In the calculus of variations and classical mechanics , the Euler–Lagrange equations are a system of second-order ordinary differential equations whose solutions are stationary points of the given action functional . The equations were discovered in the 1750s by Swiss mathematician Leonhard Euler and Italian mathematician Joseph-Louis Lagrange .
133-539: Because a differentiable functional is stationary at its local extrema , the Euler–Lagrange equation is useful for solving optimization problems in which, given some functional, one seeks the function minimizing or maximizing it. This is analogous to Fermat's theorem in calculus , stating that at any point where a differentiable function attains a local extremum its derivative is zero. In Lagrangian mechanics , according to Hamilton's principle of stationary action,
266-1092: A , x b ) {\displaystyle {\boldsymbol {q}}\in {\cal {P}}(a,b,{\boldsymbol {x}}_{a},{\boldsymbol {x}}_{b})} is a stationary point of S {\displaystyle S} if and only if ∂ L ∂ q i ( t , q ( t ) , q ˙ ( t ) ) − d d t ∂ L ∂ q ˙ i ( t , q ( t ) , q ˙ ( t ) ) = 0 , i = 1 , … , n . {\displaystyle {\frac {\partial L}{\partial q^{i}}}(t,{\boldsymbol {q}}(t),{\dot {\boldsymbol {q}}}(t))-{\frac {\mathrm {d} }{\mathrm {d} t}}{\frac {\partial L}{\partial {\dot {q}}^{i}}}(t,{\boldsymbol {q}}(t),{\dot {\boldsymbol {q}}}(t))=0,\quad i=1,\dots ,n.} Here, q ˙ ( t ) {\displaystyle {\dot {\boldsymbol {q}}}(t)}
399-1631: A b d d ε L ( x , f ( x ) + ε η ( x ) , f ′ ( x ) + ε η ′ ( x ) ) d x = ∫ a b [ η ( x ) ∂ L ∂ f ( x , f ( x ) + ε η ( x ) , f ′ ( x ) + ε η ′ ( x ) ) + η ′ ( x ) ∂ L ∂ f ′ ( x , f ( x ) + ε η ( x ) , f ′ ( x ) + ε η ′ ( x ) ) ] d x . {\displaystyle {\begin{aligned}{\frac {\mathrm {d} \Phi }{\mathrm {d} \varepsilon }}&={\frac {\mathrm {d} }{\mathrm {d} \varepsilon }}\int _{a}^{b}L(x,f(x)+\varepsilon \eta (x),f'(x)+\varepsilon \eta '(x))\,\mathrm {d} x\\&=\int _{a}^{b}{\frac {\mathrm {d} }{\mathrm {d} \varepsilon }}L(x,f(x)+\varepsilon \eta (x),f'(x)+\varepsilon \eta '(x))\,\mathrm {d} x\\&=\int _{a}^{b}\left[\eta (x){\frac {\partial L}{\partial {f}}}(x,f(x)+\varepsilon \eta (x),f'(x)+\varepsilon \eta '(x))+\eta '(x){\frac {\partial L}{\partial f'}}(x,f(x)+\varepsilon \eta (x),f'(x)+\varepsilon \eta '(x))\right]\mathrm {d} x\ .\end{aligned}}} The third line follows from
532-651: A b [ ∂ L ∂ f ( x , f ( x ) , f ′ ( x ) ) − d d x ∂ L ∂ f ′ ( x , f ( x ) , f ′ ( x ) ) ] η ( x ) d x = 0 . {\displaystyle \int _{a}^{b}\left[{\frac {\partial L}{\partial f}}(x,f(x),f'(x))-{\frac {\mathrm {d} }{\mathrm {d} x}}{\frac {\partial L}{\partial f'}}(x,f(x),f'(x))\right]\eta (x)\,\mathrm {d} x=0\,.} Applying
665-725: A b [ η ( x ) ∂ L ∂ f ( x , f ( x ) , f ′ ( x ) ) + η ′ ( x ) ∂ L ∂ f ′ ( x , f ( x ) , f ′ ( x ) ) ] d x = 0 . {\displaystyle \left.{\frac {\mathrm {d} \Phi }{\mathrm {d} \varepsilon }}\right|_{\varepsilon =0}=\int _{a}^{b}\left[\eta (x){\frac {\partial L}{\partial f}}(x,f(x),f'(x))+\eta '(x){\frac {\partial L}{\partial f'}}(x,f(x),f'(x))\,\right]\,\mathrm {d} x=0\ .} The next step
798-517: A b = 0 . {\displaystyle \int _{a}^{b}\left[{\frac {\partial L}{\partial f}}(x,f(x),f'(x))-{\frac {\mathrm {d} }{\mathrm {d} x}}{\frac {\partial L}{\partial f'}}(x,f(x),f'(x))\right]\eta (x)\,\mathrm {d} x+\left[\eta (x){\frac {\partial L}{\partial f'}}(x,f(x),f'(x))\right]_{a}^{b}=0\ .} Using the boundary conditions η ( a ) = η ( b ) = 0 {\displaystyle \eta (a)=\eta (b)=0} , ∫
931-511: A ) = x a {\displaystyle {\boldsymbol {q}}(a)={\boldsymbol {x}}_{a}} and q ( b ) = x b . {\displaystyle {\boldsymbol {q}}(b)={\boldsymbol {x}}_{b}.} The action functional S : P ( a , b , x a , x b ) → R {\displaystyle S:{\cal {P}}(a,b,{\boldsymbol {x}}_{a},{\boldsymbol {x}}_{b})\to \mathbb {R} }
1064-542: A ) = A {\displaystyle y(a)=A} and y ( b ) = B {\displaystyle y(b)=B} , we proceed by approximating the extremal curve by a polygonal line with n {\displaystyle n} segments and passing to the limit as the number of segments grows arbitrarily large. Divide the interval [ a , b ] {\displaystyle [a,b]} into n {\displaystyle n} equal segments with endpoints t 0 =
1197-432: A , t 1 , t 2 , … , t n = b {\displaystyle t_{0}=a,t_{1},t_{2},\ldots ,t_{n}=b} and let Δ t = t k − t k − 1 {\displaystyle \Delta t=t_{k}-t_{k-1}} . Rather than a smooth function y ( t ) {\displaystyle y(t)} we consider
1330-456: A Fields medal , made new impacts in mathematics by using topological quantum field theory and string theory to make predictions and provide frameworks for new rigorous mathematics, which has resulted for example in the conjectural mirror symmetry and the Seiberg–Witten invariants . Riemannian geometry studies Riemannian manifolds , smooth manifolds with a Riemannian metric . This is
1463-487: A directional derivative of a function from multivariable calculus is extended to the notion of a covariant derivative of a tensor . Many concepts of analysis and differential equations have been generalized to the setting of Riemannian manifolds. A distance-preserving diffeomorphism between Riemannian manifolds is called an isometry . This notion can also be defined locally , i.e. for small neighborhoods of points. Any two regular curves are locally isometric. However,
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#17327729040741596-527: A greatest element m , then m is a maximal element of the set, also denoted as max ( S ) {\displaystyle \max(S)} . Furthermore, if S is a subset of an ordered set T and m is the greatest element of S with (respect to order induced by T ), then m is a least upper bound of S in T . Similar results hold for least element , minimal element and greatest lower bound . The maximum and minimum function for sets are used in databases , and can be computed rapidly, since
1729-460: A mathematical discipline that studies the geometry of smooth shapes and smooth spaces, otherwise known as smooth manifolds . It uses the techniques of differential calculus , integral calculus , linear algebra and multilinear algebra . The field has its origins in the study of spherical geometry as far back as antiquity . It also relates to astronomy , the geodesy of the Earth , and later
1862-671: A real dynamical system with n {\displaystyle n} degrees of freedom. Here X {\displaystyle X} is the configuration space and L = L ( t , q ( t ) , v ( t ) ) {\displaystyle L=L(t,{\boldsymbol {q}}(t),{\boldsymbol {v}}(t))} the Lagrangian , i.e. a smooth real-valued function such that q ( t ) ∈ X , {\displaystyle {\boldsymbol {q}}(t)\in X,} and v ( t ) {\displaystyle {\boldsymbol {v}}(t)}
