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

question 41

Multiple Choice

Show that $\mathbf { F }$ 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) \(- 89\)

---

Definitions:

Significance

Significance refers to the measure of whether the results of a study or experiment are likely not due to chance, as determined by a set threshold of probability (p-value).

Alpha (α)

The threshold of significance in a statistical test; the probability of rejecting the null hypothesis when it is true, often set at 0.05 or 5%.

Population Parameter

A value that represents a characteristic of an entire population, such as its mean or standard deviation.

Statistic

A statistic is a single measure or value that is derived from a sample of data, used to estimate or infer properties of the population from which the sample was drawn.