Summary
In mathematics, the total derivative of a function f at a point is the best linear approximation near this point of the function with respect to its arguments. Unlike partial derivatives, the total derivative approximates the function with respect to all of its arguments, not just a single one. In many situations, this is the same as considering all partial derivatives simultaneously. The term "total derivative" is primarily used when f is a function of several variables, because when f is a function of a single variable, the total derivative is the same as the ordinary derivative of the function. Let be an open subset. Then a function is said to be (totally) differentiable at a point if there exists a linear transformation such that The linear map is called the (total) derivative or (total) differential of at . Other notations for the total derivative include and . A function is (totally) differentiable if its total derivative exists at every point in its domain. Conceptually, the definition of the total derivative expresses the idea that is the best linear approximation to at the point . This can be made precise by quantifying the error in the linear approximation determined by . To do so, write where equals the error in the approximation. To say that the derivative of at is is equivalent to the statement where is little-o notation and indicates that is much smaller than as . The total derivative is the unique linear transformation for which the error term is this small, and this is the sense in which it is the best linear approximation to . The function is differentiable if and only if each of its components is differentiable, so when studying total derivatives, it is often possible to work one coordinate at a time in the codomain. However, the same is not true of the coordinates in the domain. It is true that if is differentiable at , then each partial derivative exists at . The converse does not hold: it can happen that all of the partial derivatives of at exist, but is not differentiable at .
About this result
This page is automatically generated and may contain information that is not correct, complete, up-to-date, or relevant to your search query. The same applies to every other page on this website. Please make sure to verify the information with EPFL's official sources.
Related publications

No results

Related people

No results

Related units

No results

Related concepts (18)
Differentiable manifold
In mathematics, a differentiable manifold (also differential manifold) is a type of manifold that is locally similar enough to a vector space to allow one to apply calculus. Any manifold can be described by a collection of charts (atlas). One may then apply ideas from calculus while working within the individual charts, since each chart lies within a vector space to which the usual rules of calculus apply. If the charts are suitably compatible (namely, the transition from one chart to another is differentiable), then computations done in one chart are valid in any other differentiable chart.
Total derivative
In mathematics, the total derivative of a function f at a point is the best linear approximation near this point of the function with respect to its arguments. Unlike partial derivatives, the total derivative approximates the function with respect to all of its arguments, not just a single one. In many situations, this is the same as considering all partial derivatives simultaneously. The term "total derivative" is primarily used when f is a function of several variables, because when f is a function of a single variable, the total derivative is the same as the ordinary derivative of the function.
Vector-valued function
A vector-valued function, also referred to as a vector function, is a mathematical function of one or more variables whose range is a set of multidimensional vectors or infinite-dimensional vectors. The input of a vector-valued function could be a scalar or a vector (that is, the dimension of the domain could be 1 or greater than 1); the dimension of the function's domain has no relation to the dimension of its range. A common example of a vector-valued function is one that depends on a single real parameter t, often representing time, producing a vector v(t) as the result.
Show more
Related courses (62)
MATH-106(b): Analysis II
Étudier les concepts fondamentaux d'analyse et le calcul différentiel et intégral des fonctions réelles de plusieurs variables.
PHYS-202: Analytical mechanics (for SPH)
Présentation des méthodes de la mécanique analytique (équations de Lagrange et de Hamilton) et introduction aux notions de modes normaux et de stabilité.
PHYS-101(f): General physics : mechanics
Le but du cours de physique générale est de donner à l'étudiant les notions de base nécessaires à la compréhension des phénomènes physiques. L'objectif est atteint lorsque l'étudiant est capable de pr
Show more
Related lectures (628)
Partial and Total Derivatives
Covers partial and total derivatives, differentials, and temporal derivatives in functions with multiple variables.
Show more
Related MOOCs (10)
Analyse I
Le contenu de ce cours correspond à celui du cours d'Analyse I, comme il est enseigné pour les étudiantes et les étudiants de l'EPFL pendant leur premier semestre. Chaque chapitre du cours correspond
Analyse I (partie 1) : Prélude, notions de base, les nombres réels
Concepts de base de l'analyse réelle et introduction aux nombres réels.
Show more