Meta tags:

Headings (most frequently used words):

example, converse, of, the, gradient, theorem, proof, contents, examples, generalizations, see, also, references, principle,

Text of the page (most frequently used words):
the (149), mathbf (120), #theorem (49), and (39), integral (38), gradient (35), mathrm (33), int (30), function (26), left (26), right (26), that (25), this (24), cdot (24), frac (24), displaystyle (22), calculus (21), field (20), gamma (20), differential (19), derivative (19), vector (17), differentiable (17), from (16), for (16), path (16), edit (15), line (14), curve (14), converse (14), partial (13), tan (12), sin (12), varphi (12), then (11), some (11), example (11), using (10), integration (10), functions (10), limit (10), has (10), defined (10), independent (10), nabla (10), with (9), proof (9), scalar (9), potential (9), alpha (9), equation (8), fundamental (8), definition (8), conservative (8), can (8), point (8), equality (8), lim (8), aligned (8), cos (8), wikipedia (7), integrals (7), forms (7), rule (7), second (7), sum (7), also (7), over (7), any (7), here (7), above (7), theorems (6), formula (6), inverse (6), method (6), series (6), power (6), geometric (6), stokes (6), directional (6), form (6), suppose (6), closed (6), zero (6), thus (6), domain (6), work (6), where (6), end (6), tfrac (6), page (5), sidebar (5), euler (5), leibniz (5), taylor (5), first (5), differentiation (5), its (5), endpoints (5), main (5), done (5), force (5), are (5), have (5), log (5), toggle (4), contents (4), search (4), statement (4), about (4), articles (4), history (4), comparison (4), alternating (4), generalized (4), exterior (4), matrix (4), divergence (4), notation (4), one (4), see (4), powerful (4), manifolds (4), omega (4), many (4), well (4), used (4), energy (4), electric (4), charge (4), open (4), particle (4), since (4), valued (4), begin (4), only (4), through (4), hide (4), move (4), view (3), under (3), terms (3), different (3), angle (3), maclaurin (3), tensor (3), manifold (3), topics (3), list (3), derivatives (3), limits (3), secant (3), trigonometric (3), logarithmic (3), law (3), newton (3), infinitesimal (3), direct (3), binomial (3), multiple (3), multivariable (3), tangent (3), substitution (3), value (3), test (3), points (3), related (3), general (3), product (3), chain (3), generalization (3), there (3), every (3), dimensional (3), arbitrary (3), which (3), generalizations (3), easily (3), reference (3), made (3), analysis (3), electrostatic (3), subset (3), principle (3), original (3), position (3), article (3), let (3), final (3), now (3), becomes (3), just (3), because (3), denotes (3), euclidean (3), space (3), will (3), examples (3), interval (3), definitions (3), tools (3), languages (2), table (2), contact (2), privacy (2), policy (2), may (2), use (2), was (2), pages (2), short (2), description (2), wikidata (2), problem (2), bee (2), curves (2), contour (2), exponential (2), lists (2), continuity (2), approximation (2), mathematical (2), constant (2), numbers (2), special (2), tests (2), convergence (2), term (2), root (2), ratio (2), dirichlet (2), cauchy (2), condensation (2), abel (2), harmonic (2), arithmetico (2), sequence (2), advanced (2), volume (2), surface (2), jacobian (2), hessian (2), basic (2), green (2), laplacian (2), curl (2), fractions (2), half (2), parts (2), differentiating (2), sign (2), riemann (2), arc (2), antiderivative (2), stochastic (2), applications (2), other (2), techniques (2), rates (2), implicit (2), quotient (2), hôpital (2), rules (2), mean (2), order (2), rolle (2), free (2), variables (2), continuous (2), precalculus (2), williamson (2), trotter (2), 2004 (2), 978 (2), isbn (2), pearson (2), mathematics (2), references (2), independence (2), classical (2), contractible (2), result (2), exact (2), striking (2), says (2), equal (2), boundary (2), phi (2), language (2), states (2), know (2), known (2), forces (2), charges (2), cases (2), taken (2), physical (2), implies (2), contains (2), define (2), connected (2), cite (2), consequences (2), within (2), changing (2), fields (2), calculate (2), must (2), parametrize (2), ways (2), show (2), loop (2), follows (2), piecewise (2), familiar (2), formulas (2), however (2), not (2), these (2), hold (2), coulomb (2), shows (2), norm (2), like (2), note (2), simply (2), but (2), integrand (2), case (2), more (2), real (2), 5em (2), oriented (2), proved (2), smooth (2), than (2), identities (2), appearance (2), upload (2), file (2), changes (2), links (2), read (2), subsection (2), create (2), account (2), donate (2), menu (2), add, topic, mobile, cookie, statistics, developers, code, conduct, legal, safety, contacts, disclaimers, text, available, additional, apply, site, you, agree, registered, trademark, non, profit, organization, wikimedia, foundation, inc, creative, commons, attribution, sharealike, license, rendered, parsoid, last, edited, august, 2026, utc, hidden, categories, containing, proofs, child, parameter, category, retrieved, https, org, index, php, title, gradient_theorem, oldid, 1369459224, steinmetz, solid, regiomontanus, maximization, exceeds, gabriel, horn, geometry, surfaces, curvature, complex, miscellaneous, cubed, hyperbolic, irrational, algebraic, rational, mechanical, fluxions, leonhard, fluxion, isaac, gottfried, wilhelm, generality, algebra, colin, brook, adequality, stirling, natural, logarithm, bernoulli, types, telescoping, infinite, fourier, sequences, lagrange, multiplier, determinant, gauss, integro, volumes, shell, washer, trapezoidal, quadratic, properties, length, ordinary, further, maximum, minimum, extreme, stationary, linearity, operator, sided, indeterminate, slope, radian, linear, graph, bound, finite, difference, factorial, concave, 410, clegg, dan, watson, saleem, 2021, ninth, australia, boston, usa, 1185, 337, 62418, 1182, cengage, stewart, james, richard, 4th, upper, saddle, river, 374, 067276, prentice, hall, education, international, hale, mechanics, jordan, state, particular, exists, such, summarized, poincaré, lemma, dimension, notice, similarity, between, whole, orientable, compactly, supported, understood, evaluation, critical, generalize, elegantly, statements, derive, calculating, often, referred, system, assumed, infinity, limiting, indispensable, tool, systems, rigorous, unbounded, therefore, distribution, following, ideas, set, illustrate, significant, returned, assuming, present, magnetic, electromagnetism, nonzero, approaching, assume, straight, fact, represent, full, respect, two, equalities, representations, according, found, whose, desired, bigg, fix, element, originating, terminating, straightforward, alternatively, stated, dependent, finished, course, could, completed, calculation, yet, required, prove, solved, below, continuing, each, computation, vacuum, permittivity, arranged, three, located, would, travels, determine, slight, variant, single, variable, still, most, caution, passes, encloses, origin, fail, somewhat, abstract, orientation, denote, number, obtained, much, noticing, circular, counterclockwise, even, called, been, looked, far, joining, per, component, third, direction, represents, parametrizes, all, dot, circ, larger, includes, composite, multivariate, interesting, expressed, itself, both, pure, applied, physics, defining, placing, does, depend, followed, object, starts, ends, evaluated, evaluating, plane, generally, rather, nonstandard, glossary, miscellanea, variations, malliavin, fractional, specialized, distributions, fubini, clairaut, formalisms, helmholtz, decomposition, summand, risch, algorithm, reduction, formulae, heaviside, cylindrical, shells, discs, nonelementary, lebesgue, improper, transform, reynolds, faà, bruno, concepts, total, part, evaluates, encyclopedia, item, projects, printable, version, download, pdf, print, export, switch, legacy, parser, get, shortened, url, information, permanent, link, what, actions, english, talk, українська, slovenščina, português, 한국어, italiano, עברית, français, فارسی, español, esperanto, bosanski, العربية, top, personal, recent, community, portal, learn, help, contribute, random, current, events, navigation, jump, content,

Text of the page (random words):
ion 3 1 proof of the converse 3 2 example of the converse principle 4 generalizations 5 see also 6 references toggle the table of contents gradient theorem 14 languages العربية bosanski esperanto español فارسی français עברית italiano 한국어 português slovenščina українська 粵語 中文 edit links article talk english read edit view history tools tools move to sidebar hide actions read edit view history general what links here related changes upload file permanent link page information cite this page get shortened url switch to legacy parser print export download as pdf printable version in other projects wikidata item appearance move to sidebar hide from wikipedia the free encyclopedia evaluates a line integral through a gradient field using the original scalar field part of a series of articles about calculus a b f t d t f b f a displaystyle int _ a b f t dt f b f a fundamental theorem limits continuity rolle s theorem mean value theorem inverse function theorem differential definitions derivative generalizations differential infinitesimal of a function total concepts differentiation notation second