Begin2.DVI

This produces an element of area

dS = (rdθ)(2πr sin θ) = 2 πr^2 sin θ dθ for 0 ≤θ≤θ 0

The total surface area of the spherical cap is obtained by a summation of the ring

elements to produce the integral

S=

∫ θ 0

0

2 πr^2 sin θ dθ = 2πr^2 [−cos θ)]θ 00 = 2πr^2 (1 −cos θ 0 )

The solid angle subtended by this right circular cone is therefore

Ω =

S

r^2 = 2π(1 −cos θ^0 )

Potential Theory

Potential theory is concerned with the solutions of Laplace’s equation ∇^2 u= 0,

which satisfy prescribed boundary conditions. Two important problems of potential

theory are the Dirichlet problem and the Neumann problem.

The Dirichlet problem deals with finding a solution U of Laplace’s equation

throughout a region Rsuch that Utakes on certain pre assigned values on the bound-

ary of the region R.

The Neumann problem is concerned with obtaining a solution of Laplace’s equa-

tion in a region Rsuch that on the boundary of R, the normal derivative

∂U

∂n = grad U·

ˆen

has prescribed values. Here ˆen is the unit outward normal to the boundary of the

region R.

In obtaining a solution to a Dirichlet or Neumann problem in an infinite region

there is the additional requirement that U satisfy certain conditions far from the

origin.

Velocity Fields and Fluids

Let V
denote the velocity field of a fluid in motion and let ρ(x, y, z, t)denote the

density of this fluid. Place within the fluid an arbitrary closed surface and consider

an element of surface area dS on this surface. Let the mass of fluid flowing in a

normal direction across this element of surface, in a time interval ∆t, be denoted by