Page 136 - Vector Analysis

132 CHAPTER 4. Vector Calculus

Moreover, letting A denote the inverse of the Jacobian matrix of Ψ; that is, A = (∇Ψ)´1,
[] [ ]
and letting gαβ (n´1)ˆ(n´1) be the inverse matrix of gαβ (n´1)ˆ(n´1), we find that

[ n´1 ... ... n´1 ... N ]T
ψ
Aˇˇtyn=0u = ÿ g1αψ,α ¨¨¨¨¨¨¨¨¨¨¨¨ ÿ g(n´1)αψ,α ˝ .

α=1 α=1

As a consequence,

(JATen)ˇˇtyn=0u = ?g (N ˝ ψ) . (4.17)

4.6 The Divergence Theorem

Two differential operators play important roles in vector calculus. The first one is called
the divergence operator which measures the flux of a vector field, and the second one is
called the curl operator which measures the circulation (the speed of rotation) of a vector
field. We will study this two operators in the following two sections.

4.6.1 Flux integrals

Let Σ Ď R3 be an oriented surface with a fixed unit normal vector field N : Σ Ñ R3, and
u : Σ Ñ R3 be a vector-valued function. The flux integral of u over Σ with given orientation
N is the surface integral of u ¨ N over Σ.

Physical interpretation

Let Ω Ď R3 be an open set which stands for a fluid container and fully contains some liquid
such as water, and u : Ω Ñ R3 be a vector-field which stands for the fluid velocity; that is,
u(x) is the fluid velocity at point x P Ω. Furthermore, let Σ Ď Ω be a surface immersed in
the fluid with given orientation N, and c : Ω Ñ R be the concentration of certain material
dissolving in the liquid. Then the amount of the material carried across the surface in the
direction N by the fluid in a time period of ∆t is

ż
∆t ¨ cu ¨ N dS .

Σ

ż

Therefore, cu¨N dS is the instantaneous amount of the material carried across the surface

Σ

in the direction N by the fluid.