When the surface, S, is
described by z = g(x,y) (recall let u
= x and
v = y)
the position vector to
an arbitrary point on S is r =
xi + yj +
g(x,y)k
Then one can use x and y
rather than u and v as the parameters describing the surface, S.
So the normal vector
to S is N(u,v) = N(x,y) = rx
x ry which yields the
Following 3
x 3 determinant. Note: N
is not a unit vector.
i j k i j k
N(x,y) = det ∂x/∂x ∂y/∂x ∂g/∂x =
det
1 0 ∂g/∂x
∂x/∂y
∂y/∂y
∂g/∂y
0 1 ∂g/∂y
N(x,y) =
- ∂g/∂x i - ∂g/∂y j + 1 k
and | N(x,y) | = √[1 + (∂g/∂x)2
+ (∂g/∂x)2]
Thus dS = √[1 + (∂g/∂x)2
+ (∂g/∂x)2] dx dy
and
IS =
∫ ∫ f(x, y, z) dS = ∫ ∫ f(x, y, g(x,y))
√[1 + (∂g/∂x)2
+ (∂g/∂x)2] dx dy
S D
If, on the other
hand, the surface was described
by y
= g(x,z),
then D would be
the “area” projected on
to the x-z plane, etc.
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