Page 476 - Electromagnetics

A.2 Vector transport theorems
We are often interested in the time rate of change of some field integrated over a
moving volume or surface. Such a derivative may be used to describe the transport of a
physical quantity (e.g., charge, momentum, energy) through space. Many of the relevant
theorems are derived in this section. The results find application in the development of
the large-scale forms of Maxwell equations, the continuity equation, and the Poynting
theorem.

Partial, total, and material derivatives
The key to understanding transport theorems lies in the difference between the various
means of time-differentiating a field. Consider a scalar field T (r, t) (which could represent
one component of a vector or dyadic field). If we fix our position within the field and
examine how the field varies with time, we describe the partial derivative of T . However,
this may not be the most useful means of measuring the time rate of change of a field.
For instance, in mechanics we might be interested in the rate at which water cools as
it sinks to the bottom of a container. In this case, T could represent temperature. We
could create a “depth profile” at any given time (i.e., measure T (r, t 0 ) for some fixed t 0 )
by taking simultaneous data from a series of temperature probes at varying depths. We
could also create a temporal profile at any given depth (i.e., measure T (r 0 , t) for some
fixed r 0 ) by taking continuous data from a probe fixed at that depth. But neither of
these would describe how an individual sinking water particle “experiences” a change in
temperature over time.
Instead, we could use a probe that descends along with a particular water packet (i.e.,
volume element), measuring the time rate of temperature change of that element. This
rate of change is called the convective or material derivative, since it corresponds to a
situation in which a physical material quantity is followed as the derivative is calculated.
We anticipate that this quantity will depend on (1) the time rate of change of T at each
fixed point that the particle passes, and (2) the spatial rate of change of T as well as
the rapidity with which the packet of interest is swept through that space gradient. The
faster the packet descends, or the faster the temperature cools with depth, the larger the
material derivative should be.
To compute the material derivative we describe the position of a water packet by the
vector

r(t) = ˆ xx(t) + ˆ yy(t) + ˆ zz(t).
Because no two packets can occupy the same place at the same time, the specification of
r(0) = r 0 uniquely describes (or “tags”) a particular packet. The time rate of change of r
with r 0 held constant (the material derivative of the position vector) is thus the velocity
field u(r, t) of the fluid:
dr Dr

= = u. (A.57)
dt Dt
r 0
Here we use the “big D” notation to denote the material derivative, thereby avoiding
confusion with the partial and total derivatives described below.
To describe the time rate of change of the temperature of a particular water packet, we
only need to hold r 0 constant while we examine the change. If we write the temperature

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