1995-447: A vector bundle endomorphism (called an almost complex structure ) It follows from this definition that an almost complex manifold is even-dimensional. An almost complex manifold is called complex if N J = 0 {\displaystyle N_{J}=0} , where N J {\displaystyle N_{J}} is a tensor of type (2, 1) related to J {\displaystyle J} , called
2128-401: A bounded differentiable function f defined on a closed interval in the real line has a single critical point, which is a local minimum, then it is also a global minimum (use the intermediate value theorem and Rolle's theorem to prove this by contradiction ). In two and more dimensions, this argument fails. This is illustrated by the function whose only critical point is at (0,0), which is
2261-601: A concept of distance expressed by means of a smooth positive definite symmetric bilinear form defined on the tangent space at each point. Riemannian geometry generalizes Euclidean geometry to spaces that are not necessarily flat, though they still resemble Euclidean space at each point infinitesimally, i.e. in the first order of approximation . Various concepts based on length, such as the arc length of curves, area of plane regions, and volume of solids all possess natural analogues in Riemannian geometry. The notion of
2394-513: A curve on which a weighted particle will fall to a fixed point in a fixed amount of time, independent of the starting point. Lagrange solved this problem in 1755 and sent the solution to Euler. Both further developed Lagrange's method and applied it to mechanics , which led to the formulation of Lagrangian mechanics . Their correspondence ultimately led to the calculus of variations , a term coined by Euler himself in 1766. Let ( X , L ) {\displaystyle (X,L)} be
2527-426: A domain must occur at critical points (or points where the derivative equals zero). However, not all critical points are extrema. One can often distinguish whether a critical point is a local maximum, a local minimum, or neither by using the first derivative test , second derivative test , or higher-order derivative test , given sufficient differentiability. For any function that is defined piecewise , one finds
2660-401: A function. Pierre de Fermat was one of the first mathematicians to propose a general technique, adequality , for finding the maxima and minima of functions. As defined in set theory , the maximum and minimum of a set are the greatest and least elements in the set, respectively. Unbounded infinite sets , such as the set of real numbers , have no minimum or maximum. In statistics ,
2793-398: A functional J = ∫ a b L ( t , y ( t ) , y ′ ( t ) ) d t {\displaystyle J=\int _{a}^{b}L(t,y(t),y'(t))\,\mathrm {d} t} on C 1 ( [ a , b ] ) {\displaystyle C^{1}([a,b])} with the boundary conditions y (
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#17327729040742926-418: A local minimum with f (0,0) = 0. However, it cannot be a global one, because f (2,3) = −5. If the domain of a function for which an extremum is to be found consists itself of functions (i.e. if an extremum is to be found of a functional ), then the extremum is found using the calculus of variations . Maxima and minima can also be defined for sets. In general, if an ordered set S has
3059-428: A manifold, as even the notion of a topological space had not been encountered, but he did propose that it might be possible to investigate or measure the properties of the metric of spacetime through the analysis of masses within spacetime, linking with the earlier observation of Euler that masses under the effect of no forces would travel along geodesics on surfaces, and predicting Einstein's fundamental observation of
3192-443: A maximum (or minimum) by finding the maximum (or minimum) of each piece separately, and then seeing which one is greatest (or least). For a practical example, assume a situation where someone has 200 {\displaystyle 200} feet of fencing and is trying to maximize the square footage of a rectangular enclosure, where x {\displaystyle x} is the length, y {\displaystyle y}
3325-401: A maximum or a minimum. For example, the set of natural numbers has no maximum, though it has a minimum. If an infinite chain S is bounded, then the closure Cl ( S ) of the set occasionally has a minimum and a maximum, in which case they are called the greatest lower bound and the least upper bound of the set S , respectively. Differential geometry Differential geometry is
3458-428: A minimum point. An important example is a function whose domain is a closed and bounded interval of real numbers (see the graph above). Finding global maxima and minima is the goal of mathematical optimization . If a function is continuous on a closed interval, then by the extreme value theorem , global maxima and minima exist. Furthermore, a global maximum (or minimum) either must be a local maximum (or minimum) in
3591-504: A new interpretation of Euler's theorem in terms of the principle curvatures, which is the modern form of the equation. The field of differential geometry became an area of study considered in its own right, distinct from the more broad idea of analytic geometry, in the 1800s, primarily through the foundational work of Carl Friedrich Gauss and Bernhard Riemann , and also in the important contributions of Nikolai Lobachevsky on hyperbolic geometry and non-Euclidean geometry and throughout
3724-461: A nondegenerate 2- form ω , called the symplectic form . A symplectic manifold is an almost symplectic manifold for which the symplectic form ω is closed: d ω = 0 . A diffeomorphism between two symplectic manifolds which preserves the symplectic form is called a symplectomorphism . Non-degenerate skew-symmetric bilinear forms can only exist on even-dimensional vector spaces, so symplectic manifolds necessarily have even dimension. In dimension 2,
3857-658: A real function of n − 1 {\displaystyle n-1} variables given by J ( y 1 , … , y n − 1 ) ≈ ∑ k = 0 n − 1 L ( t k , y k , y k + 1 − y k Δ t ) Δ t . {\displaystyle J(y_{1},\ldots ,y_{n-1})\approx \sum _{k=0}^{n-1}L\left(t_{k},y_{k},{\frac {y_{k+1}-y_{k}}{\Delta t}}\right)\Delta t.} Extremals of this new functional defined on
