View source for Gradient theorem

{{short description|Evaluates a line integral through a gradient field using the original scalar field}}
{{Calculus|Vector}}

The '''gradient theorem''', also known as the '''fundamental theorem of calculus for line integrals''', says that a [[line integral]] through a [[Conservative vector field|gradient field]] can be evaluated by evaluating the original scalar field at the endpoints of the curve. The theorem is a generalization of the [[Fundamental theorem of calculus|second fundamental theorem of calculus]] to any curve in a plane or space (generally ''n''-dimensional) rather than just the real line.

If {{math|''φ'' : ''U'' ⊆ '''R'''<sup>''n''</sup> → '''R'''}} is a [[differentiable function]] and {{mvar|&gamma;}} a [[Differentiable function|differentiable]] curve in {{math|''U''}} which starts at a point {{math|'''p'''}} and ends at a point {{math|'''q'''}}, then

<math display="block"> \int_{\gamma} \nabla\varphi(\mathbf{r})\cdot \mathrm{d}\mathbf{r} = \varphi\left(\mathbf{q}\right) - \varphi\left(\mathbf{p}\right)</math>

where {{math|&nabla;''φ''}} denotes the gradient vector field of {{math|''φ''}}.

The gradient theorem implies that line integrals through gradient fields are [[Conservative vector field#Path independence|path-independent]]. In physics this theorem is one of the ways of defining a [[conservative force|''conservative'' force]]. By placing {{mvar|φ}} as potential, {{math|∇''φ''}} is a [[conservative field]]. [[Work (physics)|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]]. Just like the gradient theorem itself, this converse has many striking consequences and applications in both pure and applied mathematics.

==Proof==
If {{mvar|φ}} is a [[differentiable function]] from some [[Open set|open subset]] {{math|''U'' ⊆ '''R'''<sup>''n''</sup>}} to {{math|'''R'''}} and {{math|'''r'''}} is a differentiable function from some closed [[interval (mathematics)|interval]] {{math|[''a'', ''b'']}} to {{mvar|U}} (Note that {{math|'''r'''}} is differentiable at the interval endpoints {{math|''a''}} and {{math|''b''}}. To do this, {{math|'''r'''}} is defined on an interval that is larger than and includes {{math|[''a'', ''b'']}}.), then by the [[chain rule#Multivariable case|multivariate chain rule]], the [[Function composition|composite function]] {{math|''φ'' ∘ '''r'''}} is differentiable on {{math|[''a'', ''b'']}}:

<math display="block">\frac{\mathrm{d}}{\mathrm{d}t}(\varphi \circ \mathbf{r})(t)=\nabla \varphi(\mathbf{r}(t)) \cdot \mathbf{r}'(t)</math>

for all {{mvar|t}} in {{math|[''a'', ''b'']}}. Here the {{math|⋅}} denotes the [[dot product|usual inner product]].

Now suppose the domain {{mvar|U}} of {{mvar|φ}} contains the differentiable curve {{mvar|γ}} with endpoints {{math|'''p'''}} and {{math|'''q'''}}. (This is [[orientation (vector space)|oriented]] in the direction from {{math|'''p'''}} to {{math|'''q'''}}). If {{math|'''r'''}} [[Parametrization (geometry)|parametrizes]] {{mvar|γ}} for {{mvar|t}} in {{math|[''a'', ''b'']}} (i.e., {{math|'''r'''}} represents {{mvar|γ}} as a function of {{mvar|t}}), then

<math display="block">\begin{align}
\int_{\gamma} \nabla\varphi(\mathbf{r})  \cdot  \mathrm{d}\mathbf{r} &=\int_a^b \nabla\varphi(\mathbf{r}(t))  \cdot  \mathbf{r}'(t)\mathrm{d}t \\
&=\int_a^b \frac{d}{dt}\varphi(\mathbf{r}(t))\mathrm{d}t =\varphi(\mathbf{r}(b))-\varphi(\mathbf{r}(a))=\varphi\left(\mathbf{q}\right)-\varphi\left(\mathbf{p}\right) ,
\end{align} </math>

where the [[line integral#Line integral of a vector field|definition of a line integral]] is used in the first equality, the above equation is used in the second equality, and the [[Fundamental theorem of calculus#Second part|second fundamental theorem of calculus]] is used in the third equality.<ref>Williamson, Richard and Trotter, Hale. (2004). ''Multivariable Mathematics, Fourth Edition,'' p. 374. Pearson Education, Inc.</ref>

