Advanced Calculus fi..

282 Advanced Calculus, Fifth Edition- dydsFigure 5.10Tangent and normal.The line integral 1, P dx + Q dy can be interpreted as a vector integral in anotherway, namely, as*- =/v-nds =/vnds.Cwhere v is the vector Qi - Pj and n is the unit normal vector 90" behind T, so thatv, is the normal component of v. One has then (cf. Fig. 5.10)Csince the vector product a x k of a vector a in the xy-plane with k is a vector b havingthe same magnitude as a and 90" behind a. Accordingly,Thus one hasas asserted. It should be noted that for u = Pi + Qj, one has thenThe following theorem and its generalizations are fundamental in the theory of lineintegrals.4