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, alt ...
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 its ...
and as any continuous curve in which starts at a point and ends at a point , then
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
In vector calculus, a conservative vector field is a vector field that is the gradient of some function. A conservative vector field has the property that its line integral is path independent; the choice of any path between two points does not ...
.
Work
Work may refer to:
* Work (human activity), intentional activity people perform to support themselves, others, or the community
** Manual labour, physical work done by humans
** House work, housework, or homemaking
** Working animal, an animal t ...
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 (mathematics), function associating a single number to every point (geometry), point in a space (mathematics), space – possibly physical space. The scalar may either be a pure Scalar ( ...
. 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 its ...
from some
open subset 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 function (mathematics), functions and , and produces a function such that . In this operation, the function is function application, applied to the result of applying the ...
is differentiable on :
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 the direction from to ). If
parametrizes for in (i.e., represents as a function of ), then
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,
This result can be obtained much more simply by noticing that the function
has gradient
, so by the Gradient Theorem:
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 s ...
of . If is a real number, then
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
In calculus, the power rule is used to differentiate functions of the form f(x) = x^r, whenever r is a real number. Since differentiation is a linear operation on the space of differentiable functions, polynomials can also be differentiated usin ...
from single-variable calculus.
Example 3
Suppose there are
point charges arranged in three-dimensional space, and the -th point charge has
charge
Charge or charged may refer to:
Arts, entertainment, and media Films
* '' Charge, Zero Emissions/Maximum Speed'', a 2011 documentary
Music
* ''Charge'' (David Ford album)
* ''Charge'' (Machel Montano album)
* ''Charge!!'', an album by The Aqu ...
and is located at position in . We would like to calculate the
work
Work may refer to:
* Work (human activity), intentional activity people perform to support themselves, others, or the community
** Manual labour, physical work done by humans
** House work, housework, or homemaking
** Working animal, an animal t ...
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 conventiona ...
, 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 p ...
on the particle at position will be
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 s ...
of the vector in , and , where is the
vacuum permittivity.
Let be an arbitrary differentiable curve from to . Then the work done on the particle is
Now for each , direct computation shows that
Thus, continuing from above and using the gradient theorem,
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 ambe ...
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 i ...
), 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), 1969
* ''Open'' (Gotthard album), 1999
* ''Open'' (Cowboy Junkies album), 2001
* ''Open'' (YF ...
,
path-connected 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 ...
and path-independent vector field. Fix some element of , and define by
Here is any (differentiable) curve in originating at and terminating at . We know that is
well-defined
In mathematics, a well-defined expression or unambiguous expression is an expression whose definition assigns it a unique interpretation or value. Otherwise, the expression is said to be ''not well defined'', ill defined or ''ambiguous''. A funct ...
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 s ...
,
To 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
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 s ...
) where is arbitrary; for
(see its full definition above), its directional derivative with respect to is
where the first two equalities just show different representations of the directional derivative. According to
the definition of the gradient of a scalar function ,
, 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
In a physical examination, medical examination, or clinical examination, a medical practitioner examines a patient for any possible medical signs or symptoms of a medical condition. It generally co ...
consequences. In
classical electromagnetism
Classical electromagnetism or classical electrodynamics is a branch of theoretical physics that studies the interactions between electric charges and currents using an extension of the classical Newtonian model; It is, therefore, a classical fie ...
, 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 convention ...
is a path-independent force; i.e. the
work
Work may refer to:
* Work (human activity), intentional activity people perform to support themselves, others, or the community
** Manual labour, physical work done by humans
** House work, housework, or homemaking
** Working animal, an animal t ...
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 fo ...
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), 1969
* ''Open'' (Gotthard album), 1999
* ''Open'' (Cowboy Junkies album), 2001
* ''Open'' (YF ...
,
path-connected subset of that contains a
charge
Charge or charged may refer to:
Arts, entertainment, and media Films
* '' Charge, Zero Emissions/Maximum Speed'', a 2011 documentary
Music
* ''Charge'' (David Ford album)
* ''Charge'' (Machel Montano album)
* ''Charge!!'', an album by The Aqu ...
distribution). Following the ideas of the above proof, we can set some reference point in , and define a function by
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
Rigour (British English) or rigor (American English; American and British English spelling differences#-our, -or, see spelling differences) describes a condition of stiffness or strictness. These constraints may be environmentally imposed, su ...
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 mathematics, a manifold is a topological space that locally resembles Euclidean space near each point. More precisely, an n-dimensional manifold, or ''n-manifold'' for short, is a topological space with the property that each point has a Ne ...
. 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 applications, ...
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 res ...
s, the gradient theorem states that
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
In mathematics, the support of a real-valued function f is the subset of the function domain containing the elements which are not mapped to zero. If the domain of f is a topological space, then the support of f is instead defined as the smalles ...
differential form over the
boundary
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Entertainment
* ''Boundaries'' (2016 film), a 2016 Canadian film
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*Boundary (cricket), the edge of the pla ...
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 res ...
over the whole of , i.e.,
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 form is
exact
Exact may refer to:
* Exaction, a concept in real property law
* ''Ex'Act'', 2016 studio album by Exo
* Schooner Exact, the ship which carried the founders of Seattle
Companies
* Exact (company), a Dutch software company
* Exact Change, an Ameri ...
. This result is summarized by the
Poincaré lemma In mathematics, especially vector calculus and differential topology, a closed form is a differential form ''α'' whose exterior derivative is zero (), and an exact form is a differential form, ''α'', that is the exterior derivative of another ...
.
See also
*
State function
*
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
In topology, the Jordan curve theorem asserts that every '' Jordan curve'' (a plane simple closed curve) divides the plane into an " interior" region bounded by the curve and an " exterior" region containing all of the nearby and far away exteri ...
*
Differential of a function
In calculus, the differential represents the principal part of the change in a function ''y'' = ''f''(''x'') with respect to changes in the independent variable. The differential ''dy'' is defined by
:dy = f'(x)\,dx,
where f'(x) is th ...
*
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 classical ...
*
*
References
{{Calculus topics
Theorems in calculus
Articles containing proofs