Page 62 - Introduction to Continuum Mechanics

Part C Tensor-valued functions of a Scalar 47

PartC Tensor Calculus

2C1 Tensor-valued functions of a Scalar

Let T=T(f) be a tensor-valued function of a scalar t (such as time). The derivative of T with
respect to t is defined to be a second-order tensor given by

The following identities can be easily established [only Eq. (2C1.2d) will be proven here];

\ /
To prove Eq. (2C1.2d), we use the definition (2C1.1)

Thus,

Example 2C1.1

Show that in Cartesian coordinates the components ofcfT/dt, i.e., (dlldt}^ are given by the
derivatives of the components, dT^ldt.