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# 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). [1]

62 relations: Abuse of notation, Adjoint functors, Adrien-Marie Legendre, Algorithm, Alpha Chiang, Antiderivative, Arity, , Carl Gustav Jacob Jacobi, Chain rule, Cone, Conservative vector field, Constant of integration, Consumption function, Continuous function, Curl (mathematics), D'Alembert operator, Del, Derivative, Derivative test, Differential geometry, Directional derivative, Divergence, Economics, Euclidean space, Exterior derivative, Function space, Gibbs–Duhem equation, Gradient, Graph of a function, Height, Hessian matrix, Jacobian matrix and determinant, Laplace operator, Limit of a function, Marginal propensity to consume, Marquis de Condorcet, Mathematical optimization, Mathematical physics, Mathematics, MathWorld, Neighbourhood (mathematics), Open set, Order condition, Pixel, Product topology, Profit (economics), Radius, Scalar field, Seam carving, ... Expand index (12 more) »

## Abuse of notation

In mathematics, abuse of notation occurs when an author uses a mathematical notation in a way that is not formally correct but that seems likely to simplify the exposition or suggest the correct intuition (while being unlikely to introduce errors or cause confusion).

## Adjoint functors

In mathematics, specifically category theory, adjunction is a possible relationship between two functors.

## Adrien-Marie Legendre

Adrien-Marie Legendre (18 September 1752 – 10 January 1833) was a French mathematician.

## Algorithm

In mathematics and computer science, an algorithm is an unambiguous specification of how to solve a class of problems.

## Alpha Chiang

Alpha Chung-i Chiang (born 1927) is an American mathematical economist, Professor Emeritus of Economics at the University of Connecticut, and author of perhaps the most well known mathematical economics textbook; Fundamental Methods of Mathematical Economics.

## Antiderivative

In calculus, an antiderivative, primitive function, primitive integral or indefinite integral of a function is a differentiable function whose derivative is equal to the original function.

## Arity

In logic, mathematics, and computer science, the arity of a function or operation is the number of arguments or operands that the function takes.

## ∂

The character ∂ (HTML element: &amp;#8706; or &amp;part;, Unicode: U+2202) or \partial is a stylized d mainly used as a mathematical symbol to denote a partial derivative such as \frac (read as "the partial derivative of z with respect to x").

## Carl Gustav Jacob Jacobi

Carl Gustav Jacob Jacobi (10 December 1804 – 18 February 1851) was a German mathematician, who made fundamental contributions to elliptic functions, dynamics, differential equations, and number theory.

## Chain rule

In calculus, the chain rule is a formula for computing the derivative of the composition of two or more functions.

## Cone

A cone is a three-dimensional geometric shape that tapers smoothly from a flat base (frequently, though not necessarily, circular) to a point called the apex or vertex.

## Conservative vector field

In vector calculus, a conservative vector field is a vector field that is the gradient of some function, known in this context as a scalar potential.

## Constant of integration

In calculus, the indefinite integral of a given function (i.e., the set of all antiderivatives of the function) on a connected domain is only defined up to an additive constant, the constant of integration.

## Consumption function

In economics, the consumption function describes a relationship between consumption and disposable income.

## Continuous function

In mathematics, a continuous function is a function for which sufficiently small changes in the input result in arbitrarily small changes in the output.

## Curl (mathematics)

In vector calculus, the curl is a vector operator that describes the infinitesimal rotation of a vector field in three-dimensional Euclidean space.

## D'Alembert operator

In special relativity, electromagnetism and wave theory, the d'Alembert operator (represented by a box: \Box), also called the d'Alembertian, wave operator, or box operator is the Laplace operator of Minkowski space.

## Del

Del, or nabla, is an operator used in mathematics, in particular in vector calculus, as a vector differential operator, usually represented by the nabla symbol &nabla;.

## Derivative

The derivative of a function of a real variable measures the sensitivity to change of the function value (output value) with respect to a change in its argument (input value).

## Derivative test

In calculus, a derivative test uses the derivatives of a function to locate the critical points of a function and determine whether each point is a local maximum, a local minimum, or a saddle point.

## Differential geometry

Differential geometry is a mathematical discipline that uses the techniques of differential calculus, integral calculus, linear algebra and multilinear algebra to study problems in geometry.

## Directional derivative

In mathematics, the directional derivative of a multivariate differentiable function along a given vector v at a given point x intuitively represents the instantaneous rate of change of the function, moving through x with a velocity specified by v. It therefore generalizes the notion of a partial derivative, in which the rate of change is taken along one of the curvilinear coordinate curves, all other coordinates being constant.

## Divergence

In vector calculus, divergence is a vector operator that produces a scalar field, giving the quantity of a vector field's source at each point.

## Economics

Economics is the social science that studies the production, distribution, and consumption of goods and services.

## Euclidean space

In geometry, Euclidean space encompasses the two-dimensional Euclidean plane, the three-dimensional space of Euclidean geometry, and certain other spaces.

## Exterior derivative

On a differentiable manifold, the exterior derivative extends the concept of the differential of a function to differential forms of higher degree.

## Function space

In mathematics, a function space is a set of functions between two fixed sets.

## Gibbs–Duhem equation

In thermodynamics, the Gibbs–Duhem equation describes the relationship between changes in chemical potential for components in a thermodynamic system: where N_i\, is the number of moles of component i\,, \mathrm\mu_i\, the infinitesimal increase in chemical potential for this component, S\, the entropy, T\, the absolute temperature, V\, volume and p\, the pressure.

