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Differential geometry. Global analysis --- 514.757 --- 515.16 --- Differentiable mappings --- Singularities (Mathematics) --- Geometry, Algebraic --- Differentiable maps --- Mappings, Differentiable --- Differential topology --- Mappings (Mathematics) --- Differential geometry of pointwise mappings --- Topology of manifolds --- 515.16 Topology of manifolds --- 514.757 Differential geometry of pointwise mappings --- Manifolds (Mathematics) --- Variétés (mathématiques) --- Singularités (mathématiques) --- Critical point theory (Mathematical analysis) --- Points critiques, Théorie des (analyse mathématique) --- SINGULARITIES (Mathematics) --- Applications différentiables --- Applications différentiables --- Variétés (mathématiques) --- Singularités (mathématiques) --- Points critiques, Théorie des (analyse mathématique)
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The present Special Issue of Symmetry is devoted to two important areas of global Riemannian geometry, namely submanifold theory and the geometry of Lie groups and homogeneous spaces. Submanifold theory originated from the classical geometry of curves and surfaces. Homogeneous spaces are manifolds that admit a transitive Lie group action, historically related to F. Klein's Erlangen Program and S. Lie's idea to use continuous symmetries in studying differential equations. In this Special Issue, we provide a collection of papers that not only reflect some of the latest advancements in both areas, but also highlight relations between them and the use of common techniques. Applications to other areas of mathematics are also considered.
warped products --- vector equilibrium problem --- Laplace operator --- cost functional --- pointwise 1-type spherical Gauss map --- inequalities --- homogeneous manifold --- finite-type --- magnetic curves --- Sasaki-Einstein --- evolution dynamics --- non-flat complex space forms --- hyperbolic space --- compact Riemannian manifolds --- maximum principle --- submanifold integral --- Clifford torus --- D’Atri space --- 3-Sasakian manifold --- links --- isoparametric hypersurface --- Einstein manifold --- real hypersurfaces --- Kähler 2 --- *-Weyl curvature tensor --- homogeneous geodesic --- optimal control --- formality --- hadamard manifolds --- Sasakian Lorentzian manifold --- generalized convexity --- isospectral manifolds --- Legendre curves --- geodesic chord property --- spherical Gauss map --- pointwise bi-slant immersions --- mean curvature --- weakly efficient pareto points --- geodesic symmetries --- homogeneous Finsler space --- orbifolds --- slant curves --- hypersphere --- ??-space --- k-D’Atri space --- *-Ricci tensor --- homogeneous space
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This book describes the theory and applications of discrete orthogonal polynomials--polynomials that are orthogonal on a finite set. Unlike other books, Discrete Orthogonal Polynomials addresses completely general weight functions and presents a new methodology for handling the discrete weights case. J. Baik, T. Kriecherbauer, K. T.-R. McLaughlin & P. D. Miller focus on asymptotic aspects of general, nonclassical discrete orthogonal polynomials and set out applications of current interest. Topics covered include the probability theory of discrete orthogonal polynomial ensembles and the continuum limit of the Toda lattice. The primary concern throughout is the asymptotic behavior of discrete orthogonal polynomials for general, nonclassical measures, in the joint limit where the degree increases as some fraction of the total number of points of collocation. The book formulates the orthogonality conditions defining these polynomials as a kind of Riemann-Hilbert problem and then generalizes the steepest descent method for such a problem to carry out the necessary asymptotic analysis.
