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Concept of vector calculus
calculus and differential topology, a closed form is a differential form α whose exterior derivative is zero (dα = 0); and an exact form is a differential form
Closed and exact differential forms
Closed_and_exact_differential_forms
Cohomology with real coefficients computed using differential forms
are called exact and forms whose exterior derivative is 0 {\displaystyle 0} are called closed (see Closed and exact differential forms); the relationship
De_Rham_cohomology
Expression that may be integrated over a region
systolic geometry. Closed and exact differential forms Complex differential form Vector-valued differential form Equivariant differential form Calculus on Manifolds
Differential_form
Type of infinitesimal in calculus
calculus, a differential or differential form is said to be exact or perfect (exact differential), as contrasted with an inexact differential, if it is
Exact_differential
Specific mathematical differential form
heat and work, but is defined more generally within mathematics as a type of differential form. In contrast, an integral of an exact differential is always
Inexact_differential
list of real analysis topics, list of calculus topics. Closed and exact differential forms Contact (mathematics) Contour integral Contour line Critical
List of multivariable calculus topics
List_of_multivariable_calculus_topics
equation) (partial differential equations) Poincaré lemma of closed and exact differential forms Cotlar–Stein lemma Ehrling's lemma Riesz's lemma Abel's lemma
List_of_lemmas
Mathematical condition
condition for a closed differential form to be exact (while an exact form is necessarily closed). Precisely, it states that every closed p-form on an open
Poincaré_lemma
{\displaystyle \Omega _{c}^{1}(S^{1})} and d Ω 0 ( S 1 ) {\displaystyle d\Omega ^{0}(S^{1})} are the closed and exact differential forms on the circle group); in particular
Exact_category
Branch of mathematics
learning Multivector Geometric algebra Clifford algebra Closed and exact differential forms Component-free treatment of tensors Cramer's rule Dual space
Multilinear_algebra
Mathematical concept in vector calculus
and requires choosing a gauge. Fundamental theorem of vector calculus Magnetic vector potential Solenoidal vector field Closed and Exact Differential
Vector_potential
d_{\mathfrak {g}}} -closed or d g {\displaystyle d_{\mathfrak {g}}} -exact forms are called equivariantly closed or equivariantly exact. The integral of
Equivariant_differential_form
Type of functional equation (mathematics)
properties of solutions of a given differential equation may be determined without computing them exactly. Often when a closed-form expression for the solutions
Differential_equation
case X and S are affine schemes, the above definition means that Ω X / S {\displaystyle \Omega _{X/S}} is the module of Kähler differentials. The standard
Cotangent_sheaf
Differential form of degree one or section of a cotangent bundle
exterior derivative of a one-form is a two-form, of a two-form is a three-form, and so on. One-forms are widely used in differential geometry. Their duality
One-form
Differential equation containing derivatives with respect to only one variable
F(x,y_{1},\ldots ,y_{n})).} Some differential equations have solutions that can be written in an exact and closed form. Several important classes are given
Ordinary differential equation
Ordinary_differential_equation
Sequence of homomorphisms such that each kernel equals the preceding image
functors that transform exact sequences into exact sequences. The term "exact" originates from exact differential forms in the context of the de Rham complex:
Exact_sequence
Meromorphic differential form
In algebraic geometry and the theory of complex manifolds, a logarithmic differential form is a differential form with poles of a certain kind. The concept
Logarithmic_form
Differential form in commutative algebra
In mathematics, Kähler differentials provide an adaptation of differential forms to arbitrary commutative rings or schemes. The notion was introduced
