In single-variable differential calculus, the fundamental increment lemma is an immediate consequence of the definition of the derivative f'(a) of a function f at a point a:
\( f'(a)=\lim _{{h\to 0}}{\frac {f(a+h)-f(a)}{h}}. \)
The lemma asserts that the existence of this derivative implies the existence of a function\( \varphi \) such that
\( \lim _{{h\to 0}}\varphi (h)=0\qquad {\text{and}}\qquad f(a+h)=f(a)+f'(a)h+\varphi (h)h \)
for sufficiently small but non-zero h. For a proof, it suffices to define
\( \varphi (h)={\frac {f(a+h)-f(a)}{h}}-f'(a) \)
and verify this φ \varphi meets the requirements.
Differentiability in higher dimensions
In that the existence of \( \varphi \) uniquely characterises the number \( f'(a) \) , the fundamental increment lemma can be said to characterise the differentiability of single-variable functions. For this reason, a generalisation of the lemma can be used in the definition of differentiability in multivariable calculus. In particular, suppose f maps some subset of \( \mathbb {R} ^{n} \) to \( \mathbb {R} \) . Then f is said to be differentiable at a if there is a linear function
\( M:{\mathbb {R}}^{n}\to {\mathbb {R}} \)
and a function
\(, {\displaystyle \Phi :D\to \mathbb {R} ,\qquad D\subseteq \mathbb {R} ^{n}\smallsetminus \{\mathbf {0} \},} \)
such that
\( {\displaystyle \lim _{\mathbf {h} \to 0}\Phi (\mathbf {h} )=0\qquad {\text{and}}\qquad f(\mathbf {a} +\mathbf {h} )=f(\mathbf {a} )+M(\mathbf {h} )+\Phi (\mathbf {h} )\cdot \Vert \mathbf {h} \Vert } \)
for non-zero h sufficiently close to 0. In this case, M is the unique derivative (or total derivative, to distinguish from the directional and partial derivatives) of f at a. Notably, M is given by the Jacobian matrix of f evaluated at a.
See also
Generalizations of the derivative
References
Talman, Louis (2007-09-12). "Differentiability for Multivariable Functions" (PDF). Archived from the original (PDF) on 2010-06-20. Retrieved 2012-06-28.
Stewart, James (2008). Calculus (7th ed.). Cengage Learning. p. 942. ISBN 978-0538498845.
Undergraduate Texts in Mathematics
Graduate Studies in Mathematics
Hellenica World - Scientific Library
Retrieved from "http://en.wikipedia.org/"
All text is available under the terms of the GNU Free Documentation License