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.<ref>{{Cite book |last=Stewart |first=James |title=Calculus |publisher=Cengage Learning |year=2021 |isbn=978-1-337-62418-3 |edition=9th |pages=1182–1185 |language=English |chapter=16.3 The Fundamental Theorem for Line Integrals}}</ref>

==Examples==

===Example 1===
Suppose {{math|''γ'' ⊂ '''R'''<sup>2</sup>}} is the circular arc oriented counterclockwise from {{math|(5, 0)}} to {{math|(−4, 3)}}. Using the [[line integral#Line integral of a vector field|definition of a line integral]],

<math display="block">\begin{align}
  \int_{\gamma} y\, \mathrm{d}x + x\, \mathrm{d}y 
    &= \int_0^{\pi - \tan^{-1}\!\left(\frac{3}{4}\right)} ((5\sin t)(-5 \sin t) + (5 \cos t)(5 \cos t))\, \mathrm{d}t \\
    &= \int_0^{\pi - \tan^{-1}\!\left(\frac{3}{4}\right)} 25 \left(-\sin^2 t + \cos^2 t\right) \mathrm{d}t \\
    &= \int_0^{\pi - \tan^{-1}\!\left(\frac{3}{4}\right)} 25 \cos(2t) \mathrm{d}t \ =\ \left.\tfrac{25}{2}\sin(2t)\right|_0^{\pi - \tan^{-1}\!\left(\tfrac{3}{4}\right)} \\[.5em]
    &=  \tfrac{25}{2}\sin\left(2\pi - 2\tan^{-1}\!\!\left(\tfrac{3}{4}\right)\right) \\[.5em]
    &= -\tfrac{25}{2}\sin\left(2\tan^{-1}\!\!\left(\tfrac{3}{4}\right)\right) \ =\ -\frac{25(3/4)}{(3/4)^2 + 1} = -12. 
\end{align}</math>

This result can be obtained much more simply by noticing that the function <math>f(x,y)=xy</math> has gradient <math>\nabla f(x,y)=(y,x)</math>, so by the Gradient Theorem:

<math display="block">\int_{\gamma} y \,\mathrm{d}x+x \,\mathrm{d}y=\int_{\gamma}\nabla(xy) \cdot (\mathrm{d}x,\mathrm{d}y)\ =\ xy\,|_{(5,0)}^{(-4,3)}=-4 \cdot 3-5 \cdot 0=-12 .</math>

===Example 2===
For a more abstract example, suppose {{math|''γ'' ⊂ '''R'''<sup>''n''</sup>}} has endpoints {{math|'''p'''}}, {{math|'''q'''}}, with orientation from {{math|'''p'''}} to {{math|'''q'''}}. For {{math|'''u'''}} in {{math|'''R'''<sup>''n''</sup>}}, let {{math|{{abs|'''u'''}}}} denote the [[Euclidean norm]] of {{math|'''u'''}}.  If {{math|''α'' ≥ 1}} is a real number, then

<math display="block">\begin{align}
  \int_{\gamma} |\mathbf{x}|^{\alpha - 1} \mathbf{x} \cdot \mathrm{d}\mathbf{x}
    &= \frac{1}{\alpha + 1} \int_{\gamma} (\alpha + 1) |\mathbf{x}|^{(\alpha + 1) - 2} \mathbf{x} \cdot \mathrm{d}\mathbf{x} \\
    &= \frac{1}{\alpha + 1} \int_{\gamma} \nabla |\mathbf{x}|^{\alpha + 1} \cdot \mathrm{d}\mathbf{x}= \frac{|\mathbf{q}|^{\alpha + 1} - |\mathbf{p}|^{\alpha + 1}}{\alpha + 1} 
\end{align}</math>

Here the final equality follows by the gradient theorem, since the function {{math|1=''f''('''x''') = {{abs|'''x'''}}<sup>''α''+1</sup>}} is differentiable on {{math|'''R'''<sup>''n''</sup>}} if {{math|''α'' ≥ 1}}.

If {{math|''α'' < 1}} 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 {{math|{{abs|'''x'''}}<sup>''α'' − 1</sup>'''x'''}} will fail to be defined there. However, the case {{math|1=''α'' = −1}} is somewhat different; in this case, the integrand becomes {{math|1={{abs|'''x'''}}<sup>−2</sup>'''x''' = ∇(log {{abs|'''x'''}})}}, so that the final equality becomes {{math|log {{abs|'''q'''}} − log {{abs|'''p'''}}}}.

