Читать книгу An Introduction to the Finite Element Method for Differential Equations - Mohammad Asadzadeh - Страница 15

In this section, we extend the overture of the Sections 1.1 and 1.2 to higher dimensions and give definitions for linearity, nonlinearity, and superposition concepts.

We recall the common notation for the real Euclidean spaces of dimension with the elements . In most of the applications, will be , or 4 and the variables denote coordinates in space dimensions, whereas represents the time variable. In this case, we usually replace by the most common notation: . Further, we shall use the subscript notation for the partial derivatives, viz.

A more general notation for partial derivatives of a sufficiently smooth function (see Definition 1.1 below) is given by

where , denotes the partial derivative of order with respect to the variable , and is a multi‐index of integers with .