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The gradient theorem, also known as the fundamental theorem of calculus for line integrals, says that a
line integral In mathematics, a line integral is an integral where the function to be integrated is evaluated along a curve. The terms ''path integral'', ''curve integral'', and ''curvilinear integral'' are also used; '' contour integral'' is used as well, ...
through a gradient field can be evaluated by evaluating the original scalar field at the endpoints of the curve. The theorem is a generalization of the second fundamental theorem of calculus to any curve in a plane or space (generally ''n''-dimensional) rather than just the real line. For as a
differentiable function In mathematics, a differentiable function of one real variable is a function whose derivative exists at each point in its domain. In other words, the graph of a differentiable function has a non- vertical tangent line at each interior point in ...
and as any continuous curve in which starts at a point and ends at a point , then \int_ \nabla\varphi(\mathbf)\cdot \mathrm\mathbf = \varphi\left(\mathbf\right) - \varphi\left(\mathbf\right) where denotes the gradient vector field of . The gradient theorem implies that line integrals through gradient fields are path-independent. In physics this theorem is one of the ways of defining a ''conservative'' force. By placing as potential, is a conservative field. Work done by conservative forces does not depend on the path followed by the object, but only the end points, as the above equation shows. The gradient theorem also has an interesting converse: any path-independent vector field can be expressed as the gradient of a
scalar field In mathematics and physics, a scalar field is a function associating a single number to every point in a space – possibly physical space. The scalar may either be a pure mathematical number ( dimensionless) or a scalar physical quantity ...
. Just like the gradient theorem itself, this converse has many striking consequences and applications in both pure and applied mathematics.


Proof

If is a
differentiable function In mathematics, a differentiable function of one real variable is a function whose derivative exists at each point in its domain. In other words, the graph of a differentiable function has a non- vertical tangent line at each interior point in ...
from some
open subset In mathematics, open sets are a generalization of open intervals in the real line. In a metric space (a set along with a distance defined between any two points), open sets are the sets that, with every point , contain all points that are suff ...
to and is a differentiable function from some closed interval to (Note that is differentiable at the interval endpoints and . To do this, is defined on an interval that is larger than and includes .), then by the multivariate chain rule, the
composite function In mathematics, function composition is an operation that takes two functions and , and produces a function such that . In this operation, the function is applied to the result of applying the function to . That is, the functions and ...
is differentiable on : \frac(\varphi \circ \mathbf)(t)=\nabla \varphi(\mathbf(t)) \cdot \mathbf'(t) for all in . Here the denotes the usual inner product. Now suppose the domain of contains the differentiable curve with endpoints and . (This is
oriented In mathematics, orientability is a property of some topological spaces such as real vector spaces, Euclidean spaces, surfaces, and more generally manifolds that allows a consistent definition of "clockwise" and "counterclockwise". A space i ...
in the direction from to ). If parametrizes for in (i.e., represents as a function of ), then \begin \int_ \nabla\varphi(\mathbf) \cdot \mathrm\mathbf &=\int_a^b \nabla\varphi(\mathbf(t)) \cdot \mathbf'(t)\mathrmt \\ &=\int_a^b \frac\varphi(\mathbf(t))\mathrmt =\varphi(\mathbf(b))-\varphi(\mathbf(a))=\varphi\left(\mathbf\right)-\varphi\left(\mathbf\right) , \end where the definition of a line integral is used in the first equality, the above equation is used in the second equality, and the second fundamental theorem of calculus is used in the third equality. Even if the gradient theorem (also called ''fundamental theorem of calculus for line integrals'') has been proved for a differentiable (so looked as smooth) curve so far, the theorem is also proved for a piecewise-smooth curve since this curve is made by joining multiple differentiable curves so the proof for this curve is made by the proof per differentiable curve component.