derivative implicit differentiation logarithmic differentiation related rates taylor s theorem rules and identities sum product chain power quotient l hôpital s rule inverse general leibniz faà di bruno s formula reynolds integral lists of integrals integral transform leibniz integral rule definitions antiderivative integral improper riemann integral lebesgue integration contour integration integral of inverse functions nonelementary integral integration by parts discs cylindrical shells substitution trigonometric tangent half angle euler euler s formula partial fractions heaviside s method changing order reduction formulae differentiating under the integral sign risch algorithm series geometric arithmetico geometric harmonic alternating power binomial taylor convergence tests summand limit term test ratio root integral direct comparison limit comparison alternating series cauchy condensation dirichlet abel vector gradient divergence curl laplacian directional derivative identities theorems gradient green s stokes divergence generalized stokes helmholtz decomposition multivariable formalisms matrix tensor exterior geometric definitions partial derivative multiple integral line integral surface integral volume integral jacobian hessian theorems clairaut s fubini s advanced calculus on euclidean space generalized functions limit of distributions specialized fractional malliavin stochastic variations miscellanea precalculus history glossary list of topics integration bee mathematical analysis nonstandard analysis v t e 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 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 if φ u r n r is a differentiable function and γ a differentiable curve in u which starts at a point p and ends at a point q then γ φ r d r φ q φ p displaystyle int _ gamma nabla varphi mathbf r cdot mathrm d mathbf r varphi left mathbf q right varphi left mathbf p right where φ denotes the gradient vector field of φ 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 φ as potential φ 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 just like the gradient theorem itself this converse has many striking consequences and applications in both pure and applied mathematics proof edit if φ is a differentiable function from some open subset u r n to r and r is a differentiable function from some closed interval a b to u note that r is differentiable at the interval endpoints a and b to do this r is defined on an interval that is larger than and includes a b then by the multivariate chain rule the composite function φ r is differentiable on a b d d t φ r t φ r t r t displaystyle frac mathrm d mathrm d t varphi circ mathbf r t nabla varphi mathbf r t cdot mathbf r t for all t in a b here the denotes the dot product now suppose the domain u of φ contains the differentiable curve γ with endpoints p and q this is oriented in the direction from p to q if r parametrizes γ for t in a b i e r represents γ as a function of t then γ φ r d r a b φ r t r t d t a b d d t φ r t d t φ r b φ r a φ q φ p displaystyle begin aligned 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 aligned 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 1 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 2 examples edit example 1 edit suppose γ r 2 is the circular arc oriented counterclockwise from 5 0 to 4 3 using the definition of a line integral γ y d x x d y 0 π tan 1 3 4 5 sin t 5 sin t 5 cos t 5 cos t d t 0 π tan 1 3 4 25 sin 2 t cos 2 t d t 0 π tan 1 3 4 25 cos 2 t d t 25 2 sin 2 t 0 π tan 1 3 4 25 2 sin 2 π 2 tan 1 3 4 25 2 sin 2 tan 1 3 4 25 3 4 3 4 2 1 12 displaystyle begin aligned 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 aligned this result can be obtained much more simply by noticing that the function f x y x y displaystyle f x y xy has gradient f x y y x displaystyle nabla f x y y x so by the gradient theorem γ y d x x d y γ x y d x d y x y 5 0 4 3 4 3 5 0 12 displaystyle 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 example 2 edit for a more abstract example suppose γ r n has endpoints p q with orientation from p to q for u in r n let u denote the euclidean norm of u if α 1 is a real number then γ x α 1 x d x 1 α 1 γ α 1 x α 1 2 x d x 1 α 1 γ x α 1 d x q α 1 p α 1 α 1 displaystyle begin aligned 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 aligned here the final equality follows by the gradient theorem since the function f x x α 1 is differentiable on r n 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 α 1 x will fail to be defined there however the case α 1 is somewhat different in this case the integrand becomes x 2 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 