Begin2.DVI
ben green
(Ben Green)
#1
which represents a summation of the elements of surface area over the surface be-
tween appropriate limits assigned to the parameters uand v.
Note that the vectors N =∂r
∂u
×∂r
∂v
and −N =∂r
∂v
×∂r
∂u
are both normal vectors
to the surface r =r (u, v )and
ˆen=±
∂r
∂u
×∂r
∂v
∣∣
∣∣∂r
∂u
×∂r
∂v
∣∣
∣∣
=±
∂r
∂u
×∂r
∂v
√
EG −F^2
are unit normals to the surface.
In the special case the surface is defined by r =r (x, y ) = xˆe 1 +yˆe 2 +z(x, y )ˆe 3 one
can show the element of surface area is given by
dS =|ˆedx dy
n·ˆe 3 |
=√ dx dy
1 +
(
∂z
∂x
) 2
+
(
∂z
∂y
) 2
Here the surface element dS is projected onto the xy -plane to determine the limits
of integration.
In the special case the surface is defined by r =r (x, z ) = xˆe 1 +y(x, z )ˆe 2 +zˆe 3 one
can show the element of surface area is given by
dS = dx dz
|ˆen·ˆe 1 |
=√ dx dz
1 +
(
∂y
∂x
) 2
+
(
∂y
∂z
) 2
Here the surface element dS is projected onto the xz -plane to determine the limits
of integration.
In the special case the surface is defined by r =r (y, z) = x(y, z)ˆe 1 +yˆe 2 +zˆe 3 the
element of surface area is found to be given by
dS = dy dz
|ˆen·eˆ 2 |
=√ dy dz
1 +
(
∂x
∂y
) 2
+
(
∂x
∂z
) 2
In this case the surface element dS is projected onto the yz -plane to determine the
limits of integration.
