taken between appropriate limits for η. There is a similar formula for the arc of a curve η = const.
(viii.) The area of a surface z = ƒ(x, y) can be expressed by the formula
| ∫∫ { 1 + ( | ∂z | )2 + ( | ∂z | )2 }1/2 dx dy. |
| ∂x | ∂y |
When the coordinates of the points of a surface are expressed as functions of two parameters u, v, the area is expressed by the formula
| ∫∫ [ { | ∂(y, z) | }2 + { | ∂(z, x) | }2 + { | ∂(x, y) | }2 ]1/2 du dv. |
| ∂(u, v) | ∂(u, v) | ∂(u, v) |
When the surface is referred to three-dimensional polar coordinates r, θ, φ given by the equations
x = r sin θ cos φ, y = r sin θ sin φ, z = r cos θ,
and the equation of the surface is of the form r = ƒ(θ, φ), the area is expressed by the formula
| ∫∫ r [ { r2 + ( | ∂r | )2 } sin2 θ + ( | ∂r | )2 ]1/2 dθ dφ. |
| ∂θ | ∂φ |
The surface integral of a function of (θ, φ) over the surface of a sphere r = const. can be expressed in the form