $\mathbf { F } ( x , y , z ) = 6 x y \mathbf { i } + \left( 3 x ^ { 2 } + 4 y z ^ { 2 } \right) \mathbf { j } + 4 y ^ { 2 } z \mathbf { k }$ ; $A ( 0,0,0 )$ and $B ( 2 , - 2,0 )$

• A) -113
• B) -24
• C) -131
• D) -178

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Definitions:

Giovanni Arnolfini

A wealthy merchant from Renaissance Bruges, frequently associated with the famous painting by Jan van Eyck believed to depict him and his wife.

Fidelity

The degree of accuracy by which a copy reproduces its source or the loyalty to faithfulness to obligations, duties, or commitments.

Dukes of Burgundy

Titles held by the rulers of the Burgundian State, spanning parts of modern-day France, Belgium, and the Netherlands from the 9th to the 15th centuries.

Flanders

A historical region in Northern Europe that today spans parts of Belgium, France, and the Netherlands, known for its role in the medieval and early modern textile trade and art.