3990-432: A set can have at most one minimal element and at most one maximal element. Then, due to mutual comparability, the minimal element will also be the least element, and the maximal element will also be the greatest element. Thus in a totally ordered set, we can simply use the terms minimum and maximum . If a chain is finite, then it will always have a maximum and a minimum. If a chain is infinite, then it need not have
4123-523: A symplectic manifold is just a surface endowed with an area form and a symplectomorphism is an area-preserving diffeomorphism. The phase space of a mechanical system is a symplectic manifold and they made an implicit appearance already in the work of Joseph Louis Lagrange on analytical mechanics and later in Carl Gustav Jacobi 's and William Rowan Hamilton 's formulations of classical mechanics . By contrast with Riemannian geometry, where
Euler–Lagrange equation - Misplaced Pages Continue
4256-521: A theorem expressing the curvature of a space curve on a surface in terms of the principal curvatures, known as Euler's theorem . Later in the 1700s, the new French school led by Gaspard Monge began to make contributions to differential geometry. Monge made important contributions to the theory of plane curves, surfaces, and studied surfaces of revolution and envelopes of plane curves and space curves. Several students of Monge made contributions to this same theory, and for example Charles Dupin provided
4389-476: A well-known standard definition of metric and parallelism. In Riemannian geometry , the Levi-Civita connection serves a similar purpose. More generally, differential geometers consider spaces with a vector bundle and an arbitrary affine connection which is not defined in terms of a metric. In physics, the manifold may be spacetime and the bundles and connections are related to various physical fields. From
4522-423: A year later Tullio Levi-Civita and Gregorio Ricci-Curbastro produced their textbook systematically developing the theory of absolute differential calculus and tensor calculus . It was in this language that differential geometry was used by Einstein in the development of general relativity and pseudo-Riemannian geometry . The subject of modern differential geometry emerged from the early 1900s in response to
4655-416: Is a strict local maximum point if there exists some ε > 0 such that, for all x in X within distance ε of x with x ≠ x , we have f ( x ) > f ( x ) . Note that a point is a strict global maximum point if and only if it is the unique global maximum point, and similarly for minimum points. A continuous real-valued function with a compact domain always has a maximum point and
4788-465: Is a topological space , since the definition just given can be rephrased in terms of neighbourhoods. Mathematically, the given definition is written as follows: The definition of local minimum point can also proceed similarly. In both the global and local cases, the concept of a strict extremum can be defined. For example, x is a strict global maximum point if for all x in X with x ≠ x , we have f ( x ) > f ( x ) , and x
4921-737: Is a differentiable function satisfying η ( a ) = η ( b ) = 0 {\displaystyle \eta (a)=\eta (b)=0} . Then define Φ ( ε ) = J [ f + ε η ] = ∫ a b L ( x , f ( x ) + ε η ( x ) , f ′ ( x ) + ε η ′ ( x ) ) d x . {\displaystyle \Phi (\varepsilon )=J[f+\varepsilon \eta ]=\int _{a}^{b}L(x,f(x)+\varepsilon \eta (x),f'(x)+\varepsilon \eta '(x))\,\mathrm {d} x\ .} We now wish to calculate
5054-401: Is a function F : T M → [ 0 , ∞ ) {\displaystyle F:\mathrm {T} M\to [0,\infty )} such that: Symplectic geometry is the study of symplectic manifolds . An almost symplectic manifold is a differentiable manifold equipped with a smoothly varying non-degenerate skew-symmetric bilinear form on each tangent space, i.e.,
5187-421: Is a maximizer). Let f + ε η {\displaystyle f+\varepsilon \eta } be the result of such a perturbation ε η {\displaystyle \varepsilon \eta } of f {\displaystyle f} , where ε {\displaystyle \varepsilon } is small and η {\displaystyle \eta }
5320-479: Is a price to pay in technical complexity: the intrinsic definitions of curvature and connections become much less visually intuitive. These two points of view can be reconciled, i.e. the extrinsic geometry can be considered as a structure additional to the intrinsic one. (See the Nash embedding theorem .) In the formalism of geometric calculus both extrinsic and intrinsic geometry of a manifold can be characterized by
5453-434: Is an n {\displaystyle n} -dimensional "vector of speed". (For those familiar with differential geometry , X {\displaystyle X} is a smooth manifold , and L : R t × T X → R , {\displaystyle L:{\mathbb {R} }_{t}\times TX\to {\mathbb {R} },} where T X {\displaystyle TX}
Euler–Lagrange equation - Misplaced Pages Continue
5586-411: Is defined via S [ q ] = ∫ a b L ( t , q ( t ) , q ˙ ( t ) ) d t . {\displaystyle S[{\boldsymbol {q}}]=\int _{a}^{b}L(t,{\boldsymbol {q}}(t),{\dot {\boldsymbol {q}}}(t))\,dt.} A path q ∈ P ( a , b , x
5719-483: Is finding the real-valued function y ( x ) on the interval [ a , b ], such that y ( a ) = c and y ( b ) = d , for which the path length along the curve traced by y is as short as possible. the integrand function being L ( x , y , y ′ ) = 1 + y ′ 2 {\textstyle L(x,y,y')={\sqrt {1+y'^{2}}}} . The partial derivatives of L are: By substituting these into
5852-412: Is given by all the smooth complex projective varieties . CR geometry is the study of the intrinsic geometry of boundaries of domains in complex manifolds . Conformal geometry is the study of the set of angle-preserving (conformal) transformations on a space. Differential topology is the study of global geometric invariants without a metric or symplectic form. Differential topology starts from
5985-473: Is intimately linked to the development of geometry more generally, of the notion of space and shape, and of topology , especially the study of manifolds . In this section we focus primarily on the history of the application of infinitesimal methods to geometry, and later to the ideas of tangent spaces , and eventually the development of the modern formalism of the subject in terms of tensors and tensor fields . The study of differential geometry, or at least
6118-400: Is one of the classic proofs in mathematics . It relies on the fundamental lemma of calculus of variations . We wish to find a function f {\displaystyle f} which satisfies the boundary conditions f ( a ) = A {\displaystyle f(a)=A} , f ( b ) = B {\displaystyle f(b)=B} , and which extremizes
