|
|
| (One intermediate revision by one other user not shown) |
| Line 1: |
Line 1: |
| {{Calculus |Vector}}
| | Hello. Allow me introduce the writer. Her name is Refugia Shryock. Hiring is my occupation. Minnesota has always been his house but his wife wants them to transfer. One of the issues she loves most is to do aerobics and now she is trying to earn money with it.<br><br>Here is my webpage: [http://www.webmdbook.com/index.php?do=/profile-11685/info/ www.webmdbook.com] |
| | |
| The '''gradient theorem''', also known as the '''fundamental theorem of calculus for line integrals''', says that a [[line integral]] through a [[gradient]] field can be evaluated by evaluating the original scalar field at the endpoints of the curve.
| |
| | |
| Let <math> \varphi : U \subseteq \mathbb{R}^n \to \mathbb{R}</math>. Then
| |
| | |
| :<math> \varphi\left(\mathbf{q}\right)-\varphi\left(\mathbf{p}\right) = \int_{\gamma[\mathbf{p},\,\mathbf{q}]} \nabla\varphi(\mathbf{r})\cdot d\mathbf{r}. </math>
| |
| | |
| It is a generalization of the [[fundamental theorem of calculus]] to any curve in a plane or space (generally ''n''-dimensional) rather than just the real line.
| |
| | |
| 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 φ as potential, ∇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 φ is a [[differentiable function|differentiable]] function from some [[Open set|open subset]] ''U'' (of '''R'''<sup>''n''</sup>) to '''R''', and if '''r''' is a differentiable function from some closed [[interval (mathematics)|interval]] [''a,b''] to ''U'', then by the [[chain rule#The chain rule in higher dimensions|multivariate chain rule]], the [[Function composition|composite function]] φ ∘ '''r''' is differentiable on (''a'', ''b'') and
| |
| | |
| :<math>\frac{d}{dt}(\varphi \circ \mathbf{r})(t)=\nabla \varphi(\mathbf{r}(t)) \cdot \mathbf{r}'(t)</math>
| |
| | |
| for all ''t'' in (''a'', ''b''). Here the ⋅ denotes the [[dot product|usual inner product]].
| |
| | |
| Now suppose the domain ''U'' of φ contains the differentiable curve γ with endpoints '''p''' and '''q''', ([[orientation (vector space)|oriented]] in the direction from '''p''' to '''q'''). If '''r''' [[parametrization|parametrizes]] γ for ''t'' in [''a'', ''b''], then the above shows that <ref>Williamson, Richard and Trotter, Hale. (2004). ''Multivariable Mathematics, Fourth Edition,'' p. 374. Pearson Education, Inc. </ref>
| |
| | |
| :<math>\begin{align}
| |
| \int_{\gamma} \nabla\varphi(\mathbf{u}) \cdot d\mathbf{u} &=\int_a^b \nabla\varphi(\mathbf{r}(t)) \cdot \mathbf{r}'(t)dt \\
| |
| &=\int_a^b \frac{d}{dt}\varphi(\mathbf{r}(t))dt =\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 the line integral]] is used in the first equality, and the [[fundamental theorem of calculus]] is used in the third equality.
| |
| | |
| ==Examples==
| |
| ===Example 1===
| |
| Suppose γ ⊂ '''R'''<sup>2</sup> is the circular arc oriented counterclockwise from (5, 0) to (−4, 3). Using the [[line integral#Line integral of a vector field|definition of a line integral]],
| |
| | |
| :<math>\begin{align}
| |
| \int_{\gamma} y dx+x dy &=\int_0^{\pi-{\tan}^{-1}(\frac{3}{4})} (5\sin t(-5 \sin t)+5 \cos t (5 \cos t)) dt \\
| |
| &=\int_0^{\pi-{\tan}^{-1}(\frac{3}{4})} 25(-{\sin}^2 t+{\cos}^2 t) dt \\
| |
| &=\int_0^{\pi-{\tan}^{-1}(\frac{3}{4})} 25 \cos (2 t) dt \\
| |
| &=\left.\tfrac{25}{2}\sin(2t)\right|_0^{\pi-{\tan}^{-1}(\tfrac{3}{4})} \\
| |
| &=\tfrac{25}{2}\sin(2\pi-2{\tan}^{-1}(\tfrac{3}{4})) \\
| |
| &=-\tfrac{25}{2}\sin(2{\tan}^{-1}(\tfrac{3}{4})) \\
| |
| &=-\frac{25(\tfrac{3}{4})}{{(\tfrac{3}{4})}^2+1}=-12.