Kähler_differential
In general relativity, an exact solution is a (typically closed form) solution of the Einstein field equations whose derivation does not invoke simplifying
Exact solutions in general relativity
Exact_solutions_in_general_relativity
On finding a maximal set of solutions of a system of first-order homogeneous linear PDEs
manifolds. The theorem is foundational in differential topology and calculus on manifolds. Contact geometry studies 1-forms that maximally violate the assumptions
Frobenius theorem (differential topology)
Frobenius_theorem_(differential_topology)
Long exact sequence
a differential form with the Euler class e. The Gysin sequence is a long exact sequence not only for the de Rham cohomology of differential forms, but
Gysin_homomorphism
Map from multiple vectors to an underlying field of scalars, linear in each argument
a differential 0-form: f ∈ C 0 ( U ) = Ω 0 ( U ) {\displaystyle f\in C^{0}(U)=\Omega ^{0}(U)} . We first construct differential 1-forms from 0-forms and
Multilinear_form
Conformal structure admits a Hodge dual of 1-forms without even specifying a metric
In mathematics, differential forms on a Riemann surface are an important special case of the general theory of differential forms on smooth manifolds
Differential forms on a Riemann surface
Differential_forms_on_a_Riemann_surface
\Omega ^{1}(M)} . Differential k {\displaystyle k} -forms, which we refer to simply as k {\displaystyle k} -forms here, are differential forms defined on T
Exterior_calculus_identities
Group of differential equations
system of differential equations can be succinctly stated in terms of differential forms (i.e., for a form to be exact, it needs to be closed). See integrability
System of differential equations
System_of_differential_equations
Tool in homological algebra
(or closed elements), and the elements in the image of d are called (co)boundaries (or exact elements). Right from the definition of the differential, all
Chain_complex
Operation on differential forms
concept of the differential of a function to differential forms of higher degree. The exterior derivative was first described in its current form by Élie Cartan
Exterior_derivative
Differential equation that is linear with respect to the unknown function
linear differential equation is a differential equation that is linear in the unknown function and its derivatives, so it can be written in the form a 0
Linear_differential_equation
Swiss mathematician
manifold could be encoded by differential forms. As a particular form of this, he conjectured that a closed form is exact if it integrates to zero over
Georges_de_Rham
American mathematician (1925–2013)
Flanders published Differential Forms with Applications to the Physical Sciences which connected applied mathematics and differential forms. A reviewer affirmed
Harley_Flanders
Algebraic structure used in topology
constant differential C d θ {\displaystyle C\,d\theta } . Here the coboundaries are the exact differentials and the cochains are all differentials. The quotient
Cohomology
Class of ordinary differential equations
In mathematics and its applications, a Sturm–Liouville problem is a second-order linear ordinary differential equation of the form d d x [ p ( x ) d y
Sturm–Liouville_theory
Branch of mathematics
mathematical physics, operator algebras, complex analysis, and the theory of partial differential equations. K-theory is an independent discipline which draws
Homological_algebra
Equations describing classical electromagnetism
set of coupled partial differential equations that describe how electric and magnetic fields are generated by electric charges and currents. Together with
Maxwell's_equations
Methods of mathematical approximation
the exact solution of a related, simpler problem. A critical feature of the technique is a middle step that breaks the problem into "solvable" and "perturbative"
Perturbation_theory
Last letter of the Greek alphabet
islands, namely Paros, Thasos and Melos, chose the exact opposite innovation, using a broken-up circle for the short and a closed circle for the long /o/.