Note that if {{math|1=''n'' = 1}}, then this example is simply a slight variant of the familiar [[power rule]] from single-variable calculus.

===Example 3===
Suppose there are {{mvar|n}} [[Point particle#Point charge|point charges]] arranged in three-dimensional space, and the {{mvar|i}}-th point charge has [[Electric charge|charge]] {{math|''Q''<sub>''i''</sub>}} and is located at position {{math|'''p'''<sub>''i''</sub>}} in {{math|'''R'''<sup>3</sup>}}. We would like to calculate the [[work (physics)|work]] done on a particle of charge {{mvar|q}} as it travels from a point {{math|'''a'''}} to a point {{math|'''b'''}} in {{math|'''R'''<sup>3</sup>}}. Using [[Coulomb's law]], we can easily determine that the [[force]] on the particle at position {{math|'''r'''}} will be

<math display="block"> \mathbf{F}(\mathbf{r}) = kq\sum_{i=1}^n \frac{Q_i(\mathbf{r} - \mathbf{p}_i)}{\left|\mathbf{r} - \mathbf{p}_i\right|^3} </math>

Here {{math|{{abs|'''u'''}}}} denotes the [[Euclidean norm]] of the vector {{math|'''u'''}} in {{math|'''R'''<sup>3</sup>}}, and {{math|1=''k'' = 1/(4''πε''<sub>0</sub>)}}, where {{math|''ε''<sub>0</sub>}} is the [[vacuum permittivity]].

Let {{math|''γ'' ⊂ '''R'''<sup>3</sup> − {'''p'''<sub>1</sub>, ..., '''p'''<sub>''n''</sub>}{{null}}}} be an arbitrary differentiable curve from {{math|'''a'''}} to {{math|'''b'''}}. Then the work done on the particle is

<math display="block">
  W = \int_{\gamma} \mathbf{F}(\mathbf{r}) \cdot \mathrm{d}\mathbf{r}
    = \int_{\gamma} \left( kq\sum_{i=1}^n \frac{Q_i(\mathbf{r} - \mathbf{p}_i)}{\left|\mathbf{r} - \mathbf{p}_i\right|^3} \right) \cdot \mathrm{d}\mathbf{r}
    = kq \sum_{i=1}^n \left( Q_i \int_\gamma \frac{\mathbf{r} - \mathbf{p}_i}{\left|\mathbf{r} - \mathbf{p}_i\right|^3} \cdot \mathrm{d}\mathbf{r} \right)
</math>

Now for each {{mvar|i}}, direct computation shows that

<math display="block"> \frac{\mathbf{r} - \mathbf{p}_i}{\left|\mathbf{r} - \mathbf{p}_i\right|^3} = -\nabla \frac{1}{\left|\mathbf{r} - \mathbf{p}_i\right|}. </math>

Thus, continuing from above and using the gradient theorem,

<math display="block">
  W = -kq \sum_{i=1}^n \left( Q_i \int_{\gamma} \nabla \frac{1}{\left|\mathbf{r} - \mathbf{p}_i\right|} \cdot \mathrm{d}\mathbf{r} \right)
    =  kq \sum_{i=1}^n Q_i \left( \frac{1}{\left|\mathbf{a} - \mathbf{p}_i\right|} - \frac{1}{\left|\mathbf{b} - \mathbf{p}_i\right|} \right)
</math>

We are finished. Of course, we could have easily completed this calculation using the powerful language of [[Electric potential|electrostatic potential]] or [[Electric potential energy|electrostatic potential energy]] (with the familiar formulas {{math|1=''W'' = −Δ''U'' = −''q''Δ''V''}}). 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 ([[Gradient theorem#Example of the converse principle|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 {{math|'''F'''}} is the gradient of some scalar-valued function (i.e., if {{math|'''F'''}} is [[Conservative vector field|conservative]]), then {{math|'''F'''}} is a path-independent vector field (i.e., the integral of {{math|'''F'''}} over some piecewise-differentiable curve is dependent only on end points). This theorem has a powerful converse:
{{math theorem| If {{math|'''F'''}} is a path-independent vector field, then {{math|'''F'''}} is the gradient of some scalar-valued function.<ref name="wt">"Williamson, Richard and Trotter, Hale. (2004). ''Multivariable Mathematics, Fourth Edition'', p. 410. Pearson Education, Inc."</ref>}}
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 {{math|'''F'''}} over every closed loop in the domain of {{math|'''F'''}} is zero, then {{math|'''F'''}} is the gradient of some scalar-valued function.