Orthogonal polynomials --- Asymptotic theory --- Orthogonal polynomials -- Asymptotic theory. --- Polynomials. --- Civil & Environmental Engineering --- Engineering & Applied Sciences --- Operations Research --- Asymptotic theory. --- Asymptotic theory of orthogonal polynomials --- Algebra --- Airy function. --- Analytic continuation. --- Analytic function. --- Ansatz. --- Approximation error. --- Approximation theory. --- Asymptote. --- Asymptotic analysis. --- Asymptotic expansion. --- Asymptotic formula. --- Beta function. --- Boundary value problem. --- Calculation. --- Cauchy's integral formula. --- Cauchy–Riemann equations. --- Change of variables. --- Complex number. --- Complex plane. --- Correlation function. --- Degeneracy (mathematics). --- Determinant. --- Diagram (category theory). --- Discrete measure. --- Distribution function. --- Eigenvalues and eigenvectors. --- Equation. --- Estimation. --- Existential quantification. --- Explicit formulae (L-function). --- Factorization. --- Fredholm determinant. --- Functional derivative. --- Gamma function. --- Gradient descent. --- Harmonic analysis. --- Hermitian matrix. --- Homotopy. --- Hypergeometric function. --- I0. --- Identity matrix. --- Inequality (mathematics). --- Integrable system. --- Invariant measure. --- Inverse scattering transform. --- Invertible matrix. --- Jacobi matrix. --- Joint probability distribution. --- Lagrange multiplier. --- Lax equivalence theorem. --- Limit (mathematics). --- Linear programming. --- Lipschitz continuity. --- Matrix function. --- Maxima and minima. --- Monic polynomial. --- Monotonic function. --- Morera's theorem. --- Neumann series. --- Number line. --- Orthogonal polynomials. --- Orthogonality. --- Orthogonalization. --- Parameter. --- Parametrix. --- Pauli matrices. --- Pointwise convergence. --- Pointwise. --- Polynomial. --- Potential theory. --- Probability distribution. --- Probability measure. --- Probability theory. --- Probability. --- Proportionality (mathematics). --- Quantity. --- Random matrix. --- Random variable. --- Rate of convergence. --- Rectangle. --- Rhombus. --- Riemann surface. --- Special case. --- Spectral theory. --- Statistic. --- Subset. --- Theorem. --- Toda lattice. --- Trace (linear algebra). --- Trace class. --- Transition point. --- Triangular matrix. --- Trigonometric functions. --- Uniform continuity. --- Unit vector. --- Upper and lower bounds. --- Upper half-plane. --- Variational inequality. --- Weak solution. --- Weight function. --- Wishart distribution. --- Orthogonal polynomials - Asymptotic theory
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Mathematical No/ex, 27Originally published in 1981.The Princeton Legacy Library uses the latest print-on-demand technology to again make available previously out-of-print books from the distinguished backlist of Princeton University Press. These editions preserve the original texts of these important books while presenting them in durable paperback and hardcover editions. The goal of the Princeton Legacy Library is to vastly increase access to the rich scholarly heritage found in the thousands of books published by Princeton University Press since its founding in 1905.
Riemannian manifolds. --- Minimal surfaces. --- Surfaces, Minimal --- Maxima and minima --- Manifolds, Riemannian --- Riemannian space --- Space, Riemannian --- Geometry, Differential --- Manifolds (Mathematics) --- Differential geometry. Global analysis --- Addition. --- Analytic function. --- Branch point. --- Calculation. --- Cartesian coordinate system. --- Closed geodesic. --- Codimension. --- Coefficient. --- Compactness theorem. --- Compass-and-straightedge construction. --- Continuous function. --- Corollary. --- Counterexample. --- Covering space. --- Curvature. --- Curve. --- Decomposition theorem. --- Derivative. --- Differentiable manifold. --- Differential geometry. --- Disjoint union. --- Equation. --- Essential singularity. --- Estimation. --- Euclidean space. --- Existence theorem. --- Existential quantification. --- First variation. --- Flat topology. --- Fundamental group. --- Geometric measure theory. --- Great circle. --- Homology (mathematics). --- Homotopy group. --- Homotopy. --- Hyperbolic function. --- Hypersurface. --- Integer. --- Line–line intersection. --- Manifold. --- Measure (mathematics). --- Minimal surface. --- Monograph. --- Natural number. --- Open set. --- Parameter. --- Partition of unity. --- Pointwise. --- Quantity. --- Regularity theorem. --- Riemann surface. --- Riemannian manifold. --- Scalar curvature. --- Scientific notation. --- Second fundamental form. --- Sectional curvature. --- Sequence. --- Sign (mathematics). --- Simply connected space. --- Smoothness. --- Sobolev inequality. --- Solid torus. --- Subgroup. --- Submanifold. --- Summation. --- Theorem. --- Topology. --- Two-dimensional space. --- Unit sphere. --- Upper and lower bounds. --- Varifold. --- Weak topology.
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Mathematical Notes, 29Originally published in 1983.The Princeton Legacy Library uses the latest print-on-demand technology to again make available previously out-of-print books from the distinguished backlist of Princeton University Press. These editions preserve the original texts of these important books while presenting them in durable paperback and hardcover editions. The goal of the Princeton Legacy Library is to vastly increase access to the rich scholarly heritage found in the thousands of books published by Princeton University Press since its founding in 1905.
