It can be verified that a unit normal to this surface is
ˆen=
N
|N|
=
1
ar.
That is, the unit outer normal to a point P on the surface of the sphere has the
same direction as the position vector r to the point P.
Example 7-26. If the surface is described in the explicit form z=z(x, y )the
position vector to a point on the surface can be represented
r =r (x, y ) = xˆe 1 +yˆe 2 +z(x, y )ˆe 3
This vector has the partial derivatives
∂r
∂x
=eˆ 1 +∂z
∂x
ˆe 3 and ∂r
∂y
=ˆe 2 +∂z
∂y
ˆe 3
so that a normal to the surface can be calculated from the cross product
N =∂r
∂x ×
∂r
∂y =
∣∣
∣∣
∣∣
ˆe 1 ˆe 2 ˆe 3
1 0 ∂z∂x
0 1 ∂z∂y
∣∣
∣∣
∣∣=−
∂z
∂x ˆe^1 −
∂z
∂y ˆe^2 +ˆe^3
A unit normal to the surface is
ˆen=
N
|N|
=
−∂x∂z ˆe 1 −∂z∂y ˆe 2 +ˆe 3
√
(∂z
∂x
) 2
+
(∂z
∂y
) 2
+ 1
=nxˆe 1 +nyˆe 2 +nzˆe 3
where nx, n y, n zare the direction cosines of the unit normal. Note also that the vector
ˆen∗=
∂z
∂x ˆe^1 +
∂z
√ ∂y eˆ^2 −eˆ^3
(∂z
∂x
) 2
+
(
∂z
∂y
) 2
+ 1
is also normal to the surface.
