Line integrals in vector fields

You've done some work with line integral with scalar functions and you know something about parameterizing position-vector valued functions. In that case, welcome! You are now ready to explore a core tool math and physics: the line integral for vector fields. Need to know the work done as a mass is moved through a gravitational field. No sweat with line integrals.

Line integrals and vector fields

Using a line integral to find work

Line integrals in vector fields

Parametrization of a reverse path

Scalar field line integral independent of path direction

Vector field line integrals dependent on path direction

Path independence for line integrals

Closed curve line integrals of conservative vector fields

Line integrals in conservative vector fields

Example of closed line integral of conservative field

Second example of line integral of conservative vector field

Distinguishing conservative vector fields

Potential functions