2.10 The Curl of a Vector Field

The line integral of a vector field around a closed path is called the circulation of and the curl of is its measure. If we consider a small surface element bounded by a closed path ? c, we define the component of the curl parallel to the surface normal _n, in the limit ? s ? 0, as

This definition suggests that the curl of a vector field is a vector quantity. The direction of path ? c is determined by the right-hand rule. Equation (2.93) provides a complete definition of curl because it enables us to determine each of the three components of curl in any arbitrary system of orthogonal coordinates.

We begin our evaluation of the z component of curl in rectangular coordinates by defining the vector field as

at a point P within the small surface ? s bounded by path ? c, as illustrated in Figure 2.32. The line integral of along the closed path ? c consists of four separate paths:

Figure 2.32: A small surface element for defining the curl of a vector field

We now evaluate each of the four integrals of (2.94) separately. Along the path ? c ₁:

where F _x ? x is to be evaluated at y, and we have made the assumption that the component F _x