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Theory Manual Version 3.6
 Section 2.1: Vectors and Tensors Up Chapter 2: Continuum Mechanics Section 2.3: Deformation, Strain and Stress 

2.2 The Directional Derivative

In later sections the nonlinear finite element method will be formulated. Anticipating an iterative solution method to solve the nonlinear equations, it will be necessary to linearize the quantities involved. This linearization process will utilize a construction called the directional derivative [28].
The directional derivative of a function is defined as follows: The quantity may be a scalar, a vector or even a vector of unknown functions. For instance, consider a scalar function , where is the position vector in . In this case the directional derivative is given by: Here, the symbol (“nabla”) depicts the gradient operator.
The linearization of a function implies that it is approximated by a linear function. Using the directional derivative, a function can be linearized as follows: The directional derivative obeys the usual properties for derivatives.
  1. sum rule: If , then
  2. product rule: If , then
  3. chain rule: If , then
 Section 2.1: Vectors and Tensors Up Chapter 2: Continuum Mechanics Section 2.3: Deformation, Strain and Stress