Examples


Example 1

Suppose is the circular arc oriented counterclockwise from to . Using the definition of a line integral, \begin \int_ y\, \mathrmx + x\, \mathrmy &= \int_0^ ((5\sin t)(-5 \sin t) + (5 \cos t)(5 \cos t))\, \mathrmt \\ &= \int_0^ 25 \left(-\sin^2 t + \cos^2 t\right) \mathrmt \\ &= \int_0^ 25 \cos(2t) \mathrmt \ =\ \left.\tfrac\sin(2t)\_0^ \\ 5em &= \tfrac\sin\left(2\pi - 2\tan^\!\!\left(\tfrac\right)\right) \\ 5em &= -\tfrac\sin\left(2\tan^\!\!\left(\tfrac\right)\right) \ =\ -\frac = -12. \end This result can be obtained much more simply by noticing that the function f(x,y)=xy has gradient \nabla f(x,y)=(y,x), so by the Gradient Theorem: \int_ y \,\mathrmx+x \,\mathrmy=\int_\nabla(xy) \cdot (\mathrmx,\mathrmy)\ =\ xy\,, _^=-4 \cdot 3-5 \cdot 0=-12 .


Example 2

For a more abstract example, suppose has endpoints , , with orientation from to . For in , let denote the
Euclidean norm Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, that is, in Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are Euclidean ...
of . If is a real number, then \begin \int_ , \mathbf, ^ \mathbf \cdot \mathrm\mathbf &= \frac \int_ (\alpha + 1) , \mathbf, ^ \mathbf \cdot \mathrm\mathbf \\ &= \frac \int_ \nabla , \mathbf, ^ \cdot \mathrm\mathbf= \frac \end Here the final equality follows by the gradient theorem, since the function is differentiable on if . If then this equality will still hold in most cases, but caution must be taken if ''γ'' passes through or encloses the origin, because the integrand vector field will fail to be defined there. However, the case is somewhat different; in this case, the integrand becomes , so that the final equality becomes . Note that if , then this example is simply a slight variant of the familiar power rule from single-variable calculus.


Example 3

Suppose there are point charges arranged in three-dimensional space, and the -th point charge has charge and is located at position in . We would like to calculate the work done on a particle of charge as it travels from a point to a point in . Using
Coulomb's law Coulomb's inverse-square law, or simply Coulomb's law, is an experimental law of physics that quantifies the amount of force between two stationary, electrically charged particles. The electric force between charged bodies at rest is convention ...
, we can easily determine that the
force In physics, a force is an influence that can change the motion of an object. A force can cause an object with mass to change its velocity (e.g. moving from a state of rest), i.e., to accelerate. Force can also be described intuitively as a ...
on the particle at position will be \mathbf(\mathbf) = kq\sum_^n \frac Here denotes the
Euclidean norm Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, that is, in Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are Euclidean ...
of the vector in , and , where is the
vacuum permittivity Vacuum permittivity, commonly denoted (pronounced "epsilon nought" or "epsilon zero"), is the value of the absolute dielectric permittivity of classical vacuum. It may also be referred to as the permittivity of free space, the electric const ...
. Let be an arbitrary differentiable curve from to . Then the work done on the particle is W = \int_ \mathbf(\mathbf) \cdot \mathrm\mathbf = \int_ \left( kq\sum_^n \frac \right) \cdot \mathrm\mathbf = kq \sum_^n \left( Q_i \int_\gamma \frac \cdot \mathrm\mathbf \right) Now for each , direct computation shows that \frac = -\nabla \frac. Thus, continuing from above and using the gradient theorem, W = -kq \sum_^n \left( Q_i \int_ \nabla \frac \cdot \mathrm\mathbf \right) = kq \sum_^n Q_i \left( \frac - \frac \right) We are finished. Of course, we could have easily completed this calculation using the powerful language of
electrostatic potential Electrostatics is a branch of physics that studies electric charges at rest ( static electricity). Since classical times, it has been known that some materials, such as amber, attract lightweight particles after rubbing. The Greek word for a ...
or electrostatic potential energy (with the familiar formulas ). However, we have not yet ''defined'' potential or potential energy, because the ''converse'' of the gradient theorem is required to prove that these are well-defined, differentiable functions and that these formulas hold ( see below). Thus, we have solved this problem using only Coulomb's Law, the definition of work, and the gradient theorem.


