TheInfoList

In
mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and their changes (cal ...
, a partial derivative of a
function of several variables In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). I ...
is its
derivative In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities ...

with respect to one of those variables, with the others held constant (as opposed to the
total derivative In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). I ...
, in which all variables are allowed to vary). Partial derivatives are used in
vector calculus Vector calculus, or vector analysis, is concerned with differentiation Differentiation may refer to: Business * Differentiation (economics), the process of making a product different from other similar products * Product differentiation, in ...
and
differential geometry Differential geometry is a Mathematics, mathematical discipline that studies the geometry of smooth shapes and smooth spaces, otherwise known as smooth manifolds, using the techniques of differential calculus, integral calculus, linear algebra a ...
. The partial derivative of a function $f\left(x, y, \dots\right)$ with respect to the variable $x$ is variously denoted by : $f\text{'}_x$, $\partial_x f$, $\ D_xf$, $D_1f$, $\fracf$, or $\frac$. Sometimes, for $z=f\left(x, y, \ldots\right)$, 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: :$f\text{'}_x\left(x, y, \ldots\right), \frac \left(x, y, \ldots\right).$ The symbol used to denote partial derivatives is . 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 A philosopher is someone who practices philosophy Philosophy (from , ) is th ...

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 named a ...
(1786) (although he later abandoned it,
Carl Gustav Jacob Jacobi Carl Gustav Jacob Jacobi (; ; 10 December 1804 – 18 February 1851) was a Germans, German mathematician who made fundamental contributions to elliptic functions, Dynamics (mechanics), dynamics, differential equations, determinants, and number th ...
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 (music), a way to characterize harmony * Limit (song), "Limit" (song), a 2016 single by Luna Sea * Limits (Paenda song), "Limits" (Paenda song), 2019 song that represented Austria in the Eurov ...

. Let ''U'' be an
open subset Open or OPEN may refer to: Music * Open (band) Open is a band. Background Drummer Pete Neville has been involved in the Sydney/Australian music scene for a number of years. He has recently completed a Masters in screen music at the Australian ...
of $\R^n$ and $f:U\to\R$ a function. The partial derivative of ''f'' at the point $\mathbf=\left(a_1, \ldots, a_n\right) \in U$ with respect to the ''i''-th variable ''x''''i'' is defined as :$\begin \fracf\left(\mathbf\right) & = \lim_ \frac \\ & = \lim_ \frac \end$ 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 ga ...
there. However, if all partial derivatives exist in a
neighborhood A neighbourhood (British English British English (BrE) is the standard dialect A standard language (also standard variety, standard dialect, and standard) is a language variety that has undergone substantial codification of grammar ...
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 equilibrium under the action of its gravitational field and centrifugal force. It was published in 1743 by Alexis Claude Clairaut in a treatise''Théorie ...
: :$\frac = \frac .$

# Notation

For the following examples, let $f$ be a function in $x, y$ and $z$. First-order partial derivatives: :$\frac = f\text{'}_x = \partial_x f.$ Second-order partial derivatives: :$\frac = f\text{'}\text{'}_ = \partial_ f = \partial_x^2 f.$ Second-order mixed derivatives: :$\frac = \frac \left\left( \frac \right\right) = \left(f\text{'}_\right)\text{'}_ = f\text{'}\text{'}_ = \partial_ f = \partial_y \partial_x f .$ 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 Physics is the that studies , its , its and behavior through , and the related entities of and . "Physical science is that department of knowledge which relates to the order of nature, or, in other words, to the regular ...
, the partial derivative of $f$ with respect to $x$, holding $y$ and $z$ constant, is often expressed as :$\left\left( \frac \right\right)_ .$ Conventionally, for clarity and simplicity of notation, the partial derivative ''function'' and the ''value'' of the function at a specific point are conflated 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 $\left(x,y,z\right)=\left(u,v,w\right)$. However, this convention breaks down when we want to evaluate the partial derivative at a point like $\left(x,y,z\right)=\left(17, u+v, v^2\right)$. In such a case, evaluation of the function must be expressed in an unwieldy manner as :$\frac\left(17, u+v, v^2\right)$ 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 $D_1 f\left(17, u+v, v^2\right)$ 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 $D_j\left(D_i f\right)=D_ f$. 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 equilibrium under the action of its gravitational field and centrifugal force. It was published in 1743 by Alexis Claude Clairaut in a treatise''Théorie ...
implies that $D_=D_$ as long as comparatively mild regularity conditions on ''f'' are satisfied.