Omega
Operation in calculus
integral. A differential form is a mathematical concept in the fields of multivariable calculus, differential topology, and tensors. Differential forms are organized
Integral
Theorem
a closed differential form is exact if and only if the integrations of it over arbitrary cycles are all zero. For a one-form, it means that a closed one-form
De_Rham_theorem
"Construction of Exact Parametric or Closed Form Solutions of Some Unsolvable Classes of Nonlinear ODEs (Abel's Nonlinear ODEs of the First Kind and Relative
List of nonlinear ordinary differential equations
List_of_nonlinear_ordinary_differential_equations
Eigenvalue problem for the Laplace operator
partial differential equation: ∇ 2 f = − k 2 f , {\displaystyle \nabla ^{2}f=-k^{2}f,} where ∇2 is the Laplace operator, –k2 is the eigenvalue, and f is
Helmholtz_equation
{\displaystyle H_{\mathrm {c} }^{q}(X)} is the vector space of closed q-forms modulo that of exact q-forms. Despite their definition as the homology of an ascending
Cohomology with compact support
Cohomology_with_compact_support
Exterior algebraic map taking tensors from p forms to n-p forms
play a role in differential geometry when applied to the cotangent bundle of a pseudo-Riemannian manifold, and hence to differential k-forms. This allows
Hodge_star_operator
Linearization technique for nonlinear differential systems
formally represents a finite-dimensional nonlinear system of ordinary differential equations as an infinite-dimensional linear system. Truncating the resulting
Carleman_linearization
Type of electrical transformer
The linear variable differential transformer (LVDT) – also called linear variable displacement transformer, linear variable displacement transducer, or
Linear variable differential transformer
Linear_variable_differential_transformer
Mathematical object in sheaf cohomology
sheaf R {\displaystyle \mathbb {R} } by the fine sheaves of (smooth) differential forms: 0 → R → C X 0 → C X 1 → ⋯ → C X dim X → 0. {\displaystyle 0\to
Injective_sheaf
Parameter in differential equations and dynamical systems
mathematics and particularly in dynamical systems, an initial condition is the initial value (often at time t = 0 {\displaystyle t=0} ) of a differential equation
Initial_condition
Calculus of vector-valued functions
understood, in a more general form, using the machinery of differential geometry, of which vector calculus forms a subset. Grad and div generalize immediately
Vector_calculus
Theorem in complex geometry
an exact differential form in de Rham cohomology. In particular if α ∈ Ω k ( X ) {\displaystyle \alpha \in \Omega ^{k}(X)} is a closed differential k-form
Ddbar_lemma
Awarded every year by the American Mathematical Society
Lars (2003) [1963]. The Analysis of Linear Partial Differential Operators I: Distribution Theory and Fourier Analysis. Classics in Mathematics (2nd ed
Leroy_P._Steele_Prize
System where changes of output are not proportional to changes of input
problem). Second and higher order ordinary differential equations (more generally, systems of nonlinear equations) rarely yield closed-form solutions, though
Nonlinear_system
Mathematics concept
system and Langlands duality. The dimensions hp,q of spaces of harmonic (p,q)-differential forms (equivalently, the cohomology, i.e., closed forms modulo
Homological_mirror_symmetry
Method in Itô calculus
stochastic differential equation (SDE). It is an extension of the Euler method for ordinary differential equations to stochastic differential equations
Euler–Maruyama_method
Vector field that is the gradient of some function
d^{2}=0} , any exact form is closed, so any conservative vector field is irrotational. Conversely, all closed 1 {\displaystyle 1} -forms are exact if U {\displaystyle
Conservative_vector_field
Discrete (i.e., incremental) version of infinitesimal calculus
discrete (differential) forms, we refer to d {\displaystyle d} as the exterior derivative. We also use the calculus notation for the values of the forms: ω (
Discrete_calculus
Aspect of general relativity
not exact are called non-exact solutions. Such solutions mainly arise due to the difficulty of solving the EFE in closed form and often take the form of
Solutions of the Einstein field equations
Solutions_of_the_Einstein_field_equations
Branch of engineering and mathematics
the variables are expressed as vectors and the differential and algebraic equations are written in matrix form (the latter only being possible when the
Control_theory
Study of sudden qualitative behavior changes caused by small parameter changes
as the integral curves of a family of vector fields, and the solutions of a family of differential equations. Most commonly applied to the mathematical
Bifurcation_theory