6251-663: Is one which defines some notion of size, distance, shape, volume, or other rigidifying structure. For example, in Riemannian geometry distances and angles are specified, in symplectic geometry volumes may be computed, in conformal geometry only angles are specified, and in gauge theory certain fields are given over the space. Differential geometry is closely related to, and is sometimes taken to include, differential topology , which concerns itself with properties of differentiable manifolds that do not rely on any additional geometric structure (see that article for more discussion on
6384-428: Is only over μ 1 ≤ μ 2 ≤ … ≤ μ j {\displaystyle \mu _{1}\leq \mu _{2}\leq \ldots \leq \mu _{j}} in order to avoid counting the same partial derivative multiple times, for example f 12 = f 21 {\displaystyle f_{12}=f_{21}} appears only once in
6517-604: Is restricted. Since width is positive, then x > 0 {\displaystyle x>0} , and since x = 100 − y {\displaystyle x=100-y} , that implies that x < 100 {\displaystyle x<100} . Plug in critical point 50 {\displaystyle 50} , as well as endpoints 0 {\displaystyle 0} and 100 {\displaystyle 100} , into x y = x ( 100 − x ) {\displaystyle xy=x(100-x)} , and
6650-423: Is sufficient only for developing analysis on the manifold, while doing geometry requires, in addition, some way to relate the tangent spaces at different points, i.e. a notion of parallel transport . An important example is provided by affine connections . For a surface in R , tangent planes at different points can be identified using a natural path-wise parallelism induced by the ambient Euclidean space, which has
6783-417: Is the Levi-Civita connection of g {\displaystyle g} . In this case, ( J , g ) {\displaystyle (J,g)} is called a Kähler structure , and a Kähler manifold is a manifold endowed with a Kähler structure. In particular, a Kähler manifold is both a complex and a symplectic manifold . A large class of Kähler manifolds (the class of Hodge manifolds )
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#17327729040746916-457: Is the Riemannian symmetric spaces , whose curvature is not necessarily constant. These are the closest analogues to the "ordinary" plane and space considered in Euclidean and non-Euclidean geometry . Pseudo-Riemannian geometry generalizes Riemannian geometry to the case in which the metric tensor need not be positive-definite . A special case of this is a Lorentzian manifold , which is
7049-429: Is the energy functional , this leads to the soap-film minimal surface problem. If there are several unknown functions to be determined and several variables such that the system of Euler–Lagrange equations is If there is a single unknown function f to be determined that is dependent on two variables x 1 and x 2 and if the functional depends on higher derivatives of f up to n -th order such that then
7182-465: Is the tangent bundle of X ) . {\displaystyle X).} Let P ( a , b , x a , x b ) {\displaystyle {\cal {P}}(a,b,{\boldsymbol {x}}_{a},{\boldsymbol {x}}_{b})} be the set of smooth paths q : [ a , b ] → X {\displaystyle {\boldsymbol {q}}:[a,b]\to X} for which q (
7315-467: Is the study of connections on vector bundles and principal bundles, and arises out of problems in mathematical physics and physical gauge theories which underpin the standard model of particle physics . Gauge theory is concerned with the study of differential equations for connections on bundles, and the resulting geometric moduli spaces of solutions to these equations as well as the invariants that may be derived from them. These equations often arise as
7448-480: Is the tangent space at the unit endowed with the Lie bracket between left-invariant vector fields . Beside the structure theory there is also the wide field of representation theory . Geometric analysis is a mathematical discipline where tools from differential equations, especially elliptic partial differential equations are used to establish new results in differential geometry and differential topology. Gauge theory
7581-437: Is the time derivative of q ( t ) . {\displaystyle {\boldsymbol {q}}(t).} When we say stationary point, we mean a stationary point of S {\displaystyle S} with respect to any small perturbation in q {\displaystyle {\boldsymbol {q}}} . See proofs below for more rigorous detail. The derivation of the one-dimensional Euler–Lagrange equation
7714-444: Is the width, and x y {\displaystyle xy} is the area: The derivative with respect to x {\displaystyle x} is: Setting this equal to 0 {\displaystyle 0} reveals that x = 50 {\displaystyle x=50} is our only critical point . Now retrieve the endpoints by determining the interval to which x {\displaystyle x}
7847-524: Is they go from 1 to m. Then the Euler–Lagrange equation is where the summation over the μ 1 … μ j {\displaystyle \mu _{1}\dots \mu _{j}} is avoiding counting the same derivative f i , μ 1 μ 2 = f i , μ 2 μ 1 {\displaystyle f_{i,\mu _{1}\mu _{2}}=f_{i,\mu _{2}\mu _{1}}} several times, just as in
7980-712: Is to use integration by parts on the second term of the integrand, yielding ∫ a b [ ∂ L ∂ f ( x , f ( x ) , f ′ ( x ) ) − d d x ∂ L ∂ f ′ ( x , f ( x ) , f ′ ( x ) ) ] η ( x ) d x + [ η ( x ) ∂ L ∂ f ′ ( x , f ( x ) , f ′ ( x ) ) ]
8113-464: The maximum value of the function, denoted max ( f ( x ) ) {\displaystyle \max(f(x))} , and the value of the function at a minimum point is called the minimum value of the function, (denoted min ( f ( x ) ) {\displaystyle \min(f(x))} for clarity). Symbolically, this can be written as follows: The definition of global minimum point also proceeds similarly. If
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#17327729040748246-526: The Bernoulli brothers , Jacob and Johann made important early contributions to the use of infinitesimals to study geometry. In lectures by Johann Bernoulli at the time, later collated by L'Hopital into the first textbook on differential calculus , the tangents to plane curves of various types are computed using the condition d y = 0 {\displaystyle dy=0} , and similarly points of inflection are calculated. At this same time
