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

question 7

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 $B \left( x _ { 1 } , y _ { 1 } , z _ { 1 } \right)$ .
$\mathbf { F } ( x , y , z ) = 21 x ^ { 2 } y \mathbf { i } + \left( 7 x ^ { 3 } + 15 y ^ { 2 } z ^ { 2 } \right) \mathbf { j } + 10 y ^ { 3 } z \mathbf { k } ; A ( 0,0,0 )$ and $B ( - 1,0 , - 3 )$

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

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Definitions:

Streamlining Jobs

Simplifying work processes and structures to improve efficiency and reduce wastefulness.

Work Arrangements

The various formats and schedules under which employment can be structured, including full-time, part-time, remote, and flexible working hours.

Specialization

A focus on a particular area of study or work, allowing individuals or entities to become experts in a specific field.

Complexity

Describes a situation, system, or problem that has multiple parts which interact in varied and intricate ways.