| |
| \end{align}</math>
| |
| | |
| Notice all of the painstaking computations involved in directly calculating the integral. Instead, since the function ''f''(''x'', ''y'') = ''xy'' is differentiable on all of '''R'''<sup>2</sup>, we can simply use the gradient theorem to say
| |
| | |
| :<math>\int_{\gamma} y dx+x dy=\int_{\gamma}\nabla(xy) \cdot (dx,dy)=xy|_{(5,0)}^{(-4,3)}=-4 \cdot 3-5 \cdot 0=-12</math>.
| |
| | |
| Notice that either way gives the same answer, but using the latter method, most of the work is already done in the proof of the gradient theorem.
| |
| | |
| ===Example 2===
| |
| For a more abstract example, suppose γ ⊂ '''R'''<sup>''n''</sup> has endpoints '''p''', '''q''', with orientation from '''p''' to '''q'''. For '''u''' in '''R'''<sup>''n''</sup>, let |'''u'''| denote the [[Euclidean norm]] of '''u'''. If α ≥ 1 is a real number, then
| |
| | |
| :<math>\begin{align}
| |
| \int_{\gamma} |\mathbf{x}|^{\alpha - 1} \mathbf{x} \cdot d\mathbf{x} &= \frac{1}{\alpha + 1} \int_{\gamma} (\alpha + 1)|\mathbf{x}|^{(\alpha+1)-2} \mathbf{x} \cdot d\mathbf{x} \\
| |
| &=\frac{1}{\alpha + 1} \int_{\gamma} \nabla(|\mathbf{x}|^{\alpha + 1}) \cdot 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 ''f''('''x''') = |'''x'''|<sup>α + 1</sup> is differentiable on '''R'''<sup>''n''</sup> if α ≥ 1.
| |
| | |
| If 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 |'''x'''|<sup>α−1</sup>'''x''' will fail to be defined there. However, the case α = −1 is somewhat different; in this case, the integrand becomes |'''x'''|<sup>−2</sup>'''x''' = ∇(log|'''x'''|), so that the final equality becomes log|'''q'''|−log|'''p'''|.
| |
| | |
| Note that if n=1, then this example is simply a slight variant of the familiar [[Power rule]] from single-variable calculus.
| |
| | |
| ===Example 3===
| |
| Suppose there are ''n'' [[Point particle#Point charge|point charges]] arranged in three-dimensional space, and the ''i''-th point charge has [[Electric charge|charge]] ''Q<sub>i</sub>'' and is located at position '''p'''<sub>''i''</sub> in '''R'''<sup>3</sup>. We would like to calculate the [[work (physics)|work]] done on a particle of charge ''q'' as it travels from a point '''a''' to a point '''b''' in '''R'''<sup>3</sup>. Using [[Coulomb's law]], we can easily determine that the [[force]] on the particle at position '''r''' will be
| |
| | |
| :<math> \mathbf{F}(\mathbf{r}) = kq\sum_{i=1}^n \frac{Q_i(\mathbf{r}-\mathbf{p}_i)}{|\mathbf{r}-\mathbf{p}_i|^3} </math>
| |
| | |
| Here |'''u'''| denotes the [[Euclidean norm]] of the vector '''u''' in '''R'''<sup>3</sup>, and ''k'' = 1/(4πε<sub>0</sub>), where ε<sub>0</sub> is the [[Vacuum permittivity]].
