Partial Derivatives
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In
mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, a partial derivative of a
function of several variables In mathematics, a function from a set to a set assigns to each element of exactly one element of .; the words map, mapping, transformation, correspondence, and operator are often used synonymously. The set is called the domain of the functi ...
is its
derivative In mathematics, 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). Derivatives are a fundamental tool of calculus. F ...
with respect to one of those variables, with the others held constant (as opposed to the
total derivative In mathematics, the total derivative of a function 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 resp ...
, in which all variables are allowed to vary). Partial derivatives are used in
vector calculus Vector calculus, or vector analysis, is concerned with differentiation and integration of vector fields, primarily in 3-dimensional Euclidean space \mathbb^3. The term "vector calculus" is sometimes used as a synonym for the broader subject ...
and
differential geometry Differential geometry is a mathematical discipline that studies the geometry of smooth shapes and smooth spaces, otherwise known as smooth manifolds. It uses the techniques of differential calculus, integral calculus, linear algebra and multili ...
. The partial derivative of a function f(x, y, \dots) with respect to the variable x is variously denoted by It can be thought of as the rate of change of the function in the x-direction. Sometimes, for z=f(x, y, \ldots), the partial derivative of z with respect to x is denoted as \tfrac. Since a partial derivative generally has the same arguments as the original function, its functional dependence is sometimes explicitly signified by the notation, such as in: :f'_x(x, y, \ldots), \frac (x, y, \ldots). The symbol used to denote partial derivatives is
The Character (symbol), character ∂ (Unicode: U+2202) is a stylized cursive ''d'' mainly used as a Table of mathematical symbols, mathematical symbol, usually to denote a partial derivative such as / (read as "the partial derivative of ''z'' wit ...
. One of the first known uses of this symbol in mathematics is by
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 and mathematician. His ideas, including support for a liberal economy, free and equal pu ...
from 1770, who used it for partial differences. The modern partial derivative notation was created by
Adrien-Marie Legendre Adrien-Marie Legendre (; ; 18 September 1752 – 9 January 1833) was a French mathematician who made numerous contributions to mathematics. Well-known and important concepts such as the Legendre polynomials and Legendre transformation are name ...
(1786), although he later abandoned it;
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, determinants, and number theory. His name is occasiona ...
reintroduced the symbol in 1841.


Definition

Like ordinary derivatives, the partial derivative is defined as a
limit Limit or Limits may refer to: Arts and media * ''Limit'' (manga), a manga by Keiko Suenobu * ''Limit'' (film), a South Korean film * Limit (music), a way to characterize harmony * "Limit" (song), a 2016 single by Luna Sea * "Limits", a 2019 ...
. Let ''U'' be an open subset of \R^n and f:U\to\R a function. The partial derivative of ''f'' at the point \mathbf=(a_1, \ldots, a_n) \in U with respect to the ''i''-th variable ''x''''i'' is defined as :\begin \fracf(\mathbf) & = \lim_ \frac \\ & = \lim_ \frac \end Even if all partial derivatives ''∂f''/''∂x''''i''(''a'') exist at a given point ''a'', the function need not be
continuous Continuity or continuous may refer to: Mathematics * Continuity (mathematics), the opposing concept to discreteness; common examples include ** Continuous probability distribution or random variable in probability and statistics ** Continuous ...
there. However, if all partial derivatives exist in a
neighborhood A neighbourhood (British English, Irish English, Australian English and Canadian English) or neighborhood (American English; see spelling differences) is a geographically localised community within a larger city, town, suburb or rural area, ...
of ''a'' and are continuous there, then ''f'' is totally differentiable in that neighborhood and the total derivative is continuous. In this case, it is said that ''f'' is a ''C''1 function. This can be used to generalize for vector valued functions, by carefully using a componentwise argument. The partial derivative \frac can be seen as another function defined on ''U'' and can again be partially differentiated. If all mixed second order partial derivatives are continuous at a point (or on a set), ''f'' is termed a ''C''2 function at that point (or on that set); in this case, the partial derivatives can be exchanged by
Clairaut's theorem Clairaut's theorem characterizes the surface gravity on a viscous rotating ellipsoid in hydrostatic equilibrium under the action of its gravitational field and centrifugal force. It was published in 1743 by Alexis Claude Clairaut in a treatis ...
: :\frac = \frac .


