3d Vector Fields

[Emmanuel Des Meaux]

Invector calculus and physics, a vector field is an assignment of a vector to each point in a space, most commonly Euclidean space R n \displaystyle \mathbb R ^n .[1] 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 are often 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.

The elements of differential and integral calculus extend naturally to vector fields. When a vector field represents force, the line integral of a vector field represents the work done by a force moving along a path, and under this interpretation conservation of energy is exhibited as a special case of the fundamental theorem of calculus. Vector fields can usefully be thought of as representing the velocity of a moving flow in space, and this physical intuition leads to notions such as the divergence (which represents the rate of change of volume of a flow) and curl (which represents the rotation of a flow).

A vector field is a special case of a vector-valued function, whose domain's dimension has no relation to the dimension of its range; for example, the position vector of a space curve is defined only for smaller subset of the ambient space.Likewise, n coordinates, a vector field on a domain in n-dimensional Euclidean space R n \displaystyle \mathbb R ^n can be represented as a vector-valued function that associates an n-tuple of real numbers to each point of the domain. This representation of a vector field depends on the coordinate system, and there is a well-defined transformation law (covariance and contravariance of vectors) in passing from one coordinate system to the other.

Vector fields are often discussed on open subsets of Euclidean space, but also make sense on other subsets such as surfaces, where they associate an arrow tangent to the surface at each point (a tangent vector).More generally, vector fields are defined on differentiable manifolds, which are spaces that look like Euclidean space on small scales, but may have more complicated structure on larger scales. In this setting, a vector field gives a tangent vector at each point of the manifold (that is, a section of the tangent bundle to the manifold). Vector fields are one kind of tensor field.

In physics, a vector is additionally distinguished by how its coordinates change when one measures the same vector with respect to a different background coordinate system. The transformation properties of vectors distinguish a vector as a geometrically distinct entity from a simple list of scalars, or from a covector.

Such a transformation law is called contravariant. A similar transformation law characterizes vector fields in physics: specifically, a vector field is a specification of n functions in each coordinate system subject to the transformation law (1) relating the different coordinate systems.

Vector fields are thus contrasted with scalar fields, which associate a number or scalar to every point in space, and are also contrasted with simple lists of scalar fields, which do not transform under coordinate changes.

Since orthogonal transformations are actually rotations and reflections, the invariance conditions mean that vectors of a central field are always directed towards, or away from, 0; this is an alternate (and simpler) definition. A central field is always a gradient field, since defining it on one semiaxis and integrating gives an antigradient.

A common technique in physics is to integrate a vector field along a curve, also called determining its line integral. Intuitively this is summing up all vector components in line with the tangents to the curve, expressed as their scalar products. For example, given a particle in a force field (e.g. gravitation), where each vector at some point in space represents the force acting there on the particle, the line integral along a certain path is the work done on the particle, when it travels along this path. Intuitively, it is the sum of the scalar products of the force vector and the small tangent vector in each point along the curve.

with the obvious generalization to arbitrary dimensions. The divergence at a point represents the degree to which a small volume around the point is a source or a sink for the vector flow, a result which is made precise by the divergence theorem.

The curl measures the density of the angular momentum of the vector flow at a point, that is, the amount to which the flow circulates around a fixed axis. This intuitive description is made precise by Stokes' theorem.

The index of the vector field as a whole is defined when it has just finitely many zeroes. In this case, all zeroes are isolated, and the index of the vector field is defined to be the sum of the indices at all zeroes.

For an ordinary (2-dimensional) sphere in three-dimensional space, it can be shown that the index of any vector field on the sphere must be 2. This shows that every such vector field must have a zero. This implies the hairy ball theorem.

In recent decades many phenomenological formulations of irreversible dynamics and evolution equations in physics, from the mechanics of complex fluids and solids to chemical kinetics and quantum thermodynamics, have converged towards the geometric idea of "steepest entropy ascent" or "gradient flow" as a consistent universal modeling framework that guarantees compatibility with the second law of thermodynamics and extends well-known near-equilibrium results such as Onsager reciprocity to the far-nonequilibrium realm.[5]

Consider the flow of a fluid through a region of space. At any given time, any point of the fluid has a particular velocity associated with it; thus there is a vector field associated to any flow. The converse is also true: it is possible to associate a flow to a vector field having that vector field as its velocity.

By definition, a vector field on M \displaystyle M is called complete if each of its flow curves exists for all time.[6] In particular, compactly supported vector fields on a manifold are complete. If X \displaystyle X is a complete vector field on M \displaystyle M , then the one-parameter group of diffeomorphisms generated by the flow along X \displaystyle X exists for all time; it is described by a smooth mapping

The flows associated to two vector fields need not commute with each other. Their failure to commute is described by the Lie bracket of two vector fields, which is again a vector field. The Lie bracket has a simple definition in terms of the action of vector fields on smooth functions f \displaystyle f :

Algebraically, vector fields can be characterized as derivations of the algebra of smooth functions on the manifold, which leads to defining a vector field on a commutative algebra as a derivation on the algebra, which is developed in the theory of differential calculus over commutative algebras.

If we want significantly more points plotted, then it is usually best to use a computer aided graphing system such as Maple or Mathematica. Here is a sketch with many more vectors included that was generated with Mathematica.

Notice that the vectors of the vector field are all orthogonal (or perpendicular) to the contours. This will always be the case when we are dealing with the contours of a function as well as its gradient vector field.

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Single-cell (sc)RNA-seq, together with RNA velocity and metabolic labeling, reveals cellular states and transitions at unprecedented resolution. Fully exploiting these data, however, requires kinetic models capable of unveiling governing regulatory functions. Here, we introduce an analytical framework dynamo ( -release), which infers absolute RNA velocity, reconstructs continuous vector fields that predict cell fates, employs differential geometry to extract underlying regulations, and ultimately predicts optimal reprogramming paths and perturbation outcomes. We highlight dynamo's power to overcome fundamental limitations of conventional splicing-based RNA velocity analyses to enable accurate velocity estimations on a metabolically labeled human hematopoiesis scRNA-seq dataset. Furthermore, differential geometry analyses reveal mechanisms driving early megakaryocyte appearance and elucidate asymmetrical regulation within the PU.1-GATA1 circuit. Leveraging the least-action-path method, dynamo accurately predicts drivers of numerous hematopoietic transitions. Finally, in silico perturbations predict cell-fate diversions induced by gene perturbations. Dynamo, thus, represents an important step in advancing quantitative and predictive theories of cell-state transitions.

Vector fields can model velocity, magnetic force, fluid motion, and gradients. Visualize vector fields in a 2-D or 3-D view using the quiver, quiver3, and streamline functions. You can also display vectors along a horizontal axis or from the origin.

As we all know, the space of invariant vector fields on a Lie group can be identified with the tangent space at the identity (or any other point for that matter). My question is: How does this generalize to homogeneous spaces? My guess would be that one can equate the space with the tangent space at any point point. However, it is not clear to me why everything should carry over smoothly to this more general setting.

Now, on to $H$ invariant vector fields. I haven't thought much about this, but even in this case, there are (often? always?) not enough. For example, viewing $S^2 = SO(3)/SO(2)$, we see that $H$ is a circle. In this case, viewing the north pole of $S^2$ as $eSO(2)$, the only $H$ invariant vector fields are the velocity vectors of rotation through the north-south axis - i.e., the flows are lines of lattitude.

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