gradient theorem

The fundamental theorem of calculus told us that

The gradient theorem or the fundamental theorem of calculus for line integrals adapt it to the line integral.

Theorem: Suppose is a smooth curve given by , . Also suppose that is a function whose gradient vector, , is continuous on . Then,

Note that is the initial point on , and is the final point.

Implications

For gradient fields , the line integral depends only on the endpoints of the path. We call such a line integral path independent.

A special case of this is for a closed curve :

Following physics, where a conservative force does no work around a closed field, we say that the gradient field is a conservative field.

Related

• Finding Scalar Potential

References

• Calculus III - Fundamental Theorem for Line Integrals