| ∫ ( u | dx | + v | dy | + w | dz | ) ds or ∫s (u dx + v dy + w dz). |
| ds | ds | ds |
In like manner surface integrals usually present themselves in the form
∫∫ (lξ + mη + nζ) dS
where l, m, n are the direction cosines of the normal to the surface drawn in a specified sense.
The area of a bounded portion of the plane of (x, y) may be expressed either as
½ ∫ (x dy − y dx),
or as
∫∫ dx dy,
the former integral being a line integral taken round the boundary of the portion, and the latter a surface integral taken over the area within this boundary. In forming the line integral the boundary is supposed to be described in the positive sense, so that the included area is on the left hand.
53a. Theorems of Green and Stokes.We have two theorems of transformation connecting volume integrals with surface integrals and surface integrals with line integrals. The first theorem, called “Green’s theorem,” is expressed by the equation