The equation of the line through the point (x 0 , y 0 , z 0 )which is perpendicular to
the tangent plane is given by
r =r 0 +λN (7 .88)
where λis a scalar. The equation of the line can also be expressed by the parametric
equations
x=x 0 +λN 1 , y =y 0 +λN 2 , z =z 0 +λN 3 (7 .89)
where again, the normal vector N can be replaced by any of the normals previously
calculated.
Element of Surface Area
Consider the case where the surface is given in the explicit form z=z(x, y ).In
this case, the position vector of a point on the surface is given by
r =r (x, y ) = xˆe 1 +yˆe 2 +z(x, y )ˆe 3. (7 .90)
The curves
r (x, y )
y= Constant
and r (x, y )
x=Constant
are coordinate curves lying in the surface which intersect at a common point (x, y, z ).
The vectors
∂r
∂x
=eˆ 1 +∂z
∂x
ˆe 3 and ∂r
∂y
=ˆe 2 +∂z
∂y
ˆe 3
are tangent to these coordinate curves, and consequently the differential of the po-
sition vector
dr =
∂r
∂x dx +
∂r
∂y dy
lies in the tangent plane to the surface at the common point of intersection of the
coordinate curves. This differential is illustrated in figure 7-19.
Consider an element of area ∆A= ∆ x∆yin the xy plane of figure 7-19. When this
element of area is projected onto the surface z=z(x, y ),it intersects the surface in an
element of surface area ∆S. When projected onto the tangent plane to the surface
it intersects the tangent plane in an element of surface area ∆R. These projections
are illustrated in figure 7-19(c).
