and the limits of summation are determined as dx and dz range over the region
x > 0 , z > 0 ,and 2 x+z≤ 12 .This produces the surface integral
S=
∫x=6
x=0
∫z=12− 2 x
z=0
3
2 dx dz =
∫ 6
0
3
2 (2)(6 −x)dx = 54.
Similarly, if the element dS is projected onto the plane x= 0,it can be verified that
dS =
dy dz
|ˆe 1 ·ˆen|=
3
2 dydz
and the surface area is given by
S=
∫y=6
y=0
∫ 12 − 2 y
z=0
3
2
dz dy = 54.
Element of Volume
In a general (u, v, w )curvilinear coordinate system the (x, y, z )rectangular coor-
dinates of a point are given as functions of (u, v, w )and written
x=x(u, v, w ), y =y(u, v, w ), z =z(u, v, w )
so that the position vector to a point P can be written
r =r (u, v, w ) = x(u, v, w )ˆe 1 +y(u, v, w )ˆe 2 +z(u, v, w )ˆe 3
The vector ∂u∂r is tangent to the coordinate curve r =r (u, v 0 , w 0 ), the vector ∂r∂v is
tangent to the coordinate curve r =r (u 0 , v, w 0 )and the vector ∂w∂r is tangent to the
coordinate curve r =r (u 0 , v 0 , w ). Unit vectors to the coordinates curves are
ˆeu=
∂r
∂u
|∂u∂r |
, ˆev=
∂r
∂v
|∂r∂v |
, ˆew=
∂r
∂w
|∂w∂r
The magnitudes hu, h v, h wdefined by
hu=|
∂r
∂u |, h v=|
∂r
∂v |, h w=|
∂r
∂w |
are called scaled factors. The vector change
dr =∂r
∂u
du +∂r
∂v
dv +∂r
∂w
dw =hudu ˆe 1 +hvdv ˆe 2 +hwdw ˆe 3
