Mathematical Methods for Physics and Engineering : A Comprehensive Guide

VECTOR CALCULUS

For the functionφ=x^2 y+yzat the point(1, 2 ,−1), find its rate of change with distance

in the directiona=i+2j+3k. At this same point, what is the greatest possible rate of

change with distance and in which direction does it occur?

The gradient ofφis given by (10.26):

∇φ=2xyi+(x^2 +z)j+yk,

=4i+2k at the point (1, 2 ,−1).

The unit vector in the direction ofaisˆa=√ 141 (i+2j+3k), so the rate of change ofφ
with distancesin this direction is, using (10.30),

dφ

ds

=∇φ·aˆ=

1

√

14

(4+6)=

10

√

14

.

From the above discussion, at the point (1, 2 ,−1)dφ/dswill be greatest in the direction
of∇φ=4i+2kand has the value|∇φ|=

√

20 in this direction.

We can extend the above analysis to find the rate of change of a vector

field (rather than a scalar field as above) in a particular direction. The scalar

differential operatorˆa·∇can be shown to give the rate of change with distance

in the directionaˆof the quantity (vector or scalar) on which it acts. In Cartesian

coordinates it may be written as

ˆa·∇=ax

∂

∂x

+ay

∂

∂y

+az

∂

∂z

. (10.31)

Thus we can write the infinitesimal change in an electric field in moving fromr

tor+drgiven in (10.20) asdE=(dr·∇)E.

A second interesting geometrical property of∇φmay be found by considering

the surface defined byφ(x, y, z)=c,wherecis some constant. Ifˆtis a unit

tangent to this surface at some point then clearlydφ/ds= 0 in this direction

and from (10.29) we have∇φ·ˆt= 0. In other words,∇φis a vector normal to

the surfaceφ(x, y, z)=cat every point, as shown in figure 10.5. Ifnˆis a unit

normal to the surface in the direction of increasingφ(x, y, z), then the gradient is

sometimes written

∇φ≡

∂φ

∂n

nˆ, (10.32)

where∂φ/∂n≡|∇φ|is the rate of change ofφin the directionˆnand is called

thenormal derivative.

Find expressions for the equations of the tangent plane and the line normal to the surface

φ(x, y, z)=cat the pointPwith coordinatesx 0 ,y 0 ,z 0. Use the results to find the equations

of the tangent plane and the line normal to the surface of the sphereφ=x^2 +y^2 +z^2 =a^2

at the point(0, 0 ,a).

A vector normal to the surfaceφ(x, y, z)=cat the pointPis simply∇φevaluated at that
point; we denote it byn 0 .Ifr 0 is the position vector of the pointPrelative to the origin,