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In Riemannian geometry, a vector flow can be thought of as a solution to the system of differential equations induced by a vector field. [1] That is, if a (conservative) vector field is a map to the tangent space, it represents the tangent vectors to some function at each point. Splitting the tangent vectors into directional derivatives, one ...
Informally, a flow may be viewed as a continuous motion of points over time. More formally, a flow is a group action of the real numbers on a set. The idea of a vector flow, that is, the flow determined by a vector field, occurs in the areas of differential topology, Riemannian geometry and Lie groups