| |
| | |
| Let γ ⊂ '''R'''<sup>3</sup> − {'''p'''<sub>1</sub>, ..., '''p'''<sub>''n''</sub>} be an arbitrary differentiable curve from '''a''' to '''b'''. Then the work done on the particle is
| |
| | |
| :<math>W =\int_{\gamma} \mathbf{F}(\mathbf{r}) \cdot d\mathbf{r} = \int_{\gamma} \bigg( kq\sum_{i=1}^n \frac{Q_i(\mathbf{r}-\mathbf{p}_i)}{|\mathbf{r}-\mathbf{p}_i|^3} \bigg) \cdot d\mathbf{r} = kq \sum_{i=1}^n \bigg( Q_i \int_{\gamma} \frac{(\mathbf{r}-\mathbf{p}_i)}{|\mathbf{r}-\mathbf{p}_i|^3} \cdot d\mathbf{r} \bigg) </math>
| |
| | |
| Now for each ''i'', direct computation shows that
| |
| | |
| :<math> \frac{(\mathbf{r}-\mathbf{p}_i)}{|\mathbf{r}-\mathbf{p}_i|^3} = - \nabla \left(\frac{1}{|\mathbf{r}-\mathbf{p}_i|}\right). </math>
| |
| | |
| Thus, continuing from above and using the gradient theorem,
| |
| | |
| :<math> W = -kq \sum_{i=1}^n \bigg( Q_i \int_{\gamma} \nabla \left(\frac{1}{|\mathbf{r}-\mathbf{p}_i|} \right) \cdot d\mathbf{r} \bigg) = kq \sum_{i=1}^n Q_i \left( \frac{1}{|\mathbf{a}-\mathbf{p}_i|} - \frac{1}{|\mathbf{b}-\mathbf{p}_i|} \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 ''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 F is the gradient of some scalar-valued function (i.e, if F is [[Conservative vector field | conservative]]), then F is a path-independent vector field (i.e, the integral of F over some piecewise-differentiable curve is dependent only on end points). This theorem has a powerful converse; namely, if F is a path-independent vector field, then F is the gradient of some scalar-valued function.<ref name="poop">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 F over every closed loop in the domain of F is zero, then F is the gradient of some scalar-valued function.
| |
| | |
| {| class="toccolours collapsible collapsed" width="80%" style="text-align:left"
| |
| !Proof of the converse
| |
| |-
| |
| |
| |
| | |
| Suppose ''U'' is an [[Open set|open]], [[Connected space#Path connectedness|path-connected]] subset of '''R'''<sup>n</sup>, and '''F''' : ''U'' → '''R'''<sup>n</sup> is a [[Continuous function|continuous]] and path-independent vector field. Fix some element '''a''' of ''U'', and define ''f'' : ''U'' → '''R''' by
| |
| | |
| :<math> f(\mathbf{x}) := \int_{\gamma[\mathbf{a}, \mathbf{x}]} \mathbf{F}(\mathbf{u}) \cdot d\mathbf{u} </math>
| |
| | |
| Here γ['''a''', '''x'''] is any (differentiable) curve in ''U'' originating at '''a''' and terminating at '''x'''. We know that ''f'' is [[well-defined]] because '''F''' is path-independent. | |
| | |
| Let '''v''' be any nonzero vector in '''R'''<sup>n</sup>. By the definition of the [[directional derivative]],
| |
| | |
| :<math> \begin{align}
| |
| \frac{\partial f}{\partial \mathbf{v}}(\mathbf{x}) &= \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 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 d\mathbf{u}
| |
| \end{align}</math>
| |
| | |
| To calculate the integral within the final limit, we must [[parametrization|parametrize]] γ['''x''', '''x'''+''t'''''v''']. Since '''F''' is path-independent, ''U'' is open, and ''t'' is approaching zero, we may assume that this path is a straight line, and parametrize it as '''u'''(''s'') = '''x''' + ''s'''''v''' for 0 < ''s'' < ''t''. Now, since '''u''''(''s'') = '''v''', the limit becomes
| |
| | |