8379-472: The Euler–Lagrange equations and the first theory of the calculus of variations , which underpins in modern differential geometry many techniques in symplectic geometry and geometric analysis . This theory was used by Lagrange , a co-developer of the calculus of variations, to derive the first differential equation describing a minimal surface in terms of the Euler–Lagrange equation. In 1760 Euler proved
8512-462: The Euler–Lagrange equations describing the equations of motion of certain physical systems in quantum field theory , and so their study is of considerable interest in physics. The apparatus of vector bundles , principal bundles , and connections on bundles plays an extraordinarily important role in modern differential geometry. A smooth manifold always carries a natural vector bundle, the tangent bundle . Loosely speaking, this structure by itself
8645-554: The Mercator projection as a way of mapping the Earth. Mercator had an understanding of the advantages and pitfalls of his map design, and in particular was aware of the conformal nature of his projection, as well as the difference between praga , the lines of shortest distance on the Earth, and the directio , the straight line paths on his map. Mercator noted that the praga were oblique curvatur in this projection. This fact reflects
8778-546: The Nijenhuis tensor (or sometimes the torsion ). An almost complex manifold is complex if and only if it admits a holomorphic coordinate atlas . An almost Hermitian structure is given by an almost complex structure J , along with a Riemannian metric g , satisfying the compatibility condition An almost Hermitian structure defines naturally a differential two-form The following two conditions are equivalent: where ∇ {\displaystyle \nabla }
8911-506: The Poincaré conjecture . During this same period primarily due to the influence of Michael Atiyah , new links between theoretical physics and differential geometry were formed. Techniques from the study of the Yang–Mills equations and gauge theory were used by mathematicians to develop new invariants of smooth manifolds. Physicists such as Edward Witten , the only physicist to be awarded
9044-583: The Theorema Egregium of Carl Friedrich Gauss showed that for surfaces, the existence of a local isometry imposes that the Gaussian curvatures at the corresponding points must be the same. In higher dimensions, the Riemann curvature tensor is an important pointwise invariant associated with a Riemannian manifold that measures how close it is to being flat. An important class of Riemannian manifolds
9177-621: The curvature provides a local invariant of Riemannian manifolds, Darboux's theorem states that all symplectic manifolds are locally isomorphic. The only invariants of a symplectic manifold are global in nature and topological aspects play a prominent role in symplectic geometry. The first result in symplectic topology is probably the Poincaré–Birkhoff theorem , conjectured by Henri Poincaré and then proved by G.D. Birkhoff in 1912. It claims that if an area preserving map of an annulus twists each boundary component in opposite directions, then
9310-447: The equivalence principle a full 60 years before it appeared in the scientific literature. In the wake of Riemann's new description, the focus of techniques used to study differential geometry shifted from the ad hoc and extrinsic methods of the study of curves and surfaces to a more systematic approach in terms of tensor calculus and Klein's Erlangen program, and progress increased in the field. The notion of groups of transformations
9443-406: The functional derivative δ J / δ y {\displaystyle \delta J/\delta y} of the functional J {\displaystyle J} . A necessary condition for a differentiable functional to have an extremum on some function is that its functional derivative at that function vanishes, which is granted by the last equation. A standard example
9576-618: The fundamental lemma of calculus of variations now yields the Euler–Lagrange equation ∂ L ∂ f ( x , f ( x ) , f ′ ( x ) ) − d d x ∂ L ∂ f ′ ( x , f ( x ) , f ′ ( x ) ) = 0 . {\displaystyle {\frac {\partial L}{\partial f}}(x,f(x),f'(x))-{\frac {\mathrm {d} }{\mathrm {d} x}}{\frac {\partial L}{\partial f'}}(x,f(x),f'(x))=0\,.} Given
9709-401: The orthogonality between the osculating circles of a plane curve and the tangent directions is realised, and the first analytical formula for the radius of an osculating circle, essentially the first analytical formula for the notion of curvature , is written down. In the wake of the development of analytic geometry and plane curves, Alexis Clairaut began the study of space curves at just
9842-513: The total derivative of Φ {\displaystyle \Phi } with respect to ε . d Φ d ε = d d ε ∫ a b L ( x , f ( x ) + ε η ( x ) , f ′ ( x ) + ε η ′ ( x ) ) d x = ∫
9975-412: The (enlargeable) figure on the right, the necessary conditions for a local maximum are similar to those of a function with only one variable. The first partial derivatives as to z (the variable to be maximized) are zero at the maximum (the glowing dot on top in the figure). The second partial derivatives are negative. These are only necessary, not sufficient, conditions for a local maximum, because of
10108-399: The 1600s. Around this time there were only minimal overt applications of the theory of infinitesimals to the study of geometry, a precursor to the modern calculus-based study of the subject. In Euclid 's Elements the notion of tangency of a line to a circle is discussed, and Archimedes applied the method of exhaustion to compute the areas of smooth shapes such as the circle , and
10241-536: The 1860s, and Felix Klein coined the term non-Euclidean geometry in 1871, and through the Erlangen program put Euclidean and non-Euclidean geometries on the same footing. Implicitly, the spherical geometry of the Earth that had been studied since antiquity was a non-Euclidean geometry, an elliptic geometry . The development of intrinsic differential geometry in the language of Gauss was spurred on by his student, Bernhard Riemann in his Habilitationsschrift , On
10374-468: The Darboux theorem holds: all contact structures on an odd-dimensional manifold are locally isomorphic and can be brought to a certain local normal form by a suitable choice of the coordinate system. Complex differential geometry is the study of complex manifolds . An almost complex manifold is a real manifold M {\displaystyle M} , endowed with a tensor of type (1, 1), i.e.
