Page 380 - Advanced engineering mathematics

360 CHAPTER 11 Vector Differential Calculus

y
Normal to the
tangent plane at P 0

Tangent to C at P 0
Part of S

P 0

C
z

Part of the tangent
plane at P 0
x
FIGURE 11.8 Normal to a level surface.

for a ≤ t ≤ b. Then
d ∂ϕ ∂ϕ ∂ϕ

ϕ(x(t), y(t), z(t)) = 0 = x (t) + y (t) + z (t)
dt ∂x ∂y ∂z

=∇ϕ ·[x (t)i + y (t)j + z (t)k].
But x (t)i + y (t)j + z (t)k = T(t) is a tangent vector to C. Letting t = t 0 , T(t 0 ) is tangent to C at

P 0 and the last equation tells us that
∇ϕ(P 0 ) · T(t 0 ) = 0.
Therefore ∇ϕ(P 0 ) is normal to the tangent to C at P 0 .But C is any smooth curve on S passing
through P 0 . Therefore ∇ϕ(P 0 ) is normal to every tangent vector at P 0 to any curve on S through
P 0 , and is therefore normal to the tangent plane to S at P 0 .
Now we have a point P 0 on the normal plane at P 0 , and a vector ∇ϕ(P 0 ) orthogonal to this
plane. The equation of the tangent plane is
∇ϕ(P 0 ) ·[(x − x 0 )i + (y − y 0 )j + (z − z) 0 k]= 0,
or
∂ϕ ∂ϕ ∂ϕ
(P 0 )(x − x 0 ) + (P 0 )(y − y 0 ) + (P 0 )(z − z 0 ) = 0. (11.1)
∂x ∂y ∂z
A straight line through P 0 and parallel to the normal vector is called the normal line to S at
P 0 . Since the gradient of ϕ at P 0 is a normal vector, if (x, y, z) is on this normal line, then for
some scalar t,
(x − x 0 )i + (y − y 0 )j + (z − z 0 )k = t∇ϕ(P 0 ).

The parametric equations of the normal line to S at P 0 are
∂ϕ ∂ϕ ∂ϕ
x = x 0 + t (P 0 ), y = y 0 + t (P 0 ), z = z 0 + t (P 0 ). (11.2)
∂x ∂y ∂z

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October 15, 2010 16:11 THM/NEIL Page-360 27410_11_ch11_p343-366