Gradient theorem ↩

Metadata

shorthands: {}
aliases: [gradient theorem, fundamental theorem of calculus for line integrals, Fundamental theorem of calculus for line integrals]
created: 2021-10-19 20:27:02
modified: 2022-01-10 04:13:04

A line integral through a gradient field can be evaluated by evaluating the original scalar field at the endpoints of the path.

Where is any curve in the domain of that starts at and ends at .

Implies that line integrals through gradient fields are path independent.

Examples

Example 1

Let be a scalar field in , .
and
Then consider the straight line path between point and .
Let's calculate
The gradient of :
The tangent vector of the line:
(According to Calculating line or path integrals) Parametrize the line from to :
Then look at the path integral:

By using the parametrization:

The same value we got using the gradient theorem