| :<math> \lim_{t \to 0} \frac{1}{t} \int_0^t \mathbf{F}(\mathbf{u}(s)) \cdot \mathbf{u}'(s) ds = \frac{d}{dt} \int_0^t \mathbf{F}(\mathbf{x} + s\mathbf{v}) \cdot \mathbf{v} ds \bigg|_{t=0} = \mathbf{F}(\mathbf{x}) \cdot \mathbf{v} </math>
| |
| | |
| Thus we have a formula for ∂<sub>'''v'''</sub>''f'', where '''v''' is arbitrary.. Let '''x''' = (''x''<sub>1</sub>, ''x''<sub>2</sub>, ..., ''x<sub>n</sub>'') and let '''e'''<sub>''i''</sub> denote the ''i''-<sup>th</sup> [[Standard basis|standard basis vector]], so that
| |
| | |
| :<math> \nabla f(\mathbf{x}) = \bigg( \frac{\partial f(\mathbf{x})}{\partial x_1}, \frac{\partial f(\mathbf{x})}{\partial x_2}, . . . , \frac{\partial f(\mathbf{x})}{\partial x_n} \bigg) = (\mathbf{F}(\mathbf{x}) \cdot \mathbf{e}_1, \mathbf{F}(\mathbf{x}) \cdot \mathbf{e}_2, . . . , \mathbf{F}(\mathbf{x}) \cdot \mathbf{e}_n) = \mathbf{F}(\mathbf{x})</math>
| |
| | |
| Thus we have found a scalar-valued function ''f'' whose gradient is the path-independent vector field '''F''', as desired.<ref name="poop" />
| |
| |}
| |
| | |
| ===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]] '''F'''<sub>''e''</sub> : ''S'' → '''R'''<sup>3</sup> is conservative (here ''S'' is some [[Open set|open]], [[Connected space#Path connectedness|path-connected]] subset of '''R'''<sup>3</sup> that contains a [[Electric charge|charge]] distribution). Following the ideas of the above proof, we can set some reference point '''a''' in ''S'', and define a function ''U<sub>e</sub>'': ''S'' → '''R''' by
| |
| | |
| :<math> U_e(\mathbf{r}) := -\int_{\gamma[\mathbf{a},\mathbf{r}]} \mathbf{F}_e(\mathbf{u}) \cdot d\mathbf{u} </math> | |
| | |
| Using the above proof, we know ''U<sub>e</sub>'' is well-defined and differentiable, and '''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: ''W'' = −Δ''U''). This function ''U<sub>e</sub>'' is often referred to as the [[Electric potential energy|electrostatic potential energy]] of the system of charges in ''S'' (with reference to the zero-of-potential '''a'''). In many cases, the domain ''S'' is assumed to be [[Bounded set|unbounded]] and the reference point '''a''' is taken to be "infinity," which can be made [[Rigour#Mathematical rigour|rigorous]] using limiting techniques. This function ''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> \int_{\partial \gamma} \phi = \int_{\gamma} d\phi</math>
| |
| | |
| for any [[differential form|0-form]] φ defined on some differentiable curve γ ⊂ '''R'''<sup>''n''</sup> (here the integral of φ over the boundary of the γ is understood to be the evaluation of φ at the endpoints of γ).
| |
| | |
| Notice the striking similarity between this statement and the generalized version of [[Stokes' theorem]], which says that the integral of any [[compact support|compactly supported]] differential form ω over the [[boundary (topology)|boundary]] of some [[orientation (vector space)|orientable]] manifold Ω is equal to the integral of its [[exterior derivative]] dω over the whole of Ω, i.e.
| |
| | |
| :<math>\int_{\partial \Omega}\omega=\int_{\Omega}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 ω is a form defined on a [[Contractible space|contractible domain]], and the integral of ω over any closed manifold is zero. Then there exists a form ψ such that 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]]
| |
| | |
| ==References==
| |
| <references/>
| |
| | |
| [[Category:Theorems in calculus]]
| |
| [[Category:Articles containing proofs]]
| |