Vector field
Builds on Vector-valued functionEuclidean space
Summary
A vector field on a subset of assigns a vector of to each point of the subset. It models quantities with a direction and a size at each point, such as the velocity of a fluid or a force field.
Reading path
To understand this page from the ground up, read these 4 pages in order. The foundations come first.
Show the path as a graph
- no evidence level
- Euclidean space object
- Function concept
- Vector-valued function concept
- Vector field concept
Where this leads
194 pages build on this one.
- Work done by a force field
- Harmonic function
- Laplacian
- Line integral of a vector field
- Del operator
- Vector calculus identities
- Divergence
- Curl
- Conservative vector field
- Liouville's equation
- Conformal invariance of harmonic functions
- Shell theorem
- Left-invariant vector field
- Green's function for the Laplacian
- Reeb vector field
- Phragmén–Lindelöf principle for subharmonic functions
- Harmonic functions on Riemannian manifolds
- Weyl's law
- Pluriharmonic function
- Quaternionic analysis
- Cheng–Yau gradient estimate
- Removable singularities for harmonic functions
- Minimum principle for harmonic functions
- Bendixson–Dulac theorem
- Minimal surface equation
- Navier–Stokes existence and smoothness
- Superharmonic function
- Schrödinger equation
- Zaremba's punctured ball
- Harmonic at a point
- Poincaré–Hopf theorem
- Newtonian potential
- Nonlinear Schrödinger equation
- Stokes flow
- Krylov–Safonov estimate
- Neumann problem
- Schwarz integral formula
- Yamabe problem
- Harmonic majorant
- Harnack inequality for parabolic equations
- Reaction–diffusion system
- KdV equation as an integrable system
- Log-modulus of a holomorphic function is subharmonic
- Single layer potential
- Mutual absolute continuity of harmonic measures
- Lusin area integral
- Dirichlet problem
- Positive harmonic function
- Weyl's lemma
- Ginzburg–Landau equation
- Cahn–Hilliard equation
- Harmonic function on a ball
- Brelot space
- Liouville's theorem on phase-space volume
- Maximum principle for subharmonic functions
- Fundamental solution of the Laplacian
- Herglotz representation theorem
- Euler equations
- Poisson–Jensen formula
- Subharmonic function
- Poisson's equation
- Newtonian kernel
- Porous medium equation
- Ebin's slice theorem
- Smoothness and analyticity of harmonic functions
- Spectral geometry
- Gaussian free field
- Sub-mean value property
- Finite volume method
- Riesz decomposition theorem
- Kakutani's solution of the Dirichlet problem
- Clifford analysis
- Keller–Segel model
- Boundary integral equation for the Neumann problem
- Converse of the mean value property
- Gross–Pitaevskii equation
- Local existence of harmonic conjugates
- Feedback linearization
- Polyharmonic spline
- Boundary integral method for the Dirichlet problem
- Mixed finite element method
- Earnshaw's theorem
- Five-point stencil
- p-Laplacian
- Logarithmic kernel
- Spherical harmonics
- Dirichlet-to-Neumann operator
- Raviart–Thomas elements
- Plateau's problem
- Darcy's law
- Schouten–Nijenhuis bracket
- Convex functions of subharmonic functions
- Harmonic measure
- Superharmonic potential
- Resolutivity of boundary functions
- Lamé equations of elasticity
- Pólya's conjecture on Laplacian eigenvalues
- Minimal harmonic function
- Riesz measure of a subharmonic function
- Martin kernel
- Klein–Gordon equation
- Calderón projector
- Fractional Laplacian
- Obstacle problem
- Energy method
- Schiffer's conjecture
- Double layer potential
- Jump relations for layer potentials
- Discrete harmonic function
- Stefan problem as a variational inequality
- Hybrid system
- Fischer decomposition
- Navier–Stokes equations
- Douglas–Radó solution of Plateau's problem
- Koebe's theorem on the mean value property
- Structural stability
- Turing pattern
- Level-set method
- Bessel potential
- Maxwell's equations
- Perron method
- Compactness of families of harmonic functions
- Perimeter of a set
- Polyharmonic function
- Gradient estimates for harmonic functions
- p-harmonic function
- Kelvin transform
- Harnack's principle
- Lie's theorems on transformation groups
- Poisson integral formula
- Littlewood subordination theorem
- Biharmonic equation
- Dahlberg's theorem on harmonic measure
- Boundary Harnack principle
- Hörmander's condition
- Green's identities
- Least harmonic majorant
- Liouville's theorem for harmonic functions
- Cartan's magic formula
- Removable sets for bounded harmonic functions
- Calderón problem
- Hilbert's invariant integral
- Dirichlet's principle
- Harmonic functions on Riemann surfaces
- Calderón–Zygmund estimates
- Mean-field game
- Hardy spaces of harmonic functions
- Wave equation
- Herglotz–Riesz kernel
- Martin boundary
- Laplace's equation
- Caloric function
- Liouville's theorem for planar harmonic functions
- Gauss's law for potentials
- Green's function of the ball
- Maximum principle for harmonic functions
- Steklov eigenvalue problem
- Peixoto's theorem
- Hadamard three-circle theorem for subharmonic functions
- Lindelöf's maximum principle
- Condenser capacity
- Heat equation
- Chow–Rashevskii theorem
- Helmholtz equation
- Poisson kernel
- Schrödinger operator
- Hamiltonian vector field
- Harmonic measure as exit distribution
- Bateman transform
- Diffeomorphism group
- Parabolic and hyperbolic Riemann surfaces
- Poisson integral for the half-space
- Hearing the shape of a drum
- Logarithmic potential
- Sub-Riemannian geodesic problem
- Harmonic space
- Harmonic polynomial
- Hadamard's two-constants theorem
- Stability conjecture
- Schwarz reflection principle for harmonic functions
- Method of fundamental solutions
- Bôcher's theorem
- Dirichlet eigenvalue
- Harnack's inequality
- Free boundary problem
- Expansion in spherical harmonics
- Hairy ball theorem
- Monogenic function
- Allen–Cahn equation
- Hartogs lemma for subharmonic functions
- Dirac operator
- Parallelizable manifold
- Boundary integral equation for the Dirichlet problem
- Lane–Emden equation
Further reading
- Web https://en.wikipedia.org/wiki/Vector_field, Vector fields on subsets of Euclidean space
About this page
| Tag | 0299VCW1TH |
|---|---|
| URI | m2://0299VCW1TH |
| Aliases | vector-field-on-rn |
| Kind | concept |
| Writing | W1 stated |
| MSC | 26B12 |
| Uses | Vector-valued function, Euclidean space |
| Same as |
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What this page needs for W2
- edge instance_of in: at least two examples (0 of 2)
- edge not_instance_of in: at least one non-example (0 of 1)
The writing levels say how complete the text of a page is. They do not measure its truth.
What links here
Used by
- Work done by a force field
- Line integral of a vector field
- Divergence
- Curl
- Conservative vector field
- Left-invariant vector field
- Reeb vector field
- Poincaré–Hopf theorem
- Stokes flow
- Ginzburg–Landau equation
- Euler equations
- Feedback linearization
- Darcy's law
- Schouten–Nijenhuis bracket
- Lamé equations of elasticity
- Hybrid system
- Navier–Stokes equations
- Structural stability
- Maxwell's equations
- Lie's theorems on transformation groups
- Hörmander's condition
- Cartan's magic formula
- Peixoto's theorem
- Chow–Rashevskii theorem
- Hamiltonian vector field
- Diffeomorphism group
- Hairy ball theorem
- Parallelizable manifold