10507-447: The Euler–Lagrange equation is which can be represented shortly as: wherein μ 1 … μ j {\displaystyle \mu _{1}\dots \mu _{j}} are indices that span the number of variables, that is, here they go from 1 to 2. Here summation over the μ 1 … μ j {\displaystyle \mu _{1}\dots \mu _{j}} indices
10640-618: The Euler–Lagrange equation, we obtain that is, the function must have a constant first derivative, and thus its graph is a straight line . The stationary values of the functional can be obtained from the Euler–Lagrange equation under fixed boundary conditions for the function itself as well as for the first k − 1 {\displaystyle k-1} derivatives (i.e. for all f ( i ) , i ∈ { 0 , . . . , k − 1 } {\displaystyle f^{(i)},i\in \{0,...,k-1\}} ). The endpoint values of
10773-1193: The above equation by Δ t {\displaystyle \Delta t} gives ∂ J ∂ y m Δ t = L y ( t m , y m , y m + 1 − y m Δ t ) − 1 Δ t [ L y ′ ( t m , y m , y m + 1 − y m Δ t ) − L y ′ ( t m − 1 , y m − 1 , y m − y m − 1 Δ t ) ] , {\displaystyle {\frac {\partial J}{\partial y_{m}\Delta t}}=L_{y}\left(t_{m},y_{m},{\frac {y_{m+1}-y_{m}}{\Delta t}}\right)-{\frac {1}{\Delta t}}\left[L_{y'}\left(t_{m},y_{m},{\frac {y_{m+1}-y_{m}}{\Delta t}}\right)-L_{y'}\left(t_{m-1},y_{m-1},{\frac {y_{m}-y_{m-1}}{\Delta t}}\right)\right],} and taking
10906-407: The advantage that it takes the same form in any system of generalized coordinates , and it is better suited to generalizations. In classical field theory there is an analogous equation to calculate the dynamics of a field . The Euler–Lagrange equation was developed in the 1750s by Euler and Lagrange in connection with their studies of the tautochrone problem. This is the problem of determining
11039-448: The age of 16. In his book Clairaut introduced the notion of tangent and subtangent directions to space curves in relation to the directions which lie along a surface on which the space curve lies. Thus Clairaut demonstrated an implicit understanding of the tangent space of a surface and studied this idea using calculus for the first time. Importantly Clairaut introduced the terminology of curvature and double curvature , essentially
11172-403: The beginning and through the middle of the 19th century, differential geometry was studied from the extrinsic point of view: curves and surfaces were considered as lying in a Euclidean space of higher dimension (for example a surface in an ambient space of three dimensions). The simplest results are those in the differential geometry of curves and differential geometry of surfaces. Starting with
11305-402: The corresponding concept is the sample maximum and minimum . A real-valued function f defined on a domain X has a global (or absolute ) maximum point at x , if f ( x ) ≥ f ( x ) for all x in X . Similarly, the function has a global (or absolute ) minimum point at x , if f ( x ) ≤ f ( x ) for all x in X . The value of the function at a maximum point is called
11438-1382: The derivative of the 3rd argument. L ( 3rd argument ) ( y m + 1 − ( y m + Δ y m ) Δ t ) = L ( y m + 1 − y m Δ t ) − ∂ L ∂ y ′ Δ y m Δ t {\displaystyle L({\text{3rd argument}})\left({\frac {y_{m+1}-(y_{m}+\Delta y_{m})}{\Delta t}}\right)=L\left({\frac {y_{m+1}-y_{m}}{\Delta t}}\right)-{\frac {\partial L}{\partial y'}}{\frac {\Delta y_{m}}{\Delta t}}} L ( ( y m + Δ y m ) − y m − 1 Δ t ) = L ( y m − y m − 1 Δ t ) + ∂ L ∂ y ′ Δ y m Δ t {\displaystyle L\left({\frac {(y_{m}+\Delta y_{m})-y_{m-1}}{\Delta t}}\right)=L\left({\frac {y_{m}-y_{m-1}}{\Delta t}}\right)+{\frac {\partial L}{\partial y'}}{\frac {\Delta y_{m}}{\Delta t}}} Evaluating
11571-414: The development of quantum field theory and the standard model of particle physics . Outside of physics, differential geometry finds applications in chemistry , economics , engineering , control theory , computer graphics and computer vision , and recently in machine learning . The history and development of differential geometry as a subject begins at least as far back as classical antiquity . It
11704-476: The development of the modern theory, including Jean-Louis Koszul who introduced connections on vector bundles , Shiing-Shen Chern who introduced characteristic classes to the subject and began the study of complex manifolds , Sir William Vallance Douglas Hodge and Georges de Rham who expanded understanding of differential forms , Charles Ehresmann who introduced the theory of fibre bundles and Ehresmann connections , and others. Of particular importance
11837-537: The discrete points t 0 , … , t n {\displaystyle t_{0},\ldots ,t_{n}} correspond to points where ∂ J ( y 1 , … , y n ) ∂ y m = 0. {\displaystyle {\frac {\partial J(y_{1},\ldots ,y_{n})}{\partial y_{m}}}=0.} Note that change of y m {\displaystyle y_{m}} affects L not only at m but also at m-1 for
11970-443: The distinction between the two subjects). Differential geometry is also related to the geometric aspects of the theory of differential equations , otherwise known as geometric analysis . Differential geometry finds applications throughout mathematics and the natural sciences . Most prominently the language of differential geometry was used by Albert Einstein in his theory of general relativity , and subsequently by physicists in
12103-408: The domain X is a metric space , then f is said to have a local (or relative ) maximum point at the point x , if there exists some ε > 0 such that f ( x ) ≥ f ( x ) for all x in X within distance ε of x . Similarly, the function has a local minimum point at x , if f ( x ) ≤ f ( x ) for all x in X within distance ε of x . A similar definition can be used when X
12236-576: The evolution of a physical system is described by the solutions to the Euler equation for the action of the system. In this context Euler equations are usually called Lagrange equations . In classical mechanics , it is equivalent to Newton's laws of motion ; indeed, the Euler-Lagrange equations will produce the same equations as Newton's Laws. This is particularly useful when analyzing systems whose force vectors are particularly complicated. It has
12369-609: The fact that x {\displaystyle x} does not depend on ε {\displaystyle \varepsilon } , i.e. d x d ε = 0 {\displaystyle {\frac {\mathrm {d} x}{\mathrm {d} \varepsilon }}=0} . When ε = 0 {\displaystyle \varepsilon =0} , Φ {\displaystyle \Phi } has an extremum value, so that d Φ d ε | ε = 0 = ∫
12502-501: The first set of intrinsic coordinate systems on a surface, beginning the theory of intrinsic geometry upon which modern geometric ideas are based. Around this time Euler's study of mechanics in the Mechanica lead to the realization that a mass traveling along a surface not under the effect of any force would traverse a geodesic path, an early precursor to the important foundational ideas of Einstein's general relativity , and also to
12635-406: The foundational contributions of many mathematicians, including importantly the work of Henri Poincaré on the foundations of topology . At the start of the 1900s there was a major movement within mathematics to formalise the foundational aspects of the subject to avoid crises of rigour and accuracy, known as Hilbert's program . As part of this broader movement, the notion of a topological space
12768-419: The functional J [ f ] = ∫ a b L ( x , f ( x ) , f ′ ( x ) ) d x . {\displaystyle J[f]=\int _{a}^{b}L(x,f(x),f'(x))\,\mathrm {d} x\ .} We assume that L {\displaystyle L} is twice continuously differentiable. A weaker assumption can be used, but
