Page 477 - Electromagnetics

as

T (r, t) = T (r(r 0 , t), t) = T [x(r 0 , t), y(r 0 , t), z(r 0 , t), t],
then we can use the chain rule to find the time rate of change of T with r 0 held constant:

DT dT
=
Dt dt
r 0
∂T dx ∂T dy ∂T dz ∂T

= + + + .
∂x dt ∂y dt ∂z dt ∂t
r 0 r 0 r 0
We recognize the partial derivatives of the coordinates as the components of the material
velocity (A.57), and thus can write
DT ∂T ∂T ∂T ∂T ∂T
= + u x + u y + u z = + u ·∇T.
Dt ∂t ∂x ∂y ∂z ∂t
As expected, the material derivative depends on both the local time rate of change and
the spatial rate of change of temperature.
Suppose next that our probe is motorized and can travel about in the sinking water.
If the probe sinks faster than the surrounding water, the time rate of change (measured
by the probe) should exceed the material derivative. Let the probe position and velocity
be
dx(t) dy(t) dz(t)
r(t) = ˆ xx(t) + ˆ yy(t) + ˆ zz(t), v(r, t) = ˆ x + ˆ y + ˆ z .
dt dt dt
We can use the chain rule to determine the time rate of change of the temperature
observed by the probe, but in this case we do not constrain the velocity components to
represent the moving fluid. Thus, we merely obtain

dT ∂T dx ∂T dy ∂T dz ∂T
= + + +
dt ∂x dt ∂y dt ∂z dt ∂t
∂T
= + v ·∇T.
∂t
This is called the total derivative of the temperature field.
In summary, the time rate of change of a scalar field T seen by an observer moving
with arbitrary velocity v is given by the total derivative
dT ∂T
= + v ·∇T. (A.58)
dt ∂t
If the velocity of the observer happens to match the velocity u of a moving substance,
the time rate of change is the material derivative
DT ∂T
= + u ·∇T. (A.59)
Dt ∂t
We can obtain the material derivative of a vector field F by component-wise application
of (A.59):
DF D
= ˆ xF x + ˆ yF y + ˆ zF z
Dt Dt
∂F x ∂F y ∂ F z
= ˆ x + ˆ y + ˆ z + ˆ x [u · (∇F x )] + ˆ y u · (∇F y ) + ˆ z [u · (∇F z )] .
∂t ∂t ∂t

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