Everything about Vector field
Assignment of a vector to each point in a subset of Euclidean space
In vector calculus and physics, a vector field is an assignment of a vector to each point in a space, most commonly Euclidean space. A vector field on a plane can be visualized as a collection of arrows with given magnitudes and directions, each attached to a point on the plane. Vector fields often have unit of measurement, forming a vector physical quantity. They may be used to model, for example, the speed and direction of a moving fluid throughout three dimensional space, such as the wind, or the strength and direction of some force, such as the magnetic or gravitational force, as it changes from one point to another point. Wikipedia →
🧭 Start here: a 6-step reading path
- Read the overview: Vector fieldA 5-minute orientation
- Skim a recent paper: Mapping transcriptomic vector fields of single cellsInfluential work since 2022
- Pick up a classic book: Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector FieldsThe most-reprinted book on the subject
- Read the landmark paper: Inflation driven by a vector fieldCited 221 times
- Go sideways: Metric tensorA closely related topic
- Go sideways: Covariant derivativeA closely related topic
📖 Definitions
Noun a function mapping a subset of ℝn into a subset of ℝn
Noun a function which associates, to each point on a surface, a vector in the tangent plane of that point
🗂️ The subject at a glance
🏆 Landmark papers (most cited)
Inflation driven by a vector field
Vector field design on surfaces
Discrete multiscale vector field decomposition
Vector field based shape deformations
Detecting boundaries in a vector field
Visualizing nonlinear vector field topology
Vector field as a quintessence partner
Contact metric manifolds whose characteristic vector field is a harmonic vector field
🆕 Influential since 2022
📚 Books worth reading
Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector FieldsJohn Guckenheimer, Philip Holmes · 1983- 📕Rings with minimum conditionEmil Artin · 1944
Advanced engineering mathematicsGreenberg, Michael D. · 1988
Scalar, Vector, and Matrix MathematicsDennis S. Bernstein · 2018- 📕Measurements of solar vector magnetic fieldsM. J. Hagyard · 1985
Normal forms and bifurcation of planar vector fieldsShui-Nee Chow · 1994
Vector analysisN. Kemmer · 1977
Complex vector functional equationsIce Risteski · 2001
Ranked by number of editions on Open Library, a rough measure of how lasting a book is.
🔗 Related topics
- Metric tensor In the mathematical field of differential geometry, a metric tensor (or simply metric) is an additional structure on a manifold M (such as a surface) that…
- Covariant derivative In mathematics, the covariant derivative is a way of specifying a derivative along tangent vectors of a manifold. Alternatively, the covariant derivative…
- Tangent space In mathematics, the tangent space of a manifold is a generalization of tangent lines to curves in two-dimensional space and tangent planes to surfaces…
- Finsler manifold In mathematics, particularly differential geometry, a Finsler manifold is a differentiable manifold M where a (possibly asymmetric) Minkowski norm F(x…
- Tangent bundle A tangent bundle is the collection of all of the tangent spaces for all points on a manifold, structured in a way that it forms a new manifold itself.…
- Tensor field In mathematics and physics, a tensor field is a function assigning a tensor to each point of a region of a mathematical space (typically a Euclidean space…
- Differentiable manifold In mathematics, a differentiable manifold (also differential manifold) is a type of manifold that is locally similar enough to a vector space to allow…
- Normal (geometry) In geometry, a normal is an object (e.g. a line, ray, or vector) that is perpendicular to a given object. For example, the normal line to a plane curve…