Differential equations, Elliptic --- Schrödinger operator. --- Eigenfunctions. --- Functions, Proper --- Proper functions --- Boundary value problems --- Differential equations --- Integral equations --- Operator, Schrödinger --- Differential operators --- Quantum theory --- Schrödinger equation --- Numerical solutions. --- Numerical solutions --- Approximation. --- Ball (mathematics). --- Bounded function. --- Center of mass. --- Coefficient. --- Compact space. --- Complex number. --- Continuous function (set theory). --- Continuous function. --- Discrete spectrum. --- Distribution (mathematics). --- Eigenfunction. --- Eigenvalues and eigenvectors. --- Elliptic operator. --- Equation. --- Equivalence class. --- Essential spectrum. --- Estimation. --- Existential quantification. --- Exponential decay. --- Function space. --- Fundamental theorem of calculus. --- Geometry. --- Ground state. --- Infimum and supremum. --- Lebesgue measure. --- Open set. --- Pointwise. --- Quadratic form. --- Quantity. --- Restriction (mathematics). --- Riemannian manifold. --- Robert Langlands. --- Schrödinger equation. --- Self-adjoint operator. --- Self-adjoint. --- Smoothness. --- Special case. --- Subset. --- Support (mathematics). --- Theorem. --- Upper and lower bounds. --- Weak solution. --- Without loss of generality.
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This book includes 13 papers concerning some of the recent progress in the theory of function spaces and its applications. The involved function spaces include Morrey and weak Morrey spaces, Hardy-type spaces, John–Nirenberg spaces, Sobolev spaces, and Besov and Triebel–Lizorkin spaces on different underlying spaces, and they are applied in the study of problems ranging from harmonic analysis to potential analysis and partial differential equations, such as the boundedness of paraproducts and Calderón operators, the characterization of pointwise multipliers, estimates of anisotropic logarithmic potential, as well as certain Dirichlet problems for the Schrödinger equation.
expansive matrix --- (mixed-norm) Hardy space --- molecule --- Calderón–Zygmund operator --- real interpolation --- besov space --- meyer wavelet --- Euclidean space --- cube --- congruent cube --- BMO --- JNp --- (localized) John–Nirenberg–Campanato space --- Riesz–Morrey space --- vanishing John–Nirenberg space --- duality --- commutator --- commutators --- Riesz potential --- homogeneous group --- space of homogeneous type --- paraproduct --- T(1) theorem --- hardy space --- bilinear estimate --- Hajłasz–Sobolev space --- Hajłasz–Besov space --- Hajłasz–Triebel–Lizorkin space --- generalized smoothness --- Lebesgue point --- capacity --- pointwise multipliers --- Morrey spaces --- block spaces --- convexification --- Calderón operator --- Hardy’s inequality --- variable Lebesgue space --- local Morrey space --- local block space --- extrapolation --- anisotropy --- Hardy space --- continuous ellipsoid cover --- maximal function --- anisotropic log-potential --- optimal polynomial inequality --- annulus body --- dual log-mixed volume --- Sobolev spaces --- compact manifolds --- tensor bundles --- differential operators --- Triebel–Lizorkin space --- Hardy inequality --- uniform domain --- fractional Laplacian --- Schrödinger equation --- Morrey space --- Dirichlet problem --- metric measure space
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The results established in this book constitute a new departure in ergodic theory and a significant expansion of its scope. Traditional ergodic theorems focused on amenable groups, and relied on the existence of an asymptotically invariant sequence in the group, the resulting maximal inequalities based on covering arguments, and the transference principle. Here, Alexander Gorodnik and Amos Nevo develop a systematic general approach to the proof of ergodic theorems for a large class of non-amenable locally compact groups and their lattice subgroups. Simple general conditions on the spectral theory of the group and the regularity of the averaging sets are formulated, which suffice to guarantee convergence to the ergodic mean. In particular, this approach gives a complete solution to the problem of establishing mean and pointwise ergodic theorems for the natural averages on semisimple algebraic groups and on their discrete lattice subgroups. Furthermore, an explicit quantitative rate of convergence to the ergodic mean is established in many cases. The topic of this volume lies at the intersection of several mathematical fields of fundamental importance. These include ergodic theory and dynamics of non-amenable groups, harmonic analysis on semisimple algebraic groups and their homogeneous spaces, quantitative non-Euclidean lattice point counting problems and their application to number theory, as well as equidistribution and non-commutative Diophantine approximation. Many examples and applications are provided in the text, demonstrating the usefulness of the results established.
