Page 64 - Introduction to Continuum Mechanics

Part C Scalar Field, Gradient of a Scalar Function 49

Example 2C1.3

A time-dependent rigid body rotation about a fixed point can be represented by a rotation
tensor R(r), so that a position vector r 0 is transformed through rotation into r(t)=R(f)r 0. Derive
the equation

where co is the dual vector of the antisymmetric tensor

Solution. From r(t)=R(t)r 0

But, is an antisymmetric tensor (see Example 2C1.2) so that

where a» is the dual vector of

From the well-known equation in rigid body kinematics, we can identify o> as the angular
velocity of the body.

2C2 Scalar Field, Gradient of a Scalar Function
Let 0(r) be a scalar-valued function of the position vector r. That is, for each position
r r
> 0( ) gives the value of a scalar, such as density, temperature or electric potential at the point.
In other words, <p(r) describes a scalar field. Associated with a scalar field, there is a vector
field, called the gradient of 0, which is of considerable importance. The gradient of 0 at a point
ris defined to be a vector, denoted by (grad 0), or by V# such that its dot product with drgives
the difference of the values of the scalar at r+ dr and r, i.e.,

If dr denotes the magnitude of dr, and e the unit vector in the direction of dr(note: e=drfdr),
then the above equation gives, for dr\n the e direction,