Page 412 - Advanced engineering mathematics

392 CHAPTER 12 Vector Integral Calculus

Then
√
N(P 0 ) = i + 2j + k.
This result is consistent with the fact that a line from the origin through a point on this hemisphere
is normal to the hemisphere at that point.

12.5.2 Tangent Plane to a Surface
If a surface has a normal vector N(P 0 ) at a point then it has a tangent plane at P 0 .This
is the plane through P 0 : (x 0 , y 0 , z 0 ) having normal vector N(P 0 ). The equation of this tangent
plane is

N(P 0 ) ·[(x − x 0 )i + (y − y 0 )j + (z − z 0 )k]= 0,
or

∂(y, z) ∂(z, x) ∂(x, y)
(x − x 0 ) + (y − y 0 ) + (z − z 0 ) = 0.
∂(u,v) ∂(u,v) ∂(u,v)
(u 0 ,v 0 ) (u 0 ,v 0 ) (u 0 ,v 0 )
If is given by z = S(x, y), this tangent plane has equation

∂S ∂S
− (x − x 0 ) − (y − y 0 ) + z − z 0 = 0.
∂x (x 0 ,y 0 ) ∂y (x 0 ,y 0 )
EXAMPLE 12.18
√
For the elliptical cone of Example 12.16, the tangent plane at ( 3a/4,b/4,1/2) has
equation
√ √
3b 3a a b ab 1
− x − − x − + z − = 0.
4 4 4 4 2 2

EXAMPLE 12.19
√
For the hemisphere of Example 12.17, the tangent plane at (1, 2,1) has equation
√ √
(x − 1) + 2(y − 2) + (z − 1) = 0,
or
√
x + 2y + z = 4.

12.5.3 Piecewise Smooth Surfaces

A curve is smooth if it has a continuous tangent. A smooth surface is one that has a contin-
uous normal. A piecewise smooth surface is one that consists of a finite number of smooth
surfaces. For example, a sphere is smooth and the surface of a cube is piecewise smooth,
consisting of six smooth faces. The cube does not have a normal vector (or tangent plane)
along an edge.

In calculus it is shown that the area of a smooth surface given by z = S(x, y) is

2 2

∂S ∂S
area of = 1 + + dA
D ∂x ∂y

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October 14, 2010 14:53 THM/NEIL Page-392 27410_12_ch12_p367-424