Type of manifold in differential geometry
In differential geometry, a symplectic manifold is a smooth manifold, M {\displaystyle M} , equipped with a closed nondegenerate differential 2-form, ω
Symplectic_manifold
Generalized function whose value is zero everywhere except at zero
Probability and measure (2nd ed.) Boyce, William E.; DiPrima, Richard C.; Meade, Douglas B. (2017). Elementary differential equations and boundary value
Dirac_delta_function
Module over a sheaf of differential operators
exterior power of differential 1-forms on X. This bundle has a natural right action determined by ω ⋅ v := − Liev (ω), where v is a differential operator of
D-module
Relativistic vector field
forms in this decomposition, only the coexact form has any effect on the electromagnetic tensor F = d A {\displaystyle F=dA} . Exact forms are closed
Electromagnetic four-potential
Electromagnetic_four-potential
Manifold upon which it is possible to perform calculus
th cohomology group is the quotient group of the closed forms on M {\displaystyle M} by the exact forms on M {\displaystyle M} . Suppose that M {\displaystyle
Differentiable_manifold
Equation involving both integrals and derivatives of a function
u(x_{0})=u_{0},\qquad x_{0}\geq 0.} As is typical with differential equations, obtaining a closed-form solution can often be difficult. In the relatively
Integro-differential_equation
Range of physical processes in physics
coefficient and x is the distance traveled in the target. The above ordinary first-order differential equation has solutions of the form: I = I o e −
Scattering
Statement about integration on manifolds
applies to higher degree differential forms ω {\displaystyle \omega } instead of just 0-forms such as F {\displaystyle F} . A closed interval [ a , b ] {\displaystyle
Generalized_Stokes_theorem
Algebraic tool for computing topological spaces' invariants
homology. The same computation applied to the short exact sequences of vector spaces of differential forms 0 → Ω n ( X ) → Ω n ( U ) ⊕ Ω n ( V ) → Ω n ( U
Mayer–Vietoris_sequence
Mathematical formula expressing equality
of exact, analytic solutions. A partial differential equation (PDE) is a differential equation that contains unknown multivariable functions and their
Equation
Formulation to quantize gauge field theories in physics
saying that its image (the space of BRST exact forms) lies entirely within its kernel (the space of BRST closed forms). The "true" Lagrangian, presumed to
BRST_quantization
Trick relating differential forms
In differential geometry, a branch of mathematics, Moser's trick (or Moser's argument) is a method to relate two differential forms α 0 {\displaystyle
Moser's_trick
Concept in geometry
conductor that measures the wild ramification by using the sheaf of differential 1-forms. S. Bloch conjectures a formula for the Artin conductor of the ℓ-adic
Localized_Chern_class
Gives the signature of a smooth compact oriented manifold in terms of Pontryagin numbers
In differential topology, an area of mathematics, the Hirzebruch signature theorem (sometimes called the Hirzebruch index theorem) is Friedrich Hirzebruch's
Hirzebruch_signature_theorem
is the rank of the module. differential A differential graded module or dg-module is a graded module with a differential. direct sum A direct sum of
Glossary_of_module_theory
Theorem in topology
f(x_{0})=x_{0}} . The simplest forms of Brouwer's theorem are for continuous functions f {\displaystyle f} from a closed interval I {\displaystyle I} in
Brouwer_fixed-point_theorem
Physical quantity
describe all forms of energy, it is often convenient to refer to particular combinations of potential and kinetic energy as its own form. For example
Energy
number Thermodynamics Timeline of thermodynamics Triple product rule Exact differential Keenan, Thermodynamics, Wiley, New York, 1947 Physical chemistry,
Table of thermodynamic equations
Table_of_thermodynamic_equations
Branch of mathematics
culminating work of Newton and Leibniz in the later 17th century gave these strands a new and more general form. The differential and integral calculi made
Mathematical_analysis
Physics problem related to laws of motion and gravity
closed-form analytic solution. The differential equations that govern the motions of three gravitating bodies are not integrable and cannot be solved to give explicit
Three-body_problem
Atiyah–Bott fixed-point theorem (differential topology) Bing's recognition theorem (geometric topology) Birman short exact sequence (geometric topology)
List_of_theorems
Evaluates a line integral through a gradient field using the original scalar field
terms of differential forms on manifolds. In particular, suppose ω is a form defined on a contractible domain, and the integral of ω over any closed manifold
