Page 379 - Advanced engineering mathematics

11.4 The Gradient Field 359

11.4.1 Level Surfaces, Tangent Planes, and Normal Lines
Depending on ϕ and the number k, the locus of points (x, y, z) such that ϕ(x, y, z) = k may be a
surface in 3-space. Any such surface is called a level surface of ϕ. For example, if ϕ(x, y, z) =
√
2
2
2
x + y +z , then the level surface of ϕ(x, y, z)=k is a sphere of radius k if k >0, a single point
(0,0,0) if k = 0, and is vacuous if k < 0. Part of the level surface ψ(x, y, z) = z − sin(xy) = 0is
shown in Figure 11.7.
Suppose P 0 :(x 0 , y 0 , z 0 ) is on a level surface S given by ϕ(x, y, z)=k. Assume that there are
smooth (having continuous tangents) curves on the surface passing through P 0 , as typified by C
in Figure 11.8. Each such curve has a tangent vector at P 0 . The plane containing these tangent
vectors is called the tangent plane to S at P 0 . A vector orthogonal to this tangent plane at P 0 is
called a normal vector,or normal, to this tangent plane at P 0 . We will determine this tangent
plane and normal vector. The key lies in the following fact about the gradient vector.

THEOREM 11.2 Normal to a Level Surface

Let ϕ and its first partial derivatives be continuous. Then ∇ϕ(P) is normal to the level surface
ϕ(x, y, z) = k at any point P on this surface such that ∇ϕ(P) = O.

To understand this conclusion, let P 0 be on the level surface S and suppose a smooth curve C
on the surface passes through P 0 , as in Figure 11.8. Let C have parametric equations x =x(t), y =
y(t), z = z(t) for a ≤ t ≤ b. Since P 0 is on C,for some t 0 ,

x(t 0 ) = x 0 , y(t 0 ) = y 0 , z(t 0 ) = z 0 .

Furthermore, because C lies on the level surface,

ϕ(x(t), y(t), z(t)) = k

FIGURE 11.7 Part of the graph of the level
surface z = sin(xy).

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October 15, 2010 16:11 THM/NEIL Page-359 27410_11_ch11_p343-366