Converse of the gradient theorem

The gradient theorem states that if the vector field is the gradient of some scalar-valued function (i.e., if 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 in ...
), then is a path-independent vector field (i.e., the integral of over some piecewise-differentiable curve is dependent only on end points). This theorem has a powerful converse: It is straightforward to show that a vector field is path-independent if and only if the integral of the vector field over every closed loop in its domain is zero. Thus the converse can alternatively be stated as follows: If the integral of over every closed loop in the domain of is zero, then is the gradient of some scalar-valued function.


Proof of the converse

Suppose is an
open Open or OPEN may refer to: Music * Open (band), Australian pop/rock band * The Open (band), English indie rock band * Open (Blues Image album), ''Open'' (Blues Image album), 1969 * Open (Gotthard album), ''Open'' (Gotthard album), 1999 * Open (C ...
,
path-connected In topology and related branches of mathematics, a connected space is a topological space that cannot be represented as the union of two or more disjoint non-empty open subsets. Connectedness is one of the principal topological properties ...
subset of , and is a
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 g ...
and path-independent vector field. Fix some element of , and define by f(\mathbf) := \int_ \mathbf(\mathbf) \cdot \mathrm\mathbf Here is any (differentiable) curve in originating at and terminating at . We know that is well-defined because is path-independent. Let be any nonzero vector in . By the definition of the
directional derivative In mathematics, the directional derivative of a multivariable differentiable (scalar) 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 ...
, \begin \frac &= \lim_ \frac \\ &= \lim_ \frac \\ &= \lim_ \frac \int_ \mathbf(\mathbf) \cdot \mathrm\mathbf \endTo calculate the integral within the final limit, we must parametrize . Since is path-independent, is open, and is approaching zero, we may assume that this path is a straight line, and parametrize it as for . Now, since , the limit becomes \lim_ \frac \int_0^t \mathbf(\mathbf(s)) \cdot \mathbf'(s)\, \mathrms = \frac \int_0^t \mathbf(\mathbf + s\mathbf) \cdot \mathbf\, \mathrms \bigg, _ = \mathbf(\mathbf) \cdot \mathbf where the first equality is from the definition of the derivative with a fact that the integral is equal to 0 at = 0, and the second equality is from the
first fundamental theorem of calculus The fundamental theorem of calculus is a theorem that links the concept of differentiating a function (calculating its slopes, or rate of change at each time) with the concept of integrating a function (calculating the area under its graph, or ...
. Thus we have a formula for , (one of ways to represent the
directional derivative In mathematics, the directional derivative of a multivariable differentiable (scalar) 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 ...
) where is arbitrary; for f(\mathbf) := \int_ \mathbf(\mathbf) \cdot \mathrm\mathbf (see its full definition above), its directional derivative with respect to is \frac = \partial _ \mathbf f(\mathbf) = D_f(\mathbf) = \mathbf(\mathbf) \cdot \mathbf where the first two equalities just show different representations of the directional derivative. According to the definition of the gradient of a scalar function , \nabla f(\mathbf) = \mathbf(\mathbf), thus we have found a scalar-valued function whose gradient is the path-independent vector field (i.e., is a conservative vector field.), as desired.