Dynamics. --- Ergodic theory. --- Harmonic analysis. --- Lattice theory. --- Lie groups. --- Ergodic theory --- Lie groups --- Lattice theory --- Harmonic analysis --- Dynamics --- Calculus --- Mathematics --- Physical Sciences & Mathematics --- Dynamical systems --- Kinetics --- Analysis (Mathematics) --- Functions, Potential --- Potential functions --- Lattices (Mathematics) --- Space lattice (Mathematics) --- Structural analysis (Mathematics) --- Groups, Lie --- Ergodic transformations --- Mechanics, Analytic --- Force and energy --- Mechanics --- Physics --- Statics --- Banach algebras --- Mathematical analysis --- Bessel functions --- Fourier series --- Harmonic functions --- Time-series analysis --- Algebra, Abstract --- Algebra, Boolean --- Group theory --- Set theory --- Topology --- Transformations (Mathematics) --- Crystallography, Mathematical --- Lie algebras --- Symmetric spaces --- Topological groups --- Continuous groups --- Mathematical physics --- Measure theory --- Absolute continuity. --- Algebraic group. --- Amenable group. --- Asymptote. --- Asymptotic analysis. --- Asymptotic expansion. --- Automorphism. --- Borel set. --- Bounded function. --- Bounded operator. --- Bounded set (topological vector space). --- Congruence subgroup. --- Continuous function. --- Convergence of random variables. --- Convolution. --- Coset. --- Counting problem (complexity). --- Counting. --- Differentiable function. --- Dimension (vector space). --- Diophantine approximation. --- Direct integral. --- Direct product. --- Discrete group. --- Embedding. --- Equidistribution theorem. --- Ergodicity. --- Estimation. --- Explicit formulae (L-function). --- Family of sets. --- Haar measure. --- Hilbert space. --- Hyperbolic space. --- Induced representation. --- Infimum and supremum. --- Initial condition. --- Interpolation theorem. --- Invariance principle (linguistics). --- Invariant measure. --- Irreducible representation. --- Isometry group. --- Iwasawa group. --- Lattice (group). --- Lie algebra. --- Linear algebraic group. --- Linear space (geometry). --- Lipschitz continuity. --- Mass distribution. --- Mathematical induction. --- Maximal compact subgroup. --- Maximal ergodic theorem. --- Measure (mathematics). --- Mellin transform. --- Metric space. --- Monotonic function. --- Neighbourhood (mathematics). --- Normal subgroup. --- Number theory. --- One-parameter group. --- Operator norm. --- Orthogonal complement. --- P-adic number. --- Parametrization. --- Parity (mathematics). --- Pointwise convergence. --- Pointwise. --- Principal homogeneous space. --- Principal series representation. --- Probability measure. --- Probability space. --- Probability. --- Rate of convergence. --- Regular representation. --- Representation theory. --- Resolution of singularities. --- Sobolev space. --- Special case. --- Spectral gap. --- Spectral method. --- Spectral theory. --- Square (algebra). --- Subgroup. --- Subsequence. --- Subset. --- Symmetric space. --- Tensor algebra. --- Tensor product. --- Theorem. --- Transfer principle. --- Unit sphere. --- Unit vector. --- Unitary group. --- Unitary representation. --- Upper and lower bounds. --- Variable (mathematics). --- Vector group. --- Vector space. --- Volume form. --- Word metric.
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This book includes 13 papers concerning some of the recent progress in the theory of function spaces and its applications. The involved function spaces include Morrey and weak Morrey spaces, Hardy-type spaces, John–Nirenberg spaces, Sobolev spaces, and Besov and Triebel–Lizorkin spaces on different underlying spaces, and they are applied in the study of problems ranging from harmonic analysis to potential analysis and partial differential equations, such as the boundedness of paraproducts and Calderón operators, the characterization of pointwise multipliers, estimates of anisotropic logarithmic potential, as well as certain Dirichlet problems for the Schrödinger equation.