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 : $\nabla f\left(a\right) = \left\left(\frac\left(a\right), \ldots, \frac\left(a\right)\right\right).$ This vector is called the ''
gradient In vector calculus Vector calculus, or vector analysis, is concerned with differentiation Differentiation may refer to: Business * Differentiation (economics), the process of making a product different from other similar products * Prod ...

'' 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 In vector calculus Vector calculus, or vector analysis, is concerned with differentiation Differentiation may refer to: Business * Differentiation (economics), the process of making a product different from other similar products * Product ...

. A common
abuse of notation In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It h ...
is to define the
del operator Del, or nabla, is an operator used in mathematics (particularly in vector calculus Vector calculus, or vector analysis, is concerned with derivative, differentiation and integral, integration of vector fields, primarily in 3-dimensional E ...
(∇) as follows in three-dimensional
Euclidean space Euclidean space is the fundamental space of classical geometry. Originally, it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are Euclidean spaces of any nonnegative integer dimension (mathematics), dimens ...
$\R^3$ with
unit vectors In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It h ...

$\hat, \hat, \hat$: :
right Rights are legal Law is a system A system is a group of Interaction, interacting or interrelated elements that act according to a set of rules to form a unified whole. A system, surrounded and influenced by its environment, is desc ...

\hat Or, more generally, for ''n''-dimensional Euclidean space $\R^n$ with coordinates $x_1, \ldots, x_n$ and unit vectors $\hat_1, \ldots, \hat_n$: :

# Example

Suppose that ''f'' is a function of more than one variable. For instance, : $z = f\left(x,y\right) = x^2 + xy + y^2$. 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 discret ...

of this function defines a
surface File:Water droplet lying on a damask.jpg, Water droplet lying on a damask. Surface tension is high enough to prevent floating below the textile. A surface, as the term is most generally used, is the outermost or uppermost layer of a physical obje ...
in
Euclidean space Euclidean space is the fundamental space of classical geometry. Originally, it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are Euclidean spaces of any nonnegative integer dimension (mathematics), dimens ...
. To every point on this surface, there are an infinite number of
tangent line In geometry Geometry (from the grc, γεωμετρία; ''wikt:γῆ, geo-'' "earth", ''wikt:μέτρον, -metron'' "measurement") is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space ...

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 Line, lines, The Line, or LINE may refer to: Arts, entertainment, and media Films * ''Lines'' (film), a 2016 Greek film * ''The Line'' (2017 film) * ''The Line'' (2009 film) * ''The Line'', ...

. 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 $P\left(1, 1\right)$ 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 Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities ...

of the equation while assuming that $y$ is a constant, we find that the slope of ''$f$'' at the point $\left(x, y\right)$ is: : $\frac = 2x+y$. So at $\left(1, 1\right)$, by substitution, the slope is 3. Therefore, : $\frac = 3$ at the point $\left(1, 1\right)$. That is, the partial derivative of $z$ with respect to $x$ at $\left(1, 1\right)$ 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: : $f\left(x,y\right) = f_y\left(x\right) = x^2 + xy + y^2.$ In other words, every value of ''y'' defines a function, denoted ''fy'', which is a function of one variable ''x''. That is, : $f_y\left(x\right) = x^2 + xy + y^2.$ 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: : $f_a\left(x\right) = x^2 + ax + a^2$. 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: : $f_a\text{'}\left(x\right) = 2x + a$. 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: : $\frac\left(x,y\right) = 2x + y.$ 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 $f\left(x, y, ...\right)$ 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 theoremIn 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 (mathematics), funct ...
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 mathematic Mathematics (from Greek: ) includes the study of such topics as quantity Quantity is a property that can exist as a multitude or magnitude, which illustrate discontinuity and continuity. Quantities can be compared in te ...
which is used in the second order conditions in
optimization File:Nelder-Mead Simionescu.gif, Nelder-Mead minimum search of Test functions for optimization, Simionescu's function. Simplex vertices are ordered by their values, with 1 having the lowest ( best) value., alt= Mathematical optimization (alter ...

