CHAPTER 5 VECTOR INTEGRATION INTEGRAL THEOREMS AND cross-cut AB connecting the exterior and interior boundaries. The region bounded by ABCDBAEFGHA is simply-connected, and so Green’s theorem is valid. Then . dxdy ABCDBAEFGHA R But the integral on the left, leaving out the integrand is equal to      C1 C2 C Since Thus, if is the curve AEFGHA, is the curve BCDB and is the  . R C1 C2 boundary of consisting of and traversed in the positive sense, then so that   dxdy R Example 5.25 Express Green’s theorem in the plane in vector notation. We have     where and so that Mdx  Ndy   F d r F   r    Mˆ Nˆ yˆ d r   dxˆi dyˆj Also if then, F   Mˆi Nˆj   ˆ M N O  F  kˆ ˆj ˆi  So that   kˆ The Green’s theorem in the plane can be written   F d r   kˆdR R where dR=dxdy A generalization of this to surface S in space having a curve C as boundary leads quite naturally to Stokes’ theorem which is treated later.
CHAPTER 5 VECTOR INTEGRATION AND INTEGRAL THEOREMS cross-cut AB connecting the exterior and interior boundaries. The region bounded by ABCDBAEFGHA is simply-connected, and so Green’s theorem is valid. Then ABCDBAEFGHA R dxdy . But the integral on the left, leaving out the integrand is equal to      Since  . Thus, if C1 is the curve AEFGHA, C2 is the curve BCDB and C is the boundary of R consisting of C1 and C2 traversed in the positive sense, then   so that R dxdy Example 5.25 Express Green’s theorem in the plane in vector notation. We have Mdx  Ndy       F  d r where F  Mˆ  Nˆ and r   yˆ so that d r  dxˆi  dyˆj Also if F  Mˆi  Nˆj then,  F  ˆi M ˆj N kˆ O  ˆ  So that   kˆ  The Green’s theorem in the plane can be written F  d r    kˆdR R where dR=dxdy A generalization of this to surface S in space having a curve C as boundary leads quite naturally to Stokes’ theorem which is treated later.