Directional derivatives

Let \(f\) be a scalar function: \[\begin{equation} \require{physics} \vb{f}: S \rightarrow \mathbb{R}: \vb{x} \mapsto f(\vb{x}) \thinspace , \end{equation}\] in which \(S\) is an open subset of \(\mathbb{R}^n\) and \(\vb{y}\) is a vector in \(\mathbb{R}^n\). Let \(\vb{a}\) be an interior point of \(S\). We then call \[\begin{equation} \lim_{h \to 0} \qty( \frac{f(\vb{a} + h\vb{y}) - f(\vb{a})}{h} ) = f'(\vb{a};\vb{y}) \end{equation}\] the derivative of \(f\) with respect to \(\vb{y}\) at \(\vb{a}\).

If \[\begin{equation} g(t) = f(\vb{a} + t \vb{y}) \thinspace , \end{equation}\] then \[\begin{equation} g'(t) = f'(\vb{a} + t \vb{y}; \vb{y}) \thinspace . \end{equation}\]

If \(f(\vb{a} + t \vb{y})\) is differentiable for all \(t \in [0,1]\), then by the mean value theorem we have: \[\begin{equation} \exists \theta \in ]0,1]: \qquad f(\vb{a} + \vb{y}) - f(\vb{a}) = f'(\vb{a} + \theta \vb{y}; \vb{y}) \thinspace . \end{equation}\]

If \(\vb{y}\) is a unit vector (i.e. \(||\vb{y}|| = 1\)), then we call \(f'(\vb{a};\vb{y})\) a special name: the directional derivative of \(f\) w.r.t. \(\vb{y}\) in \(\vb{a}\) and we assign a new symbol to it: \[\begin{equation} \grad_{\vb{y}} f(\vb{a}) = \lim_{h \to 0} \qty( \frac{f(\vb{a} + h\vb{y}) - f(\vb{a})}{h} ) = f'(\vb{a};\vb{y}) \thinspace , \end{equation}\] which is equivalent with \[\begin{equation} \lim_{h \to 0} \qty( \frac{f(\vb{a} + h \vb{y}) - f(\vb{a}) - h \grad_{\vb{y}} f(\vb{a})}{h} ) = 0 \thinspace . \end{equation}\] The \(i\)-th partial derivative of \(f\) in \(\vb{a}\) is defined to be the directional derivative along \(\vb{e}_i\). We define a new symbol \[\begin{equation} \pdv{f(\vb{a})}{x_i} = \grad_{\vb{e}_i} f(\vb{a}) = f'(\vb{a};\vb{e}_i) = \lim_{h \to 0} \qty( \frac{f(\vb{a} + h\vb{e}_i) - f(\vb{a})}{h} ) \thinspace . \end{equation}\]