## Gradient

In mathematics, the gradient is a multi-variable generalization of the derivative.

## Graph of a function

In mathematics, the graph of a function f is, formally, the set of all ordered pairs, and, in practice, the graphical representation of this set.

## Height

Height is the measure of vertical distance, either how "tall" something or someone is, or how "high" the position is.

## Hessian matrix

In mathematics, the Hessian matrix or Hessian is a square matrix of second-order partial derivatives of a scalar-valued function, or scalar field.

## Jacobian matrix and determinant

In vector calculus, the Jacobian matrix is the matrix of all first-order partial derivatives of a vector-valued function.

## Laplace operator

In mathematics, the Laplace operator or Laplacian is a differential operator given by the divergence of the gradient of a function on Euclidean space.

## Limit of a function

Although the function (sin x)/x is not defined at zero, as x becomes closer and closer to zero, (sin x)/x becomes arbitrarily close to 1.

## Marginal propensity to consume

In economics, the marginal propensity to consume (MPC) is a metric that quantifies induced consumption, the concept that the increase in personal consumer spending (consumption) occurs with an increase in disposable income (income after taxes and transfers).

## Marquis de Condorcet

Marie Jean Antoine Nicolas de Caritat, Marquis of Condorcet (17 September 1743 – 29 March 1794), known as Nicolas de Condorcet, was a French philosopher, mathematician, and early political scientist whose Condorcet method in voting tally selects the candidate who would beat each of the other candidates in a run-off election.

## Mathematical optimization

In mathematics, computer science and operations research, mathematical optimization or mathematical programming, alternatively spelled optimisation, is the selection of a best element (with regard to some criterion) from some set of available alternatives.

## Mathematical physics

Mathematical physics refers to the development of mathematical methods for application to problems in physics.

## Mathematics

Mathematics (from Greek μάθημα máthēma, "knowledge, study, learning") is the study of such topics as quantity, structure, space, and change.

## MathWorld

MathWorld is an online mathematics reference work, created and largely written by Eric W. Weisstein.

## Neighbourhood (mathematics)

In topology and related areas of mathematics, a neighbourhood (or neighborhood) is one of the basic concepts in a topological space.

## Open set

In topology, an open set is an abstract concept generalizing the idea of an open interval in the real line.

## Order condition

The order condition is the state of a set of simultaneous equations in an econometric system such that all its parameters may be identified.

## Pixel

In digital imaging, a pixel, pel, dots, or picture element is a physical point in a raster image, or the smallest addressable element in an all points addressable display device; so it is the smallest controllable element of a picture represented on the screen.

## Product topology

In topology and related areas of mathematics, a product space is the cartesian product of a family of topological spaces equipped with a natural topology called the product topology.

## Profit (economics)

In economics, profit in the accounting sense of the excess of revenue over cost is the sum of two components: normal profit and economic profit.

## Radius

In classical geometry, a radius of a circle or sphere is any of the line segments from its center to its perimeter, and in more modern usage, it is also their length.

## Scalar field

In mathematics and physics, a scalar field associates a scalar value to every point in a space – possibly physical space.

## Seam carving

Seam carving (or liquid rescaling) is an algorithm for content-aware image resizing, developed by Shai Avidan, of Mitsubishi Electric Research Laboratories (MERL), and Ariel Shamir, of the Interdisciplinary Center and MERL.

## Slope

In mathematics, the slope or gradient of a line is a number that describes both the direction and the steepness of the line.

## Statistical mechanics

Statistical mechanics is one of the pillars of modern physics.

## Surface (topology)

In topology and differential geometry, a surface is a two-dimensional manifold, and, as such, may be an "abstract surface" not embedded in any Euclidean space.

## Symmetry of second derivatives

In mathematics, the symmetry of second derivatives (also called the equality of mixed partials) refers to the possibility under certain conditions (see below) of interchanging the order of taking partial derivatives of a function of n variables.

## System of equations

In mathematics, a set of simultaneous equations, also known as a system of equations or an equation system, is a finite set of equations for which common solutions are sought.

## Tangent

In geometry, the tangent line (or simply tangent) to a plane curve at a given point is the straight line that "just touches" the curve at that point.

## Total derivative

In the mathematical field of differential calculus, a total derivative or full derivative of a function f of several variables, e.g., t, x, y, etc., with respect to an exogenous argument, e.g., t, is the limiting ratio of the change in the function's value to the change in the exogenous argument's value (for arbitrarily small changes), taking into account the exogenous argument's direct effect as well as indirect effects via the other arguments of the function.

## Triple product rule

The triple product rule, known variously as the cyclic chain rule, cyclic relation, cyclical rule or Euler's chain rule, is a formula which relates partial derivatives of three interdependent variables.

## Unit vector

In mathematics, a unit vector in a normed vector space is a vector (often a spatial vector) of length 1.

## Vector calculus

Vector calculus, or vector analysis, is a branch of mathematics concerned with differentiation and integration of vector fields, primarily in 3-dimensional Euclidean space \mathbb^3.

## Vector field

In vector calculus and physics, a vector field is an assignment of a vector to each point in a subset of space.

## Volume

Volume is the quantity of three-dimensional space enclosed by a closed surface, for example, the space that a substance (solid, liquid, gas, or plasma) or shape occupies or contains.

## References

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