Mathematical Methods for Physics and Engineering - Matematica.NET

10.2 INTEGRATION OF VECTORSNote that the differential of a vector is also a vector. As an example, theinfinitesimal change in the position vector of a particle in an infinitesimal time dt isdr = dr dt = v dt,dtwhere v is the particle’s velocity.10.2 Integration of vectorsThe integration of a vector (or of an expression involving vectors that may itselfbe either a vector or scalar) with respect to a scalar u canberegardedastheinverse of differentiation. We must remember, however, that(i) the integral has the same nature (vector or scalar) as the integrand,(ii) the constant of integration for indefinite integrals must be of the samenature as the integral.For example, if a(u) =d[A(u)]/du then the indefinite integral of a(u) is given by∫a(u) du = A(u)+b,where b is a constant vector. The definite integral of a(u) fromu = u 1 to u = u 2is given by∫ u2u 1a(u) du = A(u 2 ) − A(u 1 ).◮A small particle of mass m orbits a much larger mass M centred at the origin O. Accordingto Newton’s law of gravitation, the position vector r of the small mass obeys the differentialequationm d2 rdt = − GMm ˆr.2 r 2Show that the vector r × dr/dt is a constant of the motion.Forming the vector product of the differential equation with r, weobtainr × d2 rdt = − GM2 r r × ˆr.2Since r and ˆr are collinear, r × ˆr = 0 and therefore we haveHowever,r × d2 r= 0. (10.10)dt2 (dr × dr )= r × d2 rdt dt dt + dr2 dt × drdt = 0,339