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

question 60

Multiple Choice

Show that $\mathrm { F }$ 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 $B \left( x _ { 1 } , y _ { 1 } , z _ { 1 } \right)$ .
$\mathbf { F } ( x , y , z ) = 27 x ^ { 2 } y \mathbf { i } + \left( 9 x ^ { 3 } + 24 y ^ { 3 } z ^ { 2 } \right) \mathbf { j } + 12 y ^ { 4 } z \mathbf { k } ; A ( 0,0,0 )$ and $B ( - 1,0,1 )$

• A) \(- 89\)
• B) 0
• C) \(- 154\)
• D) \(- 107\)

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Definitions:

Corporation

A legal entity that is separate from its owners, who own shares in the company and have limited liability.

Controller

A senior executive responsible for the accounting operations and financial reporting of a company.

Board of Directors

A group of individuals elected by shareholders to oversee the management and make key decisions for a corporation.

Declare Dividends

Declare dividends refers to the action by a company's board of directors to distribute a portion of the earnings to shareholders, typically in the form of cash or stock.