User:Cronholm144/Derivative: Difference between revisions

Revision as of 00:08, 10 June 2007

Revision as of 15:59, 10 June 2007

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The tangent line to the graph of a function generalizes to the [[tangent space]]. The tangent space is a geometric representation of the best linear approximation to the graph of a function.

The transpose of the Jacobian matrix determines a linear map from '''R'''^''m'' to '''R'''^''n''. More intrinsically, this is the dual map on [[dual space|dual vector spaces]]. This linear map is called the '''[[pullback (differential geometry)|pullback]]'''. of ''f''In atthe example above, the pullback would be written '''a'f''^*(5, -2).

The definition of the total derivative subsumes the definition of the derivative in one variable. In this case, the total derivative exists if and only if the usual derivative exists. The Jacobian matrix at the point ''a'' reduces to a 1&times;1 matrix whose only entry is the derivative ''f<nowiki>'</nowiki>(a)''. This 1&times;1 matrix satisfies the property that ''f(a + h) - f(a) - f'(a)h'' is approximately zero, in other words that