=== Proof of the converse ===
Suppose {{mvar|U}} is an [[Open set|open]], [[Connected space#Path connectedness|path-connected]] subset of {{math|'''R'''<sup>''n''</sup>}}, and {{math|'''F''' : ''U'' → '''R'''<sup>''n''</sup>}} is a [[Continuous function|continuous]] and path-independent vector field. Fix some element {{math|'''a'''}} of {{mvar|U}}, and define {{math|''f'' : ''U'' → '''R'''}} by<math display="block"> f(\mathbf{x}) := \int_{\gamma[\mathbf{a}, \mathbf{x}]} \mathbf{F}(\mathbf{u}) \cdot \mathrm{d}\mathbf{u} </math>Here {{math|''γ''['''a''', '''x''']}} is any (differentiable) curve in {{mvar|U}} originating at {{math|'''a'''}} and terminating at {{math|'''x'''}}. We know that {{math|''f''}} is [[well-defined]] because {{math|'''F'''}} is path-independent.

Let {{math|'''v'''}} be any nonzero vector in {{math|'''R'''<sup>''n''</sup>}}. By the definition of the [[directional derivative]],<math display="block"> \begin{align} 
\frac{\partial f(\mathbf{x})}{\partial \mathbf{v}} &= \lim_{t \to 0} \frac{f(\mathbf{x} + t\mathbf{v}) - f(\mathbf{x})}{t} \\
&= \lim_{t \to 0} \frac{\int_{\gamma[\mathbf{a}, \mathbf{x} + t\mathbf{v}]} \mathbf{F}(\mathbf{u}) \cdot \mathrm{d}\mathbf{u} - \int_{\gamma[\mathbf{a}, \mathbf{x}]} \mathbf{F}(\mathbf{u}) \cdot d\mathbf{u}}{t} \\
&= \lim_{t \to 0} \frac{1}{t} \int_{\gamma[\mathbf{x}, \mathbf{x} + t\mathbf{v}]} \mathbf{F}(\mathbf{u}) \cdot \mathrm{d}\mathbf{u}
\end{align}</math>To calculate the integral within the final limit, we must [[Parametrization (geometry)|parametrize]] {{math|''γ''['''x''', '''x''' + ''t'''''v''']}}. Since {{math|'''F'''}} is path-independent, {{mvar|U}} is open, and {{mvar|t}} is approaching zero, we may assume that this path is a straight line, and parametrize it as {{math|1='''u'''(''s'') = '''x''' + ''s'''''v'''}} for {{math|0 < ''s'' < ''t''}}. Now, since {{math|1='''u''''(''s'') = '''v'''}}, the limit becomes<math display="block"> \lim_{t \to 0} \frac{1}{t} \int_0^t \mathbf{F}(\mathbf{u}(s)) \cdot \mathbf{u}'(s)\, \mathrm{d}s =  \frac{\mathrm{d}}{\mathrm{d}t} \int_0^t \mathbf{F}(\mathbf{x} + s\mathbf{v}) \cdot \mathbf{v}\, \mathrm{d}s \bigg|_{t=0} = \mathbf{F}(\mathbf{x}) \cdot \mathbf{v} </math>where the first equality is from [[Derivative#Definition|the definition of the derivative]] with a fact that the integral is equal to 0 at {{mvar|t}} = 0, and the second equality is from the [[Fundamental theorem of calculus#First part|first fundamental theorem of calculus]]. Thus we have a formula for {{math|∂<sub>'''v'''</sub>''f''}}, (one of ways to represent the [[directional derivative]]) where {{math|'''v'''}} is arbitrary; for <math> f(\mathbf{x}) := \int_{\gamma[\mathbf{a}, \mathbf{x}]} \mathbf{F}(\mathbf{u}) \cdot \mathrm{d}\mathbf{u} </math> (see its full definition above), its directional derivative with respect to {{math|'''v'''}} is<math display="block"> \frac{\partial f(\mathbf{x})}{\partial \mathbf{v}} = \partial _ \mathbf{v} f(\mathbf{x}) = D_{\mathbf{v}}f(\mathbf{x}) = \mathbf{F}(\mathbf{x}) \cdot \mathbf{v} </math>where the first two equalities just show different representations of the directional derivative. According to [[Gradient#Definition|the definition of the gradient]] of a scalar function {{math|''f''}}, <math> \nabla f(\mathbf{x}) = \mathbf{F}(\mathbf{x})</math>, thus we have found a scalar-valued function {{mvar|f}} whose gradient is the path-independent vector field {{math|'''F'''}} (i.e., {{math|'''F'''}} is a conservative vector field.), as desired.<ref name="wt" />

===Example of the converse principle===
{{main| Electric potential energy}}
To illustrate the power of this converse principle, we cite an example that has significant [[physics|physical]] consequences. In [[classical electromagnetism]], the [[electric force]] is a path-independent force; i.e. the [[work (physics)|work]] done on a particle that has returned to its original position within an [[electric field]] is zero (assuming that no changing [[magnetic field]]s are present).

