Foreword
The average variation of a function depending on \(x\) when \(x\) runs between \(x\) and \(x+h\) is given by \[\Delta_h(f) = \frac{f(x+h)-f(x)}{h}.\]
Th instantaneous variation at \(x\) is \[f'(x) = \lim_{h\to 0} \frac{f(x+h)-f(x)}{h} = \frac{df}{dx}(x)\] the derivative of \(f\) at the point \(x.\)
Modelling consists in figuring out and analyzing how nature has an effect con the variation of some quantities : velocity, temperature, population, …. Thus the notion of derivative is an essential ingredient in this framework and we will stat by introducing some models described by ordinary differential equations.
When a quantity is depending on several parameters, for instance the 3 directions of the space, then the mathematical models have to involved variations in different directions, i.e. partial derivatives.
If \(f : \mathbb{R}^n \to \mathbb{R}\) is a function depending of \(x = (x_1, \ldots, x_n)\) one has : \[\frac{\partial f}{\partial x_i}(\mathbf{x}) = \lim_{h\to 0} \frac{f(x_1, \ldots, x_i+h, \ldots x_n) - f(x_1, \ldots, x_i, \ldots, x_n)}{h}.\]
It is useful to have in mind that the derivative of a function gives the linear approximations of its variation, i.e. \[f(x+h) - f(x) = f'(x)h + h\varepsilon(h)\] with \(\lim_{h\to 0} \varepsilon(h) = 0,\) and similarly when \(f : \mathbb{R}^n \to \mathbb{R}^m, x \mapsto f(x)\) , for \(h \in \mathbb{R}^n\) one has \[f(x+h) - f(x) = f'(x) h + o(h)\] where \(f'(x)\) is a linear mapping from \(\mathbb{R}^n\) into \(\mathbb{R}^m\) and \(o(h)\) a vector in \(\mathbb{R}^m\) such that \[\Vert o(h) \Vert = \Vert h\Vert \varepsilon(h), \varepsilon(h)\to 0 \; \text{when} \; h \to 0.\]
We use the same notation for the norms in \(\mathbb{R}^n\) or \(\mathbb{R}^m.\)
We will then consider several physical situations which can be described by partial differential equations.