problems.

# Antiderivative analogue

There is a concept for partial derivatives that is analogous to
antiderivative In calculus Calculus, originally called infinitesimal calculus or "the calculus of infinitesimal In mathematics, infinitesimals or infinitesimal numbers are quantities that are closer to zero than any standard real number, but are not zer ...
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): :$z = \int \frac \,dx = x^2 + xy + g\left(y\right)$ Here, the 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 $x^2 + xy + g\left(y\right)$, 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 Vector calculus, or vector analysis, is concerned with differentiation Differentiation may refer to: Business * Differentiation (economics), the process of making a product different from other similar products * Prod ...

), 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 an aesthetic Aesthetics, or esthetics (), is a branch of philosophy that deals with the nature of beauty and taste (sociology), taste, as well as the philosophy of art (its own area of philosophy that comes out of aest ...
.

# Applications

## Geometry

The
volume Volume is a scalar quantity expressing the amount Quantity or amount is a property that can exist as a multitude Multitude is a term for a group of people who cannot be classed under any other distinct category, except for their shared fact ...

''V'' of a
cone A cone is a three-dimensional Three-dimensional space (also: 3-space or, rarely, tri-dimensional space) is a geometric setting in which three values (called parameter A parameter (from the Ancient Greek language, Ancient Greek wikt:πα ...

depends on the cone's
height 200px, A cuboid demonstrating the dimensions length, width">length.html" ;"title="cuboid demonstrating the dimensions length">cuboid demonstrating the dimensions length, width, and height. Height is measure of vertical distance, either vertical ...

''h'' and its
radius In classical geometry Geometry (from the grc, γεωμετρία; ' "earth", ' "measurement") is, with , one of the oldest branches of . It is concerned with properties of space that are related with distance, shape, size, and relative ...

''r'' according to the formula :$V\left(r, h\right) = \frac.$ 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 Vector calculus, or vector analysis, is concerned with differentiation Differentiation may refer to: Business * Differentiation (economics), the process of making a product different from other similar products * Prod ...

vector :$\nabla V = \left\left(\frac,\frac\right\right) = \left\left(\frac\pi rh, \frac\pi r^2\right\right)$.

## Optimization

Partial derivatives appear in any calculus-based
optimization File:Nelder-Mead Simionescu.gif, Nelder-Mead minimum search of Test functions for optimization, Simionescu's function. Simplex vertices are ordered by their values, with 1 having the lowest ( best) value., alt= Mathematical optimization (alter ...

problem with more than one choice variable. For example, in
economics Economics () is a 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 interact ...