12901-403: The functional then the corresponding Euler–Lagrange equations are A multi-dimensional generalization comes from considering a function on n variables. If Ω {\displaystyle \Omega } is some surface, then is extremized only if f satisfies the partial differential equation When n = 2 and functional I {\displaystyle {\mathcal {I}}}
13034-422: The highest derivative f ( k ) {\displaystyle f^{(k)}} remain flexible. If the problem involves finding several functions ( f 1 , f 2 , … , f m {\displaystyle f_{1},f_{2},\dots ,f_{m}} ) of a single independent variable ( x {\displaystyle x} ) that define an extremum of
13167-417: The hypotheses which lie at the foundation of geometry . In this work Riemann introduced the notion of a Riemannian metric and the Riemannian curvature tensor for the first time, and began the systematic study of differential geometry in higher dimensions. This intrinsic point of view in terms of the Riemannian metric, denoted by d s 2 {\displaystyle ds^{2}} by Riemann,
13300-400: The interior of the domain, or must lie on the boundary of the domain. So a method of finding a global maximum (or minimum) is to look at all the local maxima (or minima) in the interior, and also look at the maxima (or minima) of the points on the boundary, and take the greatest (or least) one.Minima For differentiable functions , Fermat's theorem states that local extrema in the interior of
13433-516: The inventor of intrinsic differential geometry. In his fundamental paper Gauss introduced the Gauss map , Gaussian curvature , first and second fundamental forms , proved the Theorema Egregium showing the intrinsic nature of the Gaussian curvature, and studied geodesics, computing the area of a geodesic triangle in various non-Euclidean geometries on surfaces. At this time Gauss was already of
13566-610: The lack of a metric-preserving map of the Earth's surface onto a flat plane, a consequence of the later Theorema Egregium of Gauss . The first systematic or rigorous treatment of geometry using the theory of infinitesimals and notions from calculus began around the 1600s when calculus was first developed by Gottfried Leibniz and Isaac Newton . At this time, the recent work of René Descartes introducing analytic coordinates to geometry allowed geometric shapes of increasing complexity to be described rigorously. In particular around this time Pierre de Fermat , Newton, and Leibniz began
13699-405: The level sets of a differentiable function on M (the technical term is "completely nonintegrable tangent hyperplane distribution"). Near each point p , a hyperplane distribution is determined by a nowhere vanishing 1-form α {\displaystyle \alpha } , which is unique up to multiplication by a nowhere vanishing function: A local 1-form on M is a contact form if
13832-401: The limit as Δ t → 0 {\displaystyle \Delta t\to 0} of the right-hand side of this expression yields L y − d d t L y ′ = 0. {\displaystyle L_{y}-{\frac {\mathrm {d} }{\mathrm {d} t}}L_{y'}=0.} The left hand side of the previous equation is
13965-399: The map has at least two fixed points. Contact geometry deals with certain manifolds of odd dimension. It is close to symplectic geometry and like the latter, it originated in questions of classical mechanics. A contact structure on a (2 n + 1) -dimensional manifold M is given by a smooth hyperplane field H in the tangent bundle that is as far as possible from being associated with
14098-411: The mathematical basis of Einstein's general relativity theory of gravity . Finsler geometry has Finsler manifolds as the main object of study. This is a differential manifold with a Finsler metric , that is, a Banach norm defined on each tangent space. Riemannian manifolds are special cases of the more general Finsler manifolds. A Finsler structure on a manifold M {\displaystyle M}
14231-410: The maximum (or minimum) of a set can be computed from the maxima of a partition; formally, they are self- decomposable aggregation functions . In the case of a general partial order , the least element (i.e., one that is less than all others) should not be confused with a minimal element (nothing is lesser). Likewise, a greatest element of a partially ordered set (poset) is an upper bound of
14364-413: The natural operations such as Lie derivative of natural vector bundles and de Rham differential of forms . Beside Lie algebroids , also Courant algebroids start playing a more important role. A Lie group is a group in the category of smooth manifolds. Beside the algebraic properties this enjoys also differential geometric properties. The most obvious construction is that of a Lie algebra which
14497-633: The notation g {\displaystyle g} for a Riemannian metric, and Γ {\displaystyle \Gamma } for the Christoffel symbols, both coming from G in Gravitation . Élie Cartan helped reformulate the foundations of the differential geometry of smooth manifolds in terms of exterior calculus and the theory of moving frames , leading in the world of physics to Einstein–Cartan theory . Following this early development, many mathematicians contributed to
14630-434: The notion of principal curvatures later studied by Gauss and others. Around this same time, Leonhard Euler , originally a student of Johann Bernoulli, provided many significant contributions not just to the development of geometry, but to mathematics more broadly. In regards to differential geometry, Euler studied the notion of a geodesic on a surface deriving the first analytical geodesic equation , and later introduced
14763-476: The opinion that the standard paradigm of Euclidean geometry should be discarded, and was in possession of private manuscripts on non-Euclidean geometry which informed his study of geodesic triangles. Around this same time János Bolyai and Lobachevsky independently discovered hyperbolic geometry and thus demonstrated the existence of consistent geometries outside Euclid's paradigm. Concrete models of hyperbolic geometry were produced by Eugenio Beltrami later in
14896-1033: The partial derivative gives ∂ J ∂ y m = L y ( t m , y m , y m + 1 − y m Δ t ) Δ t + L y ′ ( t m − 1 , y m − 1 , y m − y m − 1 Δ t ) − L y ′ ( t m , y m , y m + 1 − y m Δ t ) . {\displaystyle {\frac {\partial J}{\partial y_{m}}}=L_{y}\left(t_{m},y_{m},{\frac {y_{m+1}-y_{m}}{\Delta t}}\right)\Delta t+L_{y'}\left(t_{m-1},y_{m-1},{\frac {y_{m}-y_{m-1}}{\Delta t}}\right)-L_{y'}\left(t_{m},y_{m},{\frac {y_{m+1}-y_{m}}{\Delta t}}\right).} Dividing
15029-430: The polygonal line with vertices ( t 0 , y 0 ) , … , ( t n , y n ) {\displaystyle (t_{0},y_{0}),\ldots ,(t_{n},y_{n})} , where y 0 = A {\displaystyle y_{0}=A} and y n = B {\displaystyle y_{n}=B} . Accordingly, our functional becomes
15162-433: The possibility of a saddle point . For use of these conditions to solve for a maximum, the function z must also be differentiable throughout. The second partial derivative test can help classify the point as a relative maximum or relative minimum. In contrast, there are substantial differences between functions of one variable and functions of more than one variable in the identification of global extrema. For example, if
15295-427: The previous equation. If there are p unknown functions f i to be determined that are dependent on m variables x 1 ... x m and if the functional depends on higher derivatives of the f i up to n -th order such that where μ 1 … μ j {\displaystyle \mu _{1}\dots \mu _{j}} are indices that span the number of variables, that
15428-424: The previous subsection. This can be expressed more compactly as Maxima and minima In mathematical analysis , the maximum and minimum of a function are, respectively, the greatest and least value taken by the function. Known generically as extremum , they may be defined either within a given range (the local or relative extrema) or on the entire domain (the global or absolute extrema) of
