Q:
# What is the Fundamental Theorem for Line Integrals?

The fundamental theorem for line integrals states that if a line integral passes through a gradient field, it can be evaluated by evaluating the endpoints of the curve in the initial scalar field. If a smooth curve C exists and a function with a gradient vector is continuous on curve C, then the line integral can be calculated using the fundamental theorem of calculus for single integrals.

Continue Reading
Credit:
DAJ
N/A
Getty Images

In order for the theorem to work, the line integral must be independent of the path because all that is required are the initial and final points of curve C. The gradient theorem for line integrals can only work in a conservative vector field because it is path-independent. The use of the gradient theorem simplifies the process of finding line integrals, since all that is required is to find out if the vector field is conservative.

The easiest way to determine if a vector field is conservative is by finding its potential function. An important definition relating to line integrals is a closed path, which refers to paths with endpoints that are the same, such as circles. Another important definition is a simple path, which is a path that does not cross itself, such as a circle.

Learn more about Calculus-
Q:
## What is the triangle angle bisector theorem?

A: The angle bisector theorem states that a line bisecting an angle in a triangle divides the side opposite the angle into two line segments that have lengths... Full Answer >Filed Under: -
Q:
## What is the side splitter theorem?

A: The side splitter theorem states that if a line parallel to one side of a triangle intersects with the other two sides, it divides those two sides proporti... Full Answer >Filed Under: -
Q:
## How do you calculate the midpoint Riemann sum?

A: A Riemann sum is a method of approximating the area under the curve of a function. It adds together a series of values taken at different points of that fu... Full Answer >Filed Under: -
Q:
## What is the vector equation of a line?

A: The vector equation of a line is r = a + tb. In this equation, "a" represents the vector position of some point that lies on the line, "b" represents a vec... Full Answer >Filed Under: