The Mechanics and Thermodynamics of Continua

Consider a motion χ. Since the mapping x = χ(X, t) is invertible in X for fixed t, it has an inverse X = χ−1(x, t), the reference map defined in (5.5) and (5.6). χ−1 associates with each time t and spatial point x in Bt, a material point X = χ−1(x, t) in B.

Using the reference map, we can describe the velocity as a function v(x, t) of the spatial point x and t:

The field v represents the spatial description of the velocity; v(x, t) is the velocity of the material point that at time t occupies the spatial point x.

More generally, let ϕ denote a scalar, vector, or tensor field defined on the body for all time. We generally consider ϕ to be a function ϕ(X, t) of the material point X and the time t; this is called the material description of ϕ. But, as with the velocity, we may also consider ϕ to be a function φ(x, t) of the spatial point x and t; this is called the spatial description and is related to the material description through

Similarly, a field φ(x, t) described spatially may be considered as a function ϕ(X, t) of the material point X and t; this is called the material description and is given by

• When there is no danger of confusion we use the same symbol for both the material and spatial descriptions.