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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 1.

The directional derivative of a function \(f\left(\mathbf{x}\right)\) is defined as follows:

\[ \begin{equation} Df\left(\mathbf{x}\right)\left[\mathbf{u}\right]=\left.\frac{d}{d\varepsilon}\right|_{\varepsilon=0}f\left(\mathbf{x}+\varepsilon\mathbf{u}\right)\,.\label{eq27} \end{equation} \]

The quantity \(\mathbf{x}\) may be a scalar, a vector or even a vector of unknown functions. For instance, consider a scalar function \(f\left(\mathbf{x}\right)\), where \(\mathbf{x}\) is the position vector in \(\mathbb{R}^{3}\). In this case the directional derivative is given by:

\[ \begin{equation} \begin{aligned}Df\left(\mathbf{x}\right)\left[\mathbf{u}\right] & =\left.\frac{d}{d\varepsilon}\right|_{\varepsilon=0}f\left(\mathbf{x}+\varepsilon\mathbf{u}\right)\\ & =\frac{\partial f}{\partial x_{i}}u_{i}\\ & =\nabla f\cdot\mathbf{u}\,. \end{aligned} \label{eq28} \end{equation} \]

Here, the symbol \(\nabla\)(“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 \(f\) can be linearized as follows:

\[ \begin{equation} f\left(\mathbf{x}+\mathbf{u}\right)\cong f\left(\mathbf{x}\right)+Df\left(\mathbf{x}\right)\left[\mathbf{u}\right].\label{eq29} \end{equation} \]

The directional derivative obeys the usual properties for derivatives.

  1. sum rule: If \(f=f_{1}+f_{2}\), then

    \[ \begin{equation} Df\left(\mathbf{x}\right)\left[\mathbf{u}\right]=Df_{1}\left(\mathbf{x}\right)\left[\mathbf{u}\right]+Df_{2}\left(\mathbf{x}\right)\left[\mathbf{u}\right]\,.\label{eq30} \end{equation} \]

  2. product rule: If \(f=f_{1}\cdot f_{2}\), then

    \[ \begin{equation} Df\left(\mathbf{x}\right)\left[\mathbf{u}\right]=f_{1}\left(\mathbf{x}\right)\cdot Df_{2}\left(\mathbf{x}\right)\left[\mathbf{u}\right]+f_{2}\cdot Df_{1}\left(\mathbf{x}\right)\left[\mathbf{u}\right]\,.\label{eq31} \end{equation} \]

  3. chain rule: If \(f=g\left(h\left(\mathbf{x}\right)\right)\), then

    \[ \begin{equation} Df\left(\mathbf{x}\right)\left[\mathbf{u}\right]=Dg\left(h\left(\mathbf{x}\right)\right)\left[Dh\left(\mathbf{x}\right)\left[\mathbf{u}\right]\right]\,.\label{eq32} \end{equation} \]

  1. Bonet, Javier; Wood, Richard D.. "Nonlinear continuum mechanics for finite element analysis." Cambridge University Press (1997). ↩