Question

State the Fundamental Theorem for Line Integrals.

Short answer

The integral is\(\int_{\rm{C}} {\nabla {\rm{f \times dr = }}} {\rm{f(r(b)) - f(r(a))}}\).

Step-by-step solution

Define line integral

The function to be integrated is determined along a curve in the coordinate system for a line integral. It doesn't matter if the function to be integrated is a scalar or a vector field.

Explanation

Let\({\rm{C}}\)be a smooth whose position vector function\({\rm{r(t)}}\)describes it, where\({\rm{a}} \le {\rm{t}} \le {\rm{b}}\). If f is a two- or three-variable differentiable function with a continuous gradient vector\(\nabla {\rm{f}}\)on\({\rm{C}}\), It gives,

\(\int_{\rm{C}} {\nabla {\rm{f \times dr = }}} {\rm{f(r(b)) - f(r(a))}}\)

Therefore, the line integral is \(\int_{\rm{C}} {\nabla {\rm{f \times dr = }}} {\rm{f(r(b)) - f(r(a))}}\).