Page 68 - Introduction to Continuum Mechanics

Part C Vector Field, Gradient of a Vector Field 53

But HI, being in the same direction as V0, is perpendicular to the surface of constant #. Thus,
qi is normal to the surface of constant temperature. Similarly, % is normal to the surface of
constant temperature., etc. We note that if kifa and £3 are all distinct, the equations indicate
that different thermal conductivities in the three principal directions.
(c)Since 6 — 2%i+3x2, we have

i.e.,

— A JU.
which is clearly in a different direction from the normal.

2C3 Vector Field, Gradient of a Vector Field

Let v(r) be a vector-valued function of position, describing, for example, a displacement or
a velocity field. Associated with v(r), there is a tensor field, called the gradient of v, which is
of considerable importance. The gradient of v (denoted by Vv or grad v) is defined to be the
second-order tensor which, when operating on dr gives the difference of v at r + dr and r.
That is,

Again, let dr denote \dr\ and e denote dr/dr, we have

Thus, the second-order tensor (Vv) transforms the unit vector e into the vector describing the
rate of change v in that direction.
Since

thus, in Cartesian coordinates,

That is,

Or, in general