Notation

For the following examples, let f be a function in x, y and z. First-order partial derivatives: :\frac = f'_x = \partial_x f. Second-order partial derivatives: :\frac = f''_ = \partial_ f = \partial_x^2 f. Second-order mixed derivatives: :\frac = \frac \left( \frac \right) = (f'_)'_ = f''_ = \partial_ f = \partial_y \partial_x f . Higher-order partial and mixed derivatives: :\frac = f^ = \partial_x^i \partial_y^j \partial_z^k f. When dealing with functions of multiple variables, some of these variables may be related to each other, thus it may be necessary to specify explicitly which variables are being held constant to avoid ambiguity. In fields such as
statistical mechanics In physics, statistical mechanics is a mathematical framework that applies statistical methods and probability theory to large assemblies of microscopic entities. It does not assume or postulate any natural laws, but explains the macroscopic be ...
, the partial derivative of f with respect to x, holding y and z constant, is often expressed as :\left( \frac \right)_ . Conventionally, for clarity and simplicity of notation, the partial derivative ''function'' and the ''value'' of the function at a specific point are
conflated Conflation is the merging of two or more sets of information, texts, ideas, opinions, etc., into one, often in error. Conflation is often misunderstood. It originally meant to fuse or blend, but has since come to mean the same as equate, treati ...
by including the function arguments when the partial derivative symbol (Leibniz notation) is used. Thus, an expression like :\frac is used for the function, while :\frac might be used for the value of the function at the point (x,y,z)=(u,v,w). However, this convention breaks down when we want to evaluate the partial derivative at a point like (x,y,z)=(17, u+v, v^2). In such a case, evaluation of the function must be expressed in an unwieldy manner as :\frac(17, u+v, v^2) or :\left. \frac\right , _ in order to use the Leibniz notation. Thus, in these cases, it may be preferable to use the Euler differential operator notation with D_i as the partial derivative symbol with respect to the ''i''th variable. For instance, one would write D_1 f(17, u+v, v^2) for the example described above, while the expression D_1 f represents the partial derivative ''function'' with respect to the 1st variable. For higher order partial derivatives, the partial derivative (function) of D_i f with respect to the ''j''th variable is denoted D_j(D_i f)=D_ f. That is, D_j\circ D_i =D_, so that the variables are listed in the order in which the derivatives are taken, and thus, in reverse order of how the composition of operators is usually notated. Of course,
Clairaut's theorem Clairaut's theorem characterizes the surface gravity on a viscous rotating ellipsoid in hydrostatic equilibrium under the action of its gravitational field and centrifugal force. It was published in 1743 by Alexis Claude Clairaut in a treatis ...
implies that D_=D_ as long as comparatively mild regularity conditions on ''f'' are satisfied.