15561-492: The proof becomes more difficult. If f {\displaystyle f} extremizes the functional subject to the boundary conditions, then any slight perturbation of f {\displaystyle f} that preserves the boundary values must either increase J {\displaystyle J} (if f {\displaystyle f} is a minimizer) or decrease J {\displaystyle J} (if f {\displaystyle f}
15694-614: The proof of the Atiyah–Singer index theorem . The development of complex geometry was spurred on by parallel results in algebraic geometry , and results in the geometry and global analysis of complex manifolds were proven by Shing-Tung Yau and others. In the latter half of the 20th century new analytic techniques were developed in regards to curvature flows such as the Ricci flow , which culminated in Grigori Perelman 's proof of
15827-424: The purposes of mapping the shape of the Earth. Implicitly throughout this time principles that form the foundation of differential geometry and calculus were used in geodesy , although in a much simplified form. Namely, as far back as Euclid 's Elements it was understood that a straight line could be defined by its property of providing the shortest distance between two points, and applying this same principle to
15960-417: The restriction of its exterior derivative to H is a non-degenerate two-form and thus induces a symplectic structure on H p at each point. If the distribution H can be defined by a global one-form α {\displaystyle \alpha } then this form is contact if and only if the top-dimensional form is a volume form on M , i.e. does not vanish anywhere. A contact analogue of
16093-462: The results are 2500 , 0 , {\displaystyle 2500,0,} and 0 {\displaystyle 0} respectively. Therefore, the greatest area attainable with a rectangle of 200 {\displaystyle 200} feet of fencing is 50 × 50 = 2500 {\displaystyle 50\times 50=2500} . For functions of more than one variable, similar conditions apply. For example, in
16226-486: The same period the development of projective geometry . Dubbed the single most important work in the history of differential geometry, in 1827 Gauss produced the Disquisitiones generales circa superficies curvas detailing the general theory of curved surfaces. In this work and his subsequent papers and unpublished notes on the theory of surfaces, Gauss has been dubbed the inventor of non-Euclidean geometry and
16359-477: The set which is contained within the set, whereas a maximal element m of a poset A is an element of A such that if m ≤ b (for any b in A ), then m = b . Any least element or greatest element of a poset is unique, but a poset can have several minimal or maximal elements. If a poset has more than one maximal element, then these elements will not be mutually comparable. In a totally ordered set, or chain , all elements are mutually comparable, so such
16492-494: The study of hyperbolic geometry by Lobachevsky . The simplest examples of smooth spaces are the plane and space curves and surfaces in the three-dimensional Euclidean space , and the study of these shapes formed the basis for development of modern differential geometry during the 18th and 19th centuries. Since the late 19th century, differential geometry has grown into a field concerned more generally with geometric structures on differentiable manifolds . A geometric structure
16625-435: The study of plane curves and the investigation of concepts such as points of inflection and circles of osculation , which aid in the measurement of curvature . Indeed, already in his first paper on the foundations of calculus, Leibniz notes that the infinitesimal condition d 2 y = 0 {\displaystyle d^{2}y=0} indicates the existence of an inflection point. Shortly after this time
16758-416: The study of the geometry of smooth shapes, can be traced back at least to classical antiquity . In particular, much was known about the geometry of the Earth , a spherical geometry , in the time of the ancient Greek mathematicians. Famously, Eratosthenes calculated the circumference of the Earth around 200 BC, and around 150 AD Ptolemy in his Geography introduced the stereographic projection for
16891-442: The surface of the Earth leads to the conclusion that great circles , which are only locally similar to straight lines in a flat plane, provide the shortest path between two points on the Earth's surface. Indeed, the measurements of distance along such geodesic paths by Eratosthenes and others can be considered a rudimentary measure of arclength of curves, a concept which did not see a rigorous definition in terms of calculus until
17024-464: The volumes of smooth three-dimensional solids such as the sphere, cones, and cylinders. There was little development in the theory of differential geometry between antiquity and the beginning of the Renaissance . Before the development of calculus by Newton and Leibniz , the most significant development in the understanding of differential geometry came from Gerardus Mercator 's development of
17157-482: The work of Riemann , the intrinsic point of view was developed, in which one cannot speak of moving "outside" the geometric object because it is considered to be given in a free-standing way. The fundamental result here is Gauss's theorema egregium , to the effect that Gaussian curvature is an intrinsic invariant. The intrinsic point of view is more flexible. For example, it is useful in relativity where space-time cannot naturally be taken as extrinsic. However, there
17290-696: Was Hermann Weyl who made important contributions to the foundations of general relativity, introduced the Weyl tensor providing insight into conformal geometry , and first defined the notion of a gauge leading to the development of gauge theory in physics and mathematics . In the middle and late 20th century differential geometry as a subject expanded in scope and developed links to other areas of mathematics and physics. The development of gauge theory and Yang–Mills theory in physics brought bundles and connections into focus, leading to developments in gauge theory . Many analytical results were investigated including
17423-617: Was developed by Sophus Lie and Jean Gaston Darboux , leading to important results in the theory of Lie groups and symplectic geometry . The notion of differential calculus on curved spaces was studied by Elwin Christoffel , who introduced the Christoffel symbols which describe the covariant derivative in 1868, and by others including Eugenio Beltrami who studied many analytic questions on manifolds. In 1899 Luigi Bianchi produced his Lectures on differential geometry which studied differential geometry from Riemann's perspective, and
17556-468: Was distilled in by Felix Hausdorff in 1914, and by 1942 there were many different notions of manifold of a combinatorial and differential-geometric nature. Interest in the subject was also focused by the emergence of Einstein's theory of general relativity and the importance of the Einstein Field equations. Einstein's theory popularised the tensor calculus of Ricci and Levi-Civita and introduced
17689-416: Was the development of an idea of Gauss's about the linear element d s {\displaystyle ds} of a surface. At this time Riemann began to introduce the systematic use of linear algebra and multilinear algebra into the subject, making great use of the theory of quadratic forms in his investigation of metrics and curvature. At this time Riemann did not yet develop the modern notion of
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