Example of the converse principle

To illustrate the power of this converse principle, we cite an example that has significant
physical Physical may refer to: * Physical examination, a regular overall check-up with a doctor * ''Physical'' (Olivia Newton-John album), 1981 ** "Physical" (Olivia Newton-John song) * ''Physical'' (Gabe Gurnsey album) * "Physical" (Alcazar song) (2004) * ...
consequences. In classical electromagnetism, the
electric force Coulomb's inverse-square law, or simply Coulomb's law, is an experimental law of physics that quantifies the amount of force between two stationary, electrically charged particles. The electric force between charged bodies at rest is conventio ...
is a path-independent force; i.e. the work done on a particle that has returned to its original position within an
electric field An electric field (sometimes E-field) is the physical field that surrounds electrically charged particles and exerts force on all other charged particles in the field, either attracting or repelling them. It also refers to the physical field ...
is zero (assuming that no changing
magnetic field A magnetic field is a vector field that describes the magnetic influence on moving electric charges, electric currents, and magnetic materials. A moving charge in a magnetic field experiences a force perpendicular to its own velocity and to ...
s are present). Therefore, the above theorem implies that the electric force field is conservative (here is some
open Open or OPEN may refer to: Music * Open (band), Australian pop/rock band * The Open (band), English indie rock band * Open (Blues Image album), ''Open'' (Blues Image album), 1969 * Open (Gotthard album), ''Open'' (Gotthard album), 1999 * Open (C ...
,
path-connected In topology and related branches of mathematics, a connected space is a topological space that cannot be represented as the union of two or more disjoint non-empty open subsets. Connectedness is one of the principal topological properties ...
subset of that contains a charge distribution). Following the ideas of the above proof, we can set some reference point in , and define a function by U_e(\mathbf) := -\int_ \mathbf_e(\mathbf) \cdot \mathrm\mathbf Using the above proof, we know is well-defined and differentiable, and (from this formula we can use the gradient theorem to easily derive the well-known formula for calculating work done by conservative forces: ). This function is often referred to as the electrostatic potential energy of the system of charges in (with reference to the zero-of-potential ). In many cases, the domain is assumed to be unbounded and the reference point is taken to be "infinity", which can be made rigorous using limiting techniques. This function is an indispensable tool used in the analysis of many physical systems.


Generalizations

Many of the critical theorems of vector calculus generalize elegantly to statements about the integration of differential forms on manifolds. In the language of
differential form In mathematics, differential forms provide a unified approach to define integrands over curves, surfaces, solids, and higher-dimensional manifolds. The modern notion of differential forms was pioneered by Élie Cartan. It has many application ...
s and
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 re ...
s, the gradient theorem states that \int_ \phi = \int_ \mathrm\phi for any 0-form, , defined on some differentiable curve (here the integral of over the boundary of the is understood to be the evaluation of at the endpoints of ''γ''). Notice the striking similarity between this statement and the generalized version of
Stokes' theorem Stokes's theorem, also known as the Kelvin–Stokes theorem Nagayoshi Iwahori, et al.:"Bi-Bun-Seki-Bun-Gaku" Sho-Ka-Bou(jp) 1983/12Written in Japanese)Atsuo Fujimoto;"Vector-Kai-Seki Gendai su-gaku rekucha zu. C(1)" :ja:培風館, Bai-Fu-Kan( ...
, which says that the integral of any compactly supported differential form over the boundary of some
orientable In mathematics, orientability is a property of some topological spaces such as real vector spaces, Euclidean spaces, surfaces, and more generally manifolds that allows a consistent definition of "clockwise" and "counterclockwise". A space i ...
manifold is equal to the integral of its
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 re ...
over the whole of , i.e., \int_\omega=\int_\mathrm\omega This powerful statement is a generalization of the gradient theorem from 1-forms defined on one-dimensional manifolds to differential forms defined on manifolds of arbitrary dimension. The converse statement of the gradient theorem also has a powerful generalization in terms of differential forms on manifolds. In particular, suppose is a form defined on a contractible domain, and the integral of over any closed manifold is zero. Then there exists a form such that . Thus, on a contractible domain, every
closed Closed may refer to: Mathematics * Closure (mathematics), a set, along with operations, for which applying those operations on members always results in a member of the set * Closed set, a set which contains all its limit points * Closed interval, ...
form is exact. This result is summarized by the Poincaré lemma.


See also

*
State function In the thermodynamics of equilibrium, a state function, function of state, or point function for a thermodynamic system is a mathematical function relating several state variables or state quantities (that describe equilibrium states of a system ...
*
Scalar potential In mathematical physics, scalar potential, simply stated, describes the situation where the difference in the potential energies of an object in two different positions depends only on the positions, not upon the path taken by the object in trav ...
* Jordan curve theorem * Differential of a function *
Classical mechanics Classical mechanics is a physical theory describing the motion of macroscopic objects, from projectiles to parts of machinery, and astronomical objects, such as spacecraft, planets, stars, and galaxies. For objects governed by classi ...
* *


References

{{Calculus topics Theorems in calculus Articles containing proofs