Similar to the integral of a scalar field over a curve, which we called the line integral, we can define the integral of a vector-field over a surface.

Let be a surface in space with finite surface area. Let be a continuous scalar-field defined on the surface .We can subdivide into smaller portions, say having areas , and form the sum

Figure: Subdivision of the surface

where , is selected arbitrarily. By refining the patches into more smaller patches such that , if approaches a limit, we call it the surface integral of over , and denote it by