edit suppose there are n point charges arranged in three dimensional space and the i th point charge has charge q i and is located at position p i in r 3 we would like to calculate the work done on a particle of charge q as it travels from a point a to a point b in r 3 using coulomb s law we can easily determine that the force on the particle at position r will be f r k q i 1 n q i r p i r p i 3 displaystyle 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 here u denotes the euclidean norm of the vector u in r 3 and k 1 4 πε 0 where ε 0 is the vacuum permittivity let γ r 3 p 1 p n be an arbitrary differentiable curve from a to b then the work done on the particle is w γ f r d r γ k q i 1 n q i r p i r p i 3 d r k q i 1 n q i γ r p i r p i 3 d r displaystyle 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 now for each i direct computation shows that r p i r p i 3 1 r p i displaystyle frac mathbf r mathbf p _ i left mathbf r mathbf p _ i right 3 nabla frac 1 left mathbf r mathbf p _ i right thus continuing from above and using the gradient theorem w k q i 1 n q i γ 1 r p i d r k q i 1 n q i 1 a p i 1 b p i displaystyle 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 we are finished of course we could have easily completed this calculation using the powerful language of electrostatic potential or 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 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 edit the gradient theorem states that if the vector field f is the gradient of some scalar valued function i e if f is 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 theorem if f is a path independent vector field then f is the gradient of some scalar valued function 3 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 proof of the converse edit suppose u is an open path connected subset of r n and f u r n is a continuous and path independent vector field fix some element a of u and define f u r by f x γ a x f u d u displaystyle f mathbf x int _ gamma mathbf a mathbf x mathbf f mathbf u cdot mathrm d mathbf u 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 n by the definition of the directional derivative f x v lim t 0 f x t v f x t lim t 0 γ a x t v f u d u γ a x f u d u t lim t 0 1 t γ x x t v f u d u displaystyle begin aligned 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 aligned to calculate the integral within the final limit we must 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 lim t 0 1 t 0 t f u s u s d s d d t 0 t f x s v v d s t 0 f x v displaystyle 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 where the first equality is from the definition of the derivative with a fact that the integral is equal to 0 at t 0 and the second equality is from the first fundamental theorem of calculus thus we have a formula for v f one of ways to represent the directional derivative where v is arbitrary for f x γ a x f u d u displaystyle f mathbf x int _ gamma mathbf a mathbf x mathbf f mathbf u cdot mathrm d mathbf u see its full definition above its directional derivative with respect to v is f x v v f x d v f x f x v displaystyle 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 where the first two equalities just show different representations of the directional derivative according to the definition of the gradient of a scalar function f f x f x displaystyle nabla f mathbf x mathbf f mathbf x thus we have found a scalar valued function f whose gradient is the path independent vector field f i e f is a conservative vector field as desired 3 example of the converse principle edit main article electric potential energy to illustrate the power of this converse principle we cite an example that has significant physical consequences in classical electromagnetism the electric force is a path independent force i e the work done on a particle that has returned to its original position within an electric field is zero assuming that no changing magnetic fields are present therefore the above theorem implies that the electric force field f e s r 3 is conservative here s is some open path connected subset of r 3 that contains a charge distribution following the ideas of the above proof we can set some reference point a in s and define a function u e s r by u e r γ a r f e u d u displaystyle u_ e mathbf r int _ gamma mathbf a mathbf r mathbf f _ e mathbf u cdot mathrm d mathbf u using the above proof we know u e is well defined and differentiable and f e u e 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 e is often referred to as the 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 unbounded and the reference point a is taken to be infinity which can be...