Jacobianen

Jacobianen is a term used in some mathematical literature to refer to Jacobians in plural, encompassing the Jacobian matrices and determinants associated with differentiable vector-valued functions. In this sense, Jacobianen encompasses the local linear approximations that a function induces near a point and the corresponding scalar measure of how volumes change under the mapping.

For a differentiable function f: R^n -> R^m, the Jacobian matrix J_f(x) is the m-by-n matrix whose entries are the partial derivatives ∂f_i/∂x_j. When n = m, the Jacobian determinant det(J_f(x)) provides a scalar factor for volume distortion under the map near x. A full-rank Jacobian indicates local invertibility, linked to the inverse function theorem, while a singular Jacobian (determinant zero) signals potential critical points or dimensional reduction.

The chain rule is naturally expressed through Jacobians: for g ∘ f, J_{g∘f}(x) = J_g(f(x)) · J_f(x). This rule also underpins change of variables in multiple integrals, where the absolute value of the Jacobian determinant accounts for area or volume scaling during the transformation.

Applications of the concept span several fields, including analysis, differential geometry, and applied domains. In calculus, Jacobians facilitate integration under coordinate changes. In dynamical systems and differential equations, they help analyze stability and local behavior. In robotics, computer vision, and computer graphics, Jacobians relate to motion, projection, and control problems.

Notationally, Jacobianen is more common in some languages as a plural form for Jacobians or Jacobian determinants; in English, the terms Jacobian matrices or Jacobian determinants are typically used.