CHAPTER 5 VECTOR INTEGRATION INTEGRAL THEOREMS AND LINE INTEGRALS  iˆ  kˆ  (x,y,z,) Let where is the position vector of define a curve    rt xt yt zt rt ˆj C joining points P1 and P2 where t=t1 and t=t2 respectively. C The curve is composed of a finite number of curves (see figure) for each of which  has a continuous derivative. rt P2 rd r d r C O r P1 x, Let be a vector function of P2    F y, z F1iˆ F2 ˆj F3kˆ d r n C. position defined and continuous along Then the d r integral of the tangential component of along C from F P1 P2 to C dr2 dr1 P1 P2 C, is an example of a line integral. If is a force on a particle moving along this line F integral represents the work done by the force. If C is a simple closed curve, the C integral around is often denoted by F d r  In general, any integral, which is to be evaluated along a curve is called a line integral. If is a scalar field, then the line integral of V along the curve C from P1 to P2 is   d r
CHAPTER 5 VECTOR INTEGRATION AND INTEGRAL THEOREMS LINE INTEGRALS Let rt   xt iˆ  yt  ˆj  zt kˆ where rt  is the position vector of (x,y,z,) define a curve C joining points P1 and P2 where t=t1 and t=t2 respectively. The curve C is composed of a finite number of curves (see figure) for each of which rt  has a continuous derivative. P2 rd r d r C O r P1 Let F x, y, z  F1iˆ  F2 ˆj  F3kˆ be a vector function of P2 d r n position defined and continuous along C. Then the d r integral of the tangential component of F along C from P1 to P2 dr2 dr1 P1 C P2 is an example of a line integral. If F is a force on a particle moving along C, this line integral represents the work done by the force. If C is a simple closed curve, the integral around C is often denoted by F  d r In general, any integral, which is to be evaluated along a curve is called a line integral. If  is a scalar field, then the line integral of V along the curve C from P1 to P2 is  d r