Therefore, the above theorem implies that the electric [[Force field (physics)|force field]] {{math|'''F'''<sub>''e''</sub> : ''S'' → '''R'''<sup>3</sup>}} is conservative (here {{mvar|S}} is some [[Open set|open]], [[Connected space#Path connectedness|path-connected]] subset of {{math|'''R'''<sup>3</sup>}} that contains a [[Electric charge|charge]] distribution). Following the ideas of the above proof, we can set some reference point {{math|'''a'''}} in {{mvar|S}}, and define a function {{math|''U<sub>e</sub>'': ''S'' → '''R'''}} by

<math display="block"> U_e(\mathbf{r}) :=  -\int_{\gamma[\mathbf{a},\mathbf{r}]} \mathbf{F}_e(\mathbf{u}) \cdot \mathrm{d}\mathbf{u} </math>

Using the above proof, we know {{math|''U''<sub>''e''</sub>}} is well-defined and differentiable, and {{math|1='''F'''<sub>''e''</sub> = −∇''U<sub>e</sub>''}} (from this formula we can use the gradient theorem to easily derive the well-known formula for calculating work done by conservative forces: {{math|1=''W'' = −Δ''U''}}). This function {{math|''U''<sub>''e''</sub>}} is often referred to as the [[Electric potential energy|electrostatic potential energy]] of the system of charges in {{mvar|S}} (with reference to the zero-of-potential {{math|'''a'''}}). In many cases, the domain {{mvar|S}} is assumed to be [[Bounded set|unbounded]] and the reference point {{math|'''a'''}} is taken to be "infinity", which can be made [[Rigour#Mathematical rigour|rigorous]] using limiting techniques. This function {{math|''U''<sub>''e''</sub>}} is an indispensable tool used in the analysis of many physical systems.

==Generalizations==
{{main|Stokes' theorem|Closed and exact differential forms}}

Many of the critical theorems of vector calculus generalize elegantly to statements about the [[Differential form#Integration|integration of differential forms]] on [[Differentiable manifold|manifolds]]. In the language of [[differential form]]s and [[exterior derivative]]s, the gradient theorem states that

<math display="block"> \int_{\partial \gamma} \phi = \int_{\gamma} \mathrm{d}\phi</math>

for any [[differential form|0-form]], {{mvar|ϕ}}, defined on some differentiable curve {{math|''γ'' ⊂ '''R'''<sup>''n''</sup>}} (here the integral of {{math|''ϕ''}} over the boundary of the {{mvar|γ}} is understood to be the evaluation of {{math|''ϕ''}} at the endpoints of ''γ'').

Notice the striking similarity between this statement and the [[Generalized Stokes theorem|generalized Stokes’ theorem]], which says that the integral of any [[compact support|compactly supported]] differential form {{mvar|ω}} over the [[boundary (topology)|boundary]] of some [[orientation (vector space)|orientable]] manifold {{math|Ω}} is equal to the integral of its [[exterior derivative]] {{math|d''ω''}} over the whole of {{math|Ω}}, i.e.,

<math display="block">\int_{\partial \Omega}\omega=\int_{\Omega}\mathrm{d}\omega</math>

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 {{mvar|ω}} is a form defined on a [[Contractible space|contractible domain]], and the integral of {{mvar|ω}} over any closed manifold is zero. Then there exists a form {{mvar|ψ}} such that {{math|1=''ω'' = d''ψ''}}. Thus, on a contractible domain, every [[Closed and exact differential forms|closed]] form is [[Closed and exact differential forms|exact]]. This result is summarized by the [[Closed and exact differential forms#Poincaré lemma|Poincaré lemma]].

==See also==
*[[State function]]
*[[Scalar potential]]
*[[Jordan curve theorem]]
*[[Differential of a function]]
*[[Classical mechanics]]
*{{section link|Line integral|Path independence}}
*{{section link|Conservative vector field|Path independence}}

==References==
{{reflist}}

{{Calculus topics}}

[[Category:Theorems in calculus]]
[[Category:Articles containing proofs]]