Research & information: general --- Mathematics & science --- expansive matrix --- (mixed-norm) Hardy space --- molecule --- Calderón–Zygmund operator --- real interpolation --- besov space --- meyer wavelet --- Euclidean space --- cube --- congruent cube --- BMO --- JNp --- (localized) John–Nirenberg–Campanato space --- Riesz–Morrey space --- vanishing John–Nirenberg space --- duality --- commutator --- commutators --- Riesz potential --- homogeneous group --- space of homogeneous type --- paraproduct --- T(1) theorem --- hardy space --- bilinear estimate --- Hajłasz–Sobolev space --- Hajłasz–Besov space --- Hajłasz–Triebel–Lizorkin space --- generalized smoothness --- Lebesgue point --- capacity --- pointwise multipliers --- Morrey spaces --- block spaces --- convexification --- Calderón operator --- Hardy’s inequality --- variable Lebesgue space --- local Morrey space --- local block space --- extrapolation --- anisotropy --- Hardy space --- continuous ellipsoid cover --- maximal function --- anisotropic log-potential --- optimal polynomial inequality --- annulus body --- dual log-mixed volume --- Sobolev spaces --- compact manifolds --- tensor bundles --- differential operators --- Triebel–Lizorkin space --- Hardy inequality --- uniform domain --- fractional Laplacian --- Schrödinger equation --- Morrey space --- Dirichlet problem --- metric measure space --- expansive matrix --- (mixed-norm) Hardy space --- molecule --- Calderón–Zygmund operator --- real interpolation --- besov space --- meyer wavelet --- Euclidean space --- cube --- congruent cube --- BMO --- JNp --- (localized) John–Nirenberg–Campanato space --- Riesz–Morrey space --- vanishing John–Nirenberg space --- duality --- commutator --- commutators --- Riesz potential --- homogeneous group --- space of homogeneous type --- paraproduct --- T(1) theorem --- hardy space --- bilinear estimate --- Hajłasz–Sobolev space --- Hajłasz–Besov space --- Hajłasz–Triebel–Lizorkin space --- generalized smoothness --- Lebesgue point --- capacity --- pointwise multipliers --- Morrey spaces --- block spaces --- convexification --- Calderón operator --- Hardy’s inequality --- variable Lebesgue space --- local Morrey space --- local block space --- extrapolation --- anisotropy --- Hardy space --- continuous ellipsoid cover --- maximal function --- anisotropic log-potential --- optimal polynomial inequality --- annulus body --- dual log-mixed volume --- Sobolev spaces --- compact manifolds --- tensor bundles --- differential operators --- Triebel–Lizorkin space --- Hardy inequality --- uniform domain --- fractional Laplacian --- Schrödinger equation --- Morrey space --- Dirichlet problem --- metric measure space
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The ∂̄ Neumann problem is probably the most important and natural example of a non-elliptic boundary value problem, arising as it does from the Cauchy-Riemann equations. It has been known for some time how to prove solvability and regularity by the use of L2 methods. In this monograph the authors apply recent methods involving the Heisenberg group to obtain parametricies and to give sharp estimates in various function spaces, leading to a better understanding of the ∂̄ Neumann problem. The authors have added substantial background material to make the monograph more accessible to students.Originally published in 1977.The Princeton Legacy Library uses the latest print-on-demand technology to again make available previously out-of-print books from the distinguished backlist of Princeton University Press. These editions preserve the original texts of these important books while presenting them in durable paperback and hardcover editions. The goal of the Princeton Legacy Library is to vastly increase access to the rich scholarly heritage found in the thousands of books published by Princeton University Press since its founding in 1905.
Partial differential equations --- Neumann problem. --- Neumann problem --- Mathematics --- Physical Sciences & Mathematics --- Calculus --- Boundary value problems --- Differential equations, Partial --- A priori estimate. --- Abuse of notation. --- Analytic continuation. --- Analytic function. --- Approximation. --- Asymptotic expansion. --- Asymptotic formula. --- Basis (linear algebra). --- Besov space. --- Boundary (topology). --- Boundary value problem. --- Boundedness. --- Calculation. --- Cauchy's integral formula. --- Cauchy–Riemann equations. --- Change of variables. --- Characterization (mathematics). --- Combination. --- Commutative property. --- Commutator. --- Complex analysis. --- Complex manifold. --- Complex number. --- Computation. --- Convolution. --- Coordinate system. --- Corollary. --- Counterexample. --- Derivative. --- Determinant. --- Differential equation. --- Dimension (vector space). --- Dimension. --- Dimensional analysis. --- Dirichlet boundary condition. --- Eigenvalues and eigenvectors. --- Elliptic boundary value problem. --- Equation. --- Error term. --- Estimation. --- Even and odd functions. --- Existential quantification. --- Function space. --- Fundamental solution. --- Green's theorem. --- Half-space (geometry). --- Hardy's inequality. --- Heisenberg