Gradient_theorem
Restatement of Newton's law of universal gravitation
,} which is the differential form of Gauss's law for gravity. It is possible to derive the integral form from the differential form using the reverse
Gauss's_law_for_gravity
removed. A closed 1-form ω is exact if and only if ∫γ ω = 0 for every closed Jordan curve γ. This follows from the Poincaré lemma for 1-forms and the fact
Planar_Riemann_surface
Roughly, the number of k-dimensional holes on a topological surface
they predict the dimensions of vector spaces of closed differential forms modulo exact differential forms. The connection with the definition given above
Betti_number
Manifold with Riemannian, complex and symplectic structure
In mathematics and especially differential geometry, a Kähler manifold is a manifold with three mutually compatible structures: a complex structure, a
Kähler_manifold
Concept in mathematics
small perturbations (to be exact C1-small perturbations). Examples of such qualitative properties are numbers of fixed points and periodic orbits (but not
Structural_stability
Mathematical term
differential forms of degree ( p , q ) {\displaystyle (p,q)} . In the article on complex forms, the Dolbeault operator is defined as a differential operator
Dolbeault_cohomology
Former natural arch in Hawaiʻi
the NPS dates to about 550 years ago. The park attributes its shape to differential erosion, in which waves wear away lava layers of differing hardness at
Hōlei_Sea_Arch
Abel, is any ordinary differential equation that is cubic in the unknown function. In other words, it is an equation of the form y ′ = f 3 ( x ) y 3 +
Abel equation of the first kind
Abel_equation_of_the_first_kind
Mathematics award
awarded in 1936 to Finnish mathematician Lars Ahlfors and American mathematician Jesse Douglas, and it has been awarded every four years since 1950. In
Fields_Medal
Stochastic process modeling random walk with friction
common representation for the Ornstein–Uhlenbeck process and similar stochastic differential equations by tacitly assuming that the noise term is a derivative
Ornstein–Uhlenbeck_process
Algebra based on a vector space with a quadratic form
applications of the exterior algebra is in differential geometry where it is used to define the bundle of differential forms on a smooth manifold. In the case
Clifford_algebra
Award of the American Mathematical Society
Smale "for his contributions to various aspects of differential topology." 1966 Morton Brown and Barry Mazur "for their work on the generalized Schoenflies
Oswald Veblen Prize in Geometry
Oswald_Veblen_Prize_in_Geometry
{\mathcal {A}}_{M,\mathbb {C} }^{k}} is the sheaf of germs of smooth differential k-forms tensored with C {\displaystyle \mathbb {C} } . So, we write H D ∗
Bundle_gerbe
Russian-Italian mathematician (1938–2019)
the algebraic theory of differential operators, homological algebra, differential geometry and algebraic topology, mechanics and mathematical physics, the
Alexandre Mikhailovich Vinogradov
Alexandre_Mikhailovich_Vinogradov
Branch of mathematics
problems depend not on the exact shape of the objects involved, but rather on the way they are put together. For example, the square and the circle have many
Topology
Construct in algebraic geometry
the exact sequence above: There is a new term on the left, the conormal sheaf of f, and the relative differentials ΩX/Y have vanished because a closed immersion
Cotangent_complex
Model of the early universe
exterior derivative and ∧ {\displaystyle \wedge } the wedge product of differential forms. The 1-forms σ i {\displaystyle \sigma _{i}} form a left-invariant
Mixmaster_universe
Symplectic topology tool
theory, in that it is generated by certain collections of closed Reeb orbits and its differential counts certain holomorphic curves with ends at certain
Floer_homology
Type of option contract in finance
This approach gives explicit (closed form) prices to barrier options. Yet another method is the partial differential equation (PDE) approach. The PDE
Barrier_option
It is a generalization of the Lie bracket from an operation on the tangent bundle
{\displaystyle {\mathbf {T} }^{*}} of M {\displaystyle M} is the bundle of differential one-forms. In the case p = 1 {\displaystyle p=1} the Courant bracket maps
Courant_bracket
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CLOSED AND-EXACT-DIFFERENTIAL-FORMS
CLOSED AND-EXACT-DIFFERENTIAL-FORMS
CLOSED AND-EXACT-DIFFERENTIAL-FORMS
CLOSED AND-EXACT-DIFFERENTIAL-FORMS
CLOSED AND-EXACT-DIFFERENTIAL-FORMS
CLOSED AND-EXACT-DIFFERENTIAL-FORMS
CLOSED AND-EXACT-DIFFERENTIAL-FORMS
CLOSED AND-EXACT-DIFFERENTIAL-FORMS
CLOSED AND-EXACT-DIFFERENTIAL-FORMS
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