Mathematical Methods for Physics and Engineering : A Comprehensive Guide

10.7 VECTOR OPERATORS

x

y

z

nˆ 0

(0, 0 ,a)

O

a

φ=x^2 +y^2 +z^2 =a^2

z=a

Figure 10.6 The tangent plane and the normal to the surface of the sphere

φ=x^2 +y^2 +z^2 =a^2 at the pointr 0 with coordinates (0, 0 ,a).

andris the position vector of any point on the tangent plane, then the vector equation of
the tangent plane is, from (7.41),

(r−r 0 )·n 0 =0.

Similarly, ifris the position vector of any point on the straight line passing throughP
(with position vectorr 0 ) in the direction of the normaln 0 then the vector equation of this
line is, from subsection 7.7.1,

(r−r 0 )×n 0 = 0.

For the surface of the sphereφ=x^2 +y^2 +z^2 =a^2 ,

∇φ=2xi+2yj+2zk

=2ak at the point (0, 0 ,a).

Therefore the equation of the tangent plane to the sphere at this point is

(r−r 0 )· 2 ak=0.

This gives 2a(z−a)=0orz=a, as expected. The equation of the line normal to the
sphere at the point (0, 0 ,a)is

(r−r 0 )× 2 ak= 0 ,

which gives 2ayi− 2 axj= 0 orx=y=0,i.e.thez-axis, as expected. The tangent plane
and normal to the surface of the sphere at this point are shown in figure 10.6.

Further properties of the gradient operation, which are analogous to those of

the ordinary derivative, are listed in subsection 10.8.1 and may be easily proved.