group. --- Holomorphic function. --- Infimum and supremum. --- Integer. --- Integral curve. --- Integral expression. --- Inverse function. --- Invertible matrix. --- Iteration. --- Laplace's equation. --- Left inverse. --- Lie algebra. --- Lie group. --- Linear combination. --- Logarithm. --- Lp space. --- Mathematical induction. --- Neumann boundary condition. --- Notation. --- Open problem. --- Orthogonal complement. --- Orthogonality. --- Parametrix. --- Partial derivative. --- Pointwise. --- Polynomial. --- Principal branch. --- Principal part. --- Projection (linear algebra). --- Pseudo-differential operator. --- Quantity. --- Recursive definition. --- Schwartz space. --- Scientific notation. --- Second derivative. --- Self-adjoint. --- Singular value. --- Sobolev space. --- Special case. --- Standard basis. --- Stein manifold. --- Subgroup. --- Subset. --- Summation. --- Support (mathematics). --- Tangent bundle. --- Theorem. --- Theory. --- Upper half-plane. --- Variable (mathematics). --- Vector field. --- Volume element. --- Weak solution. --- Neumann, Problème de --- Equations aux derivees partielles --- Problemes aux limites
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This book introduces new methods in the theory of partial differential equations derivable from a Lagrangian. These methods constitute, in part, an extension to partial differential equations of the methods of symplectic geometry and Hamilton-Jacobi theory for Lagrangian systems of ordinary differential equations. A distinguishing characteristic of this approach is that one considers, at once, entire families of solutions of the Euler-Lagrange equations, rather than restricting attention to single solutions at a time. The second part of the book develops a general theory of integral identities, the theory of "compatible currents," which extends the work of E. Noether. Finally, the third part introduces a new general definition of hyperbolicity, based on a quadratic form associated with the Lagrangian, which overcomes the obstacles arising from singularities of the characteristic variety that were encountered in previous approaches. On the basis of the new definition, the domain-of-dependence theorem and stability properties of solutions are derived. Applications to continuum mechanics are discussed throughout the book. The last chapter is devoted to the electrodynamics of nonlinear continuous media.
Differentiaalvergelijkingen [Hyperbolische ] --- Differential equations [Hyperbolic] --- Equations différentielles hyperboliques --- Symplectic manifolds --- Differential equations, Hyperbolic. --- Symplectic manifolds. --- Variétés symplectiques --- Equations différentielles hyperboliques --- Variétés symplectiques --- Manifolds, Symplectic --- Geometry, Differential --- Manifolds (Mathematics) --- Hyperbolic differential equations --- Differential equations, Partial --- Action (physics). --- Boundary value problem. --- Canonical form. --- Causal structure. --- Classical mechanics. --- Complex analysis. --- Configuration space. --- Conservative vector field. --- Conserved current. --- Conserved quantity. --- Continuum mechanics. --- Derivative. --- Diffeomorphism. --- Differentiable manifold. --- Differential geometry. --- Dimension. --- Dimensional analysis. --- Dirichlet's principle. --- Einstein field equations. --- Electromagnetic field. --- Equation. --- Equations of motion. --- Equivalence class. --- Error term. --- Euclidean space. --- Euler system. --- Euler's equations (rigid body dynamics). --- Euler–Lagrange equation. --- Existence theorem. --- Existential quantification. --- Exponential map (Lie theory). --- Exponential map (Riemannian geometry). --- Exterior derivative. --- Fiber bundle. --- Foliation. --- Fritz John. --- General relativity. --- Hamiltonian mechanics. --- Hamilton–Jacobi equation. --- Harmonic map. --- Hessian matrix. --- Holomorphic function. --- Hyperbolic partial differential equation. --- Hyperplane. --- Hypersurface. --- Identity element. --- Iteration. --- Iterative method. --- Lagrangian (field theory). --- Lagrangian. --- Legendre transformation. --- Lie algebra. --- Linear approximation. --- Linear differential equation. --- Linear map. --- Linear span. --- Linearity. --- Linearization. --- Maximum principle. --- Maxwell's equations. --- Nonlinear system. --- Open set. --- Ordinary differential equation. --- Orthogonal complement. --- Parameter. --- Partial differential equation. --- Phase space. --- Pointwise. --- Poisson bracket. --- Polynomial. --- Principal part. --- Principle of least action. --- Probability. --- Pullback bundle. --- Pullback. --- Quadratic form. --- Quantity. --- Requirement. --- Riemannian manifold. --- Second derivative. --- Simultaneous equations. --- Special case. --- State function. --- Stokes' theorem. --- Subset. --- Surjective function. --- Symplectic geometry. --- Tangent bundle. --- Tangent vector. --- Theorem. --- Theoretical physics. --- Theory. --- Underdetermined system. --- Variable (mathematics). --- Vector bundle. --- Vector field. --- Vector space. --- Volume form. --- Zero of a function. --- Zero set.
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