Gradient

An important example of a function of several variables is the case of a scalar-valued function ''f''(''x''1, ..., ''xn'') on a domain in Euclidean space \R^n (e.g., on \R^2 or \R^3). In this case ''f'' has a partial derivative ''∂f''/''∂xj'' with respect to each variable ''x''''j''. At the point ''a'', these partial derivatives define the vector : \nabla f(a) = \left(\frac(a), \ldots, \frac(a)\right). This vector is called the ''
gradient In vector calculus, the gradient of a scalar-valued differentiable function of several variables is the vector field (or vector-valued function) \nabla f whose value at a point p is the "direction and rate of fastest increase". If the gradi ...
'' of ''f'' at ''a''. If ''f'' is differentiable at every point in some domain, then the gradient is a vector-valued function ∇''f'' which takes the point ''a'' to the vector ∇''f''(''a''). Consequently, the gradient produces a vector field. A common
abuse of notation In mathematics, abuse of notation occurs when an author uses a mathematical notation in a way that is not entirely formally correct, but which might help simplify the exposition or suggest the correct intuition (while possibly minimizing errors a ...
is to define the
del operator Del, or nabla, is an operator used in mathematics (particularly in vector calculus) as a vector differential operator, usually represented by the nabla symbol ∇. When applied to a function defined on a one-dimensional domain, it denotes t ...
(∇) as follows in three-dimensional
Euclidean space Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, that is, in Euclid's Elements, Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics ther ...
\R^3 with
unit vectors In mathematics, a unit vector in a normed vector space is a vector (often a spatial vector) of length 1. A unit vector is often denoted by a lowercase letter with a circumflex, or "hat", as in \hat (pronounced "v-hat"). The term ''direction vec ...
\hat, \hat, \hat: : \nabla = \left \right\hat + \left \right\hat + \left
right Rights are law, legal, social, or ethics, ethical principles of Liberty, freedom or entitlement; that is, rights are the fundamental normative rules about what is allowed of people or owed to people according to some legal system, social convent ...
\hat Or, more generally, for ''n''-dimensional Euclidean space \R^n with coordinates x_1, \ldots, x_n and unit vectors \hat_1, \ldots, \hat_n: : \nabla = \sum_^n \left frac \right\hat_j = \left frac \right\hat_1 + \left frac \right\hat_2 + \dots + \left frac \right\hat_n


Directional derivative


Example

Suppose that ''f'' is a function of more than one variable. For instance, : z = f(x,y) = x^2 + xy + y^2. The
graph Graph may refer to: Mathematics *Graph (discrete mathematics), a structure made of vertices and edges **Graph theory, the study of such graphs and their properties *Graph (topology), a topological space resembling a graph in the sense of discre ...
of this function defines a
surface A surface, as the term is most generally used, is the outermost or uppermost layer of a physical object or space. It is the portion or region of the object that can first be perceived by an observer using the senses of sight and touch, and is ...
in
Euclidean space Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, that is, in Euclid's Elements, Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics ther ...
. To every point on this surface, there are an infinite number of
tangent line 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. Leibniz defined it as the line through a pair of infinitely close points on the curve. More ...
s. Partial differentiation is the act of choosing one of these lines and finding its
slope In mathematics, the slope or gradient of a line is a number that describes both the ''direction'' and the ''steepness'' of the line. Slope is often denoted by the letter ''m''; there is no clear answer to the question why the letter ''m'' is use ...
. Usually, the lines of most interest are those that are parallel to the xz-plane, and those that are parallel to the yz-plane (which result from holding either y or x constant, respectively). To find the slope of the line tangent to the function at P(1, 1) and parallel to the xz-plane, we treat y as a constant. The graph and this plane are shown on the right. Below, we see how the function looks on the plane y = 1. By finding the
derivative In mathematics, 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). Derivatives are a fundamental tool of calculus. F ...
of the equation while assuming that y is a constant, we find that the slope of ''f'' at the point (x, y) is: : \frac = 2x+y. So at (1, 1), by substitution, the slope is 3. Therefore, : \frac = 3 at the point (1, 1). That is, the partial derivative of z with respect to x at (1, 1) is 3, as shown in the graph. The function ''f'' can be reinterpreted as a family of functions of one variable indexed by the other variables: : f(x,y) = f_y(x) = x^2 + xy + y^2. In other words, every value of ''y'' defines a function, denoted ''fy'', which is a function of one variable ''x''. That is, : f_y(x) = x^2 + xy + y^2. In this section the subscript notation ''fy'' denotes a function contingent on a fixed value of ''y'', and not a partial derivative. Once a value of ''y'' is chosen, say ''a'', then ''f''(''x'',''y'') determines a function ''fa'' which traces a curve ''x''2 + ''ax'' + ''a''2 on the xz-plane: : f_a(x) = x^2 + ax + a^2. In this expression, ''a'' is a ''constant'', not a ''variable'', so ''fa'' is a function of only one real variable, that being ''x''. Consequently, the definition of the derivative for a function of one variable applies: : f_a'(x) = 2x + a. The above procedure can be performed for any choice of ''a''. Assembling the derivatives together into a function gives a function which describes the variation of ''f'' in the ''x'' direction: : \frac(x,y) = 2x + y. This is the partial derivative of ''f'' with respect to ''x''. Here ''∂'' is a rounded ''d'' called the '' partial derivative symbol''; to distinguish it from the letter ''d'', ''∂'' is sometimes pronounced "partial".


Higher order partial derivatives

Second and higher order partial derivatives are defined analogously to the higher order derivatives of univariate functions. For the function f(x, y, ...) the "own" second partial derivative with respect to ''x'' is simply the partial derivative of the partial derivative (both with respect to ''x''): Chiang, Alpha C. ''Fundamental Methods of Mathematical Economics'', McGraw-Hill, third edition, 1984. :\frac \equiv \partial \frac \equiv \frac \equiv f_. The cross partial derivative with respect to ''x'' and ''y'' is obtained by taking the partial derivative of ''f'' with respect to ''x'', and then taking the partial derivative of the result with respect to ''y'', to obtain :\frac \equiv \partial \frac \equiv \frac \equiv f_.
Schwarz's theorem In mathematics, the symmetry of second derivatives (also called the equality of mixed partials) refers to the possibility of interchanging the order of taking partial derivatives of a function :f\left(x_1,\, x_2,\, \ldots,\, x_n\right) of ''n'' ...
states that if the second derivatives are continuous, the expression for the cross partial derivative is unaffected by which variable the partial derivative is taken with respect to first and which is taken second. That is, :\frac = \frac or equivalently f_ = f_. Own and cross partial derivatives appear in the
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. It describes the local curvature of a function of many variables. The Hessian matrix was developed ...
which is used in the second order conditions in
optimization Mathematical optimization (alternatively spelled ''optimisation'') or mathematical programming is the selection of a best element, with regard to some criterion, from some set of available alternatives. It is generally divided into two subfi ...
problems. The higher order partial derivatives can be obtained by successive differentiation


Antiderivative analogue

There is a concept for partial derivatives that is analogous to
antiderivative In calculus, an antiderivative, inverse derivative, primitive function, primitive integral or indefinite integral of a function is a differentiable function whose derivative is equal to the original function . This can be stated symbolically ...
s for regular derivatives. Given a partial derivative, it allows for the partial recovery of the original function. Consider the example of :\frac = 2x+y. The "partial" integral can be taken with respect to ''x'' (treating ''y'' as constant, in a similar manner to partial differentiation): :z = \int \frac \,dx = x^2 + xy + g(y). Here, the "constant" of integration is no longer a constant, but instead a function of all the variables of the original function except ''x''. The reason for this is that all the other variables are treated as constant when taking the partial derivative, so any function which does not involve x will disappear when taking the partial derivative, and we have to account for this when we take the antiderivative. The most general way to represent this is to have the "constant" represent an unknown function of all the other variables. Thus the set of functions x^2 + xy + g(y), where ''g'' is any one-argument function, represents the entire set of functions in variables ''x'',''y'' that could have produced the ''x''-partial derivative 2x + y. If all the partial derivatives of a function are known (for example, with the
gradient In vector calculus, the gradient of a scalar-valued differentiable function of several variables is the vector field (or vector-valued function) \nabla f whose value at a point p is the "direction and rate of fastest increase". If the gradi ...
), then the antiderivatives can be matched via the above process to reconstruct the original function up to a constant. Unlike in the single-variable case, however, not every set of functions can be the set of all (first) partial derivatives of a single function. In other words, not every vector field is
conservative Conservatism is a cultural, social, and political philosophy that seeks to promote and to preserve traditional institutions, practices, and values. The central tenets of conservatism may vary in relation to the culture and civilization i ...
.


Applications


Geometry

The
volume Volume is a measure of occupied three-dimensional space. It is often quantified numerically using SI derived units (such as the cubic metre and litre) or by various imperial or US customary units (such as the gallon, quart, cubic inch). The de ...
''V'' of a
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. A cone is formed by a set of line segments, half-lines, or lines con ...
depends on the cone's
height Height is measure of vertical distance, either vertical extent (how "tall" something or someone is) or vertical position (how "high" a point is). For example, "The height of that building is 50 m" or "The height of an airplane in-flight is abou ...
''h'' and its
radius In classical geometry, a radius ( : radii) 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. The name comes from the latin ''radius'', meaning ray but also the ...
''r'' according to the formula :V(r, h) = \frac. The partial derivative of ''V'' with respect to ''r'' is :\frac = \frac, which represents 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 equals \frac, which represents the rate with which the volume changes if its height is varied and its radius is kept constant. By contrast, the ''total'' derivative of ''V'' with respect to ''r'' and ''h'' are respectively :\frac = \overbrace^\frac + \overbrace^\frac\frac and :\frac = \overbrace^\frac + \overbrace^\frac\frac The difference between the total and partial derivative is the elimination of indirect dependencies between variables in partial derivatives. If (for some arbitrary reason) the cone's proportions have to stay the same, and the height and radius are in a fixed ratio ''k'', :k = \frac = \frac. This gives the total derivative with respect to ''r'': :\frac = \frac + \frack which simplifies to: :\frac = k \pi r^2 Similarly, the total derivative with respect to ''h'' is: :\frac = \pi r^2 The total derivative with respect to ''both'' r and h of the volume intended as scalar function of these two variables is given by the
gradient In vector calculus, the gradient of a scalar-valued differentiable function of several variables is the vector field (or vector-valued function) \nabla f whose value at a point p is the "direction and rate of fastest increase". If the gradi ...
vector :\nabla V = \left(\frac,\frac\right) = \left(\frac\pi rh, \frac\pi r^2\right).


Optimization

Partial derivatives appear in any calculus-based
optimization Mathematical optimization (alternatively spelled ''optimisation'') or mathematical programming is the selection of a best element, with regard to some criterion, from some set of available alternatives. It is generally divided into two subfi ...
problem with more than one choice variable. For example, in
economics Economics () is the social science that studies the Production (economics), production, distribution (economics), distribution, and Consumption (economics), consumption of goods and services. Economics focuses on the behaviour and intera ...
a firm may wish to maximize profit π(''x'', ''y'') with respect to the choice of the quantities ''x'' and ''y'' of two different types of output. The
first order condition 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. Derivative tests can also give information abou ...
s for this optimization are π''x'' = 0 = π''y''. Since both partial derivatives π''x'' and π''y'' will generally themselves be functions of both arguments ''x'' and ''y'', these two first order conditions form a system of two equations in two unknowns.


Thermodynamics, quantum mechanics and mathematical physics

Partial derivatives appear in thermodynamic equations like Gibbs-Duhem equation, in quantum mechanics as Schrodinger wave equation as well in other equations from
mathematical physics Mathematical physics refers to the development of mathematics, mathematical methods for application to problems in physics. The ''Journal of Mathematical Physics'' defines the field as "the application of mathematics to problems in physics and t ...
. Here the variables being held constant in partial derivatives can be ratio of simple variables like
mole fraction In chemistry, the mole fraction or molar fraction (''xi'' or ) is defined as unit of the amount of a constituent (expressed in moles), ''ni'', divided by the total amount of all constituents in a mixture (also expressed in moles), ''n''tot. This ex ...
s ''xi'' in the following example involving the Gibbs energies in a ternary mixture system: :\bar= G + (1-x_2) \left(\frac\right)_ Express
mole fraction In chemistry, the mole fraction or molar fraction (''xi'' or ) is defined as unit of the amount of a constituent (expressed in moles), ''ni'', divided by the total amount of all constituents in a mixture (also expressed in moles), ''n''tot. This ex ...
s of a component as functions of other components' mole fraction and binary mole ratios: :x_1 = \frac :x_3 = \frac Differential quotients can be formed at constant ratios like those above: :\left(\frac\right)_ = - \frac :\left(\frac\right)_ = - \frac Ratios X, Y, Z of mole fractions can be written for ternary and multicomponent systems: :X = \frac :Y = \frac :Z = \frac which can be used for solving
partial differential equation In mathematics, a partial differential equation (PDE) is an equation which imposes relations between the various partial derivatives of a Multivariable calculus, multivariable function. The function is often thought of as an "unknown" to be sol ...
s like: :\left(\frac\right)_ = \left(\frac\right)_ This equality can be rearranged to have differential quotient of mole fractions on one side.


Image resizing

Partial derivatives are key to target-aware image resizing algorithms. Widely known as
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. It functions by e ...
, these algorithms require each
pixel In digital imaging, a pixel (abbreviated px), pel, or picture element is the smallest addressable element in a raster image, or the smallest point in an all points addressable display device. In most digital display devices, pixels are the smal ...
in an image to be assigned a numerical 'energy' to describe their dissimilarity against orthogonal adjacent pixels. The
algorithm In mathematics and computer science, an algorithm () is a finite sequence of rigorous instructions, typically used to solve a class of specific Computational problem, problems or to perform a computation. Algorithms are used as specificat ...
then progressively removes rows or columns with the lowest energy. The formula established to determine a pixel's energy (magnitude of
gradient In vector calculus, the gradient of a scalar-valued differentiable function of several variables is the vector field (or vector-valued function) \nabla f whose value at a point p is the "direction and rate of fastest increase". If the gradi ...
at a pixel) depends heavily on the constructs of partial derivatives.


Economics

Partial derivatives play a prominent role in
economics Economics () is the social science that studies the Production (economics), production, distribution (economics), distribution, and Consumption (economics), consumption of goods and services. Economics focuses on the behaviour and intera ...
, in which most functions describing economic behaviour posit that the behaviour depends on more than one variable. For example, a societal
consumption function In economics, the consumption function describes a relationship between consumption and disposable income. The concept is believed to have been introduced into macroeconomics by John Maynard Keynes in 1936, who used it to develop the notion of a ...
may describe the amount spent on consumer goods as depending on both income and wealth; the
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 t ...
is then the partial derivative of the consumption function with respect to income.


See also

*
d'Alembertian operator In special relativity, electromagnetism and wave theory, the d'Alembert operator (denoted by a box: \Box), also called the d'Alembertian, wave operator, box operator or sometimes quabla operator (''cf''. nabla symbol) is the Laplace operator of ...
*
Chain rule In calculus, the chain rule is a formula that expresses the derivative of the composition of two differentiable functions and in terms of the derivatives of and . More precisely, if h=f\circ g is the function such that h(x)=f(g(x)) for every , ...
*
Curl (mathematics) In vector calculus, the curl is a vector operator that describes the infinitesimal circulation of a vector field in three-dimensional Euclidean space. The curl at a point in the field is represented by a vector whose length and direction denot ...
*
Divergence In vector calculus, divergence is a vector operator that operates on a vector field, producing a scalar field giving the quantity of the vector field's source at each point. More technically, the divergence represents the volume density of the ...
*
Exterior derivative On a differentiable manifold, the exterior derivative extends the concept of the differential of a function to differential forms of higher degree. The exterior derivative was first described in its current form by Élie Cartan in 1899. The res ...
*
Iterated integral In multivariable calculus, an iterated integral is the result of applying integrals to a function of more than one variable (for example f(x,y) or f(x,y,z)) in a way that each of the integrals considers some of the variables as given constants. ...
*
Jacobian matrix and determinant In vector calculus, the Jacobian matrix (, ) of a vector-valued function of several variables is the matrix of all its first-order partial derivatives. When this matrix is square, that is, when the function takes the same number of variables as ...
*
Laplacian In mathematics, the Laplace operator or Laplacian is a differential operator given by the divergence of the gradient of a scalar function on Euclidean space. It is usually denoted by the symbols \nabla\cdot\nabla, \nabla^2 (where \nabla is the ...
*
Multivariable calculus Multivariable calculus (also known as multivariate calculus) is the extension of calculus in one variable to calculus with functions of several variables: the differentiation and integration of functions involving several variables, rather th ...
*
Symmetry of second derivatives In mathematics, the symmetry of second derivatives (also called the equality of mixed partials) refers to the possibility of interchanging the order of taking partial derivatives of a function :f\left(x_1,\, x_2,\, \ldots,\, x_n\right) of ''n'' ...
* Triple product rule, also known as the cyclic chain rule.


Notes


References


External links

*
Partial Derivatives
at
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