Mathematical Methods for Physics and Engineering : A Comprehensive Guide

10.1 DIFFERENTIATION OF VECTORS

a(u)

a(u+∆u)

∆a=a(u+∆u)−a(u)

Figure 10.1 A small change in a vectora(u) resulting from a small change

inu.

assuming that the limit exists, in which casea(u) is said to be differentiable at

that point. Note thatda/duis also a vector, which is not, in general, parallel to

a(u). In Cartesian coordinates, the derivative of the vectora(u)=axi+ayj+azk

is given by

da

du

=

dax

du

i+

day

du

j+

daz

du

k.

Perhaps the simplest application of the above is to finding the velocity and

acceleration of a particle in classical mechanics. If the time-dependent position

vector of the particle with respect to the origin in Cartesian coordinates is given

byr(t)=x(t)i+y(t)j+z(t)kthen the velocity of the particle is given by the vector

v(t)=

dr

dt

=

dx

dt

i+

dy

dt

j+

dz

dt

k.

The direction of the velocity vector is along the tangent to the pathr(t)atthe

instantaneous position of the particle, and its magnitude|v(t)|is equal to the

speed of the particle. The acceleration of the particle is given in a similar manner

by

a(t)=

dv

dt

=

d^2 x

dt^2

i+

d^2 y

dt^2

j+

d^2 z

dt^2

k.

The position vector of a particle at timetin Cartesian coordinates is given byr(t)=

2 t^2 i+(3t−2)j+(3t^2 −1)k. Find the speed of the particle att=1and the component of

its acceleration in the directions=i+2j+k.

The velocity and acceleration of the particle are given by

v(t)=

dr

dt

=4ti+3j+6tk,

a(t)=

dv

dt

=4i+6k.