G Some Tools of Vector Analysis
663
G.3 Scalar, Vector, and Tensor Fields
A field is any quantity defined at each point in a given space. A scalar is a tensor of
rank zero, so its value is left unchanged by a rotation of coordinates. A scalar field
is a scalar defined at each point is space. The temperature across your body forms a
scalar field.
A vector field is a field of vectors of a particular kind, defined over a space. Vector
fields naturally occur in biological systems. The field of velocities in a flowing fluid,
such as blood or protoplasm, is a good example. The flux of heat from inside your
body to your skin is a vector field. The electric and magnetic fields within and
across cells is another. The proportionality coefficients relating stress to strain in
your bones forms a tensor field across the bone.
If a field smoothly changes across space, then it is possible to define spatial
derivatives of that field.
G.4 Grad Operation on a Scalar Field
The differential of a scalar field f (x 1 , x 2 , x 3 ), i.e. the change of this function for a
small shift in the coordinates, is given by
df =
i
∂f
∂x i
dx i .
(G.25)
The ‘grad’ operation on a scalar field f , denoted by ∇f , is defined through
df = ∇f · dr .
(G.26)
In curvilinear coordinate, we must have
∇f =
i
1
h i
∂f
∂x i
e i .
(G.27)
G.5 The Divergence of a Vector Field and Gauss’ Theorem
Let J represent the current density for the flow of some fluid entity Q, such as mass,
volume, or charge. Then the amount of that entity passing through a given area A
per unit time can be represented as
dQ
dt
=
A
J · dS ,
(G.28)