Page 65 - Introduction to Continuum Mechanics

§0 Tensors Calculus

That is, the component of V0 in the direction of e gives the rate of change of <p in that direction
(the directional derivative). In particular, the components of V<p in the *i direction is given by

Similarly,

Therefore, the Cartesian components of V0 are that is,

The gradient vector has a simple geometrical interpretation. For example, if $(r) describes
a temperature field, then, on a surface of constant temperature (i.e., isothermal surface), 0 ~
a constant. Let r be a point on this surface. Then, for any and all neighboring point rf dron
the same isothermal surface, d^»=0. Thus, V<j>-dr**Q. In other words, V0 is a vector, perpen-
dicular to the surface at the point r. On the other hand, the dot product V0 -dr is a maximum
when dris in the same direction as V0. In other words, V0 is greatest ifdris normal to the
surface of constant <j> and in this case,

for dr in the normal direction.

Example 2C2.1
If <j>-xiX2+xi, find a unit vector n normal to the surface of a constant 0 passing through
(2,1,0).
Solution, We have

At the point (2,1,0), V0=e 1+2e 2+e 3. Thu: