Editing Conservative vector field (section)

=== Path independence ===
A line integral of a vector field <math>\mathbf{v}</math> is said to be path-independent if it depends on only two integral path endpoints regardless of which path between them is chosen:<ref name=":0">{{Cite book |last=Stewart |first=James |title=Calculus |publisher=Cengage Learning |year=2015 |isbn=978-1-285-74062-1 |edition=8th |pages=1127–1134 |language=English |chapter=16.3 The Fundamental Theorem of Line Integrals"}}</ref> 
<math display="block">\int_{P_1} \mathbf{v} \cdot d \mathbf{r} = \int_{P_2} \mathbf{v} \cdot d \mathbf{r}</math>

for any pair of integral paths <math>P_1</math> and <math>P_2</math> between a given pair of path endpoints in <math>U</math>.

The path independence is also equivalently expressed as

<math display="block">\int_{P_c} \mathbf{v} \cdot d \mathbf{r} = 0</math>
for any [[piecewise]] smooth closed path <math>P_c</math> in <math>U</math> where the two endpoints are coincident. Two expressions are equivalent since any closed path <math>P_c</math> can be made by two path; <math>P_1</math> from an endpoint <math>A</math> to another endpoint <math>B</math>, and <math>P_2</math> from <math>B</math> to <math>A</math>, so
<math display="block">\int_{P_c} \mathbf{v} \cdot d \mathbf{r} = \int_{P_1} \mathbf{v} \cdot d \mathbf{r} + \int_{P_2} \mathbf{v} \cdot d \mathbf{r} = \int_{P_1} \mathbf{v} \cdot d \mathbf{r} - \int_{-P_2} \mathbf{v} \cdot d \mathbf{r} = 0</math>
where <math>-P_2</math> is the reverse of <math>P_2</math> and the last equality holds due to the path independence <math display="inline">\displaystyle \int_{P_1} \mathbf{v} \cdot d \mathbf{r} = \int_{-P_2} \mathbf{v} \cdot d \mathbf{r}.</math>