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A determinant whose elements are functions. Functional determinants of specific types play an important role in mathematical analysis. In the first place this refers to the [[Jacobian|Jacobian]] and the [[Wronskian|Wronskian]]. The concept of a Jacobian is used in an essential way when studying differentiable mappings between domains of Euclidean spaces <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f042/f042050/f0420501.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f042/f042050/f0420502.png" />; when changing the variable in multiple integrals; when clarifying conditions for determining an implicit function by a system of equations or when a system of given functions is dependent; etc. The concept of the Wronskian is extensively applied in the theory of linear ordinary differential equations.