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Encyclopedia > Partial derivative

In mathematics, a partial derivative of a function of several variables is its derivative with respect to one of those variables with the others held constant (as opposed to the total derivative, in which all variables are allowed to vary). They are useful in vector calculus and differential geometry. Euclid, Greek mathematician, 3rd century BC, known today as the father of geometry; shown here in a detail of The School of Athens by Raphael. ... Partial plot of a function f. ... In mathematics, the derivative is defined as the instantaneous rate of change of a function. ... In mathematics, a total derivative is a combination of partial derivatives. ... Vector calculus (also called vector analysis) is a field of mathematics concerned with multivariate real analysis of vectors in two or more dimensions. ... In mathematics, differential topology is the field dealing with differentiable functions on differentiable manifolds. ...


The partial derivative of a function f with respect to the variable x is represented as frac{ partial f }{ partial x } or partial_xf or fx (where partial is a rounded 'd' known as the 'partial derivative symbol,' which coincides with the cursive Cyrillic letter "de" and is pronounced as its English counterpart "d" - that incidentally was the notation first introduced by Legendre). The Cyrillic alphabet (pronounced , also called azbuka, from the old name of the first two letters) is an alphabet used for several East and South Slavic languages; (Belarusian, Bulgarian, Macedonian, Russian, Rusyn, Serbian, and Ukrainian) and many other languages of the former Soviet Union, Asia and Eastern Europe. ... De (Д, д) is a letter of the Cyrillic alphabet. ... Adrien-Marie Legendre (September 18, 1752–January 10, 1833) was a French mathematician. ...

Contents

Examples

Consider the volume V of a cone; it depends on the cone's height h and its radius r according to the formula This article needs to be cleaned up to conform to a higher standard of quality. ...

The partial derivative of V with respect to r is

it describes the rate with which a cone's volume changes if its radius is varied and its height is kept constant. The partial derivative with respect to h is This article needs to be cleaned up to conform to a higher standard of quality. ...

frac{ partial V}{partial h} = frac{ r^2 pi }{3}

and represents the rate with which the volume changes if its height is varied and its radius is kept constant.


Equations involving an unknown function's partial derivatives are called partial differential equations and are common in physics and engineering. In mathematics, a partial differential equation (PDE) is a relation involving an unknown function of several independent variables and its partial derivatives with respect to those variables. ...


Notation

For the following examples, let f be a function in x, y and z.


First-order partial derivatives:

frac{ partial f}{ partial x} = f_x = partial_x f

Second-order partial derivatives:

frac{ partial^2 f}{ partial x^2} = f_{xx} = partial_{xx} f

Second-order mixed derivatives: In mathematics, the Hessian matrix is the square matrix of second partial derivatives of a scalar-valued function. ...

frac{ partial^2 f}{partial x,partial y} = f_{xy} = f_{yx} = partial_{xy} f = partial_{yx} f

Higher-order partial and mixed derivatives:

frac{ partial^{i+j+k} f}{ partial x^i, partial y^j, partial z^k } = f^{(i, j, k)}

When dealing with functions of multiple variables, some of these variables may be related to each other, and it may be necessary to specify explicitly which variables are being held constant. In fields such as statistical mechanics, the partial derivative of f with respect to x, holding y and z constant, is often expressed as Statistical mechanics is the application of statistics, which includes mathematical tools for dealing with large populations, to the field of mechanics, which is concerned with the motion of particles or objects when subjected to a force. ...

left( frac{partial f}{partial x} right)_{y,z}

Formal definition and properties

Like ordinary derivatives, the partial derivative is defined as a limit. Let U be an open subset of Rn and f : UR a function. We define the partial derivative of f at the point a = (a1, ..., an) ∈ U with respect to the ith variable xi as In mathematics, the concept of a limit is used to describe the behavior of a function as its argument either gets close to some point, or as it becomes arbitrarily large; or the behavior of a sequences elements, as their index increases indefinitely. ... In topology and related fields of mathematics, a set U is called open if, intuitively speaking, you can wiggle or change any point x in U by a small amount in any direction and still be inside U. In other words, if x is surrounded only by elements of U...

frac{ partial }{partial x_i }f(mathbf{a}) = lim_{h rightarrow 0}{ f(a_1, dots , a_{i-1}, a_i+h, a_{i+1}, dots ,a_n) - f(a_1, dots ,a_n) over h }

Even if all partial derivatives ∂f/∂xi(a) exist at a given point a, the function need not be continuous there. However, if all partial derivatives exist in a neighborhood of a and are continuous there, then f is totally differentiable in that neighborhood and the total derivative is continuous. In this case, we say that f is a C1 function. In mathematics, a continuous function is a function for which, intuitively, small changes in the input result in small changes in the output. ... This is a glossary of some terms used in the branch of mathematics known as topology. ... In mathematics, the derivative is defined as the instantaneous rate of change of a function. ...


The partial derivative ∂f/∂xi can be seen as another function defined on U and can again be partially differentiated. If all mixed partial derivatives exist and are continuous, we call f a C2 function; in this case, the partial derivatives can be exchanged by Clairaut's theorem: In mathematical analysis, Clairauts theorem states that if has continuous second partial derivatives at then for In words, the partial derivatives of this function commute. ...

frac{partial^2f}{partial x_i, partial x_j} = frac{partial^2f} {partial x_j, partial x_i}.

See also


  Results from FactBites:
 
PlanetMath: derivative notation (249 words)
is the derivative with respect to the first variable of the derivative with respect to the second variable.
The second of these notations represents the derivative matrix, which in most cases is the Jacobian, but in some cases, does not exist, even though the Jacobian exists.
This is version 10 of derivative notation, born on 2001-11-14, modified 2006-08-10.
  More results at FactBites »

 
 

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