a firm may wish to maximize
profit Profit may refer to: Business and law * Profit (accounting) Profit, in accounting Accounting or Accountancy is the measurement, processing, and communication of financial and non financial information about economic entity, economic en ...
π(''x'', ''y'') with respect to the choice of the quantities ''x'' and ''y'' of two different types of output. The
first order conditionIn calculus, a derivative test uses the derivatives of a function (mathematics), function to locate the stationary point, critical points of a function and determine whether each point is a local maximum, a local minimum, or a saddle point. Derivativ ...
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 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 the developme ...
. Here the variables being held constant in partial derivatives can be ratio of simple variables like
mole fraction In chemistry Chemistry is the scientific discipline involved with Chemical element, elements and chemical compound, compounds composed of atoms, molecules and ions: their composition, structure, properties, behavior and the changes they underg ...
s ''xi'' in the following example involving the Gibbs energies in a ternary mixture system: :$\bar= G + \left(1-x_2\right) \left\left(\frac\right\right)_$ Express
mole fraction In chemistry Chemistry is the scientific discipline involved with Chemical element, elements and chemical compound, compounds composed of atoms, molecules and ions: their composition, structure, properties, behavior and the changes they underg ...
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: :$\left\left(\frac\right\right)_ = - \frac$ :$\left\left(\frac\right\right)_ = - \frac$ 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: :$\left\left(\frac\right\right)_ = \left\left(\frac\right\right)_$ 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, these algorithms require each
pixel In digital imaging Digital imaging or digital image acquisition is the creation of a representation of the visual characteristics of an object, such as a physical scene or the interior structure of an object. The term is often assumed to imp ...

in an image to be assigned a numerical 'energy' to describe their dissimilarity against orthogonal adjacent pixels. The
algorithm In and , an algorithm () is a finite sequence of , computer-implementable instructions, typically to solve a class of problems or to perform a computation. Algorithms are always and are used as specifications for performing s, , , and other ...

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 Vector calculus, or vector analysis, is concerned with differentiation Differentiation may refer to: Business * Differentiation (economics), the process of making a product different from other similar products * Prod ...

at a pixel) depends heavily on the constructs of partial derivatives.

## Economics

Partial derivatives play a prominent role in
economics Economics () is a 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 interact ...

, 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 Economics () is the social science that studies how people interact with value; in particular, the Production (economics), production, distribution (economics), distribution, and Consumption (economics), consumption of goods a ...
may describe the amount spent on consumer goods as depending on both income and wealth; the
marginal propensity to consume In economics Economics () is the social science that studies how people interact with value; in particular, the Production (economics), production, distribution (economics), distribution, and Consumption (economics), consumption of goods an ...
is then the partial derivative of the consumption function with respect to income.

*
d'Alembertian operator In special relativity, electromagnetism and Wave, wave theory, the d'Alembert operator (denoted by a box: \Box), also called the d'Alembertian, wave operator, or box operator is the Laplace operator of Minkowski space. The operator is named after F ...
*
Chain rule In calculus Calculus, originally called infinitesimal calculus or "the calculus of infinitesimals", is the mathematics, mathematical study of continuous change, in the same way that geometry is the study of shape and algebra is the study of ...
*
Curl (mathematics) In vector calculus Vector calculus, or vector analysis, is concerned with differentiation Differentiation may refer to: Business * Differentiation (economics), the process of making a product different from other similar products * Produc ...
*
Divergence In vector calculus Vector calculus, or vector analysis, is concerned with derivative, differentiation and integral, integration of vector fields, primarily in 3-dimensional Euclidean space \mathbb^3. The term "vector calculus" is sometimes ...

*
Exterior derivative On a differentiable manifold In mathematics, a differentiable manifold (also differential manifold) is a type of manifold The real projective plane is a two-dimensional manifold that cannot be realized in three dimensions without self-interse ...
*
Iterated integral In multivariable calculus, an iterated integral is the result of applying integrals to a Function (mathematics), function of Function of several real variables, more than one variable (for example f(x,y) or f(x,y,z)) in a way that each of the integr ...
*
Jacobian matrix and determinant In vector calculus, the Jacobian matrix (, ) of a vector-valued function in several variables is the matrix of all its first-order partial derivative In mathematics, a partial derivative of a function (mathematics)#MULTIVARIATE FUNCTION, fun ...
*
Laplacian In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities an ...
*
Multivariable calculus Multivariable calculus (also known as multivariate calculus) is the extension of calculus in one Variable (mathematics), variable to calculus with function of several variables, functions of several variables: the Differential calculus, different ...
*
Symmetry of second derivativesIn mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ha ...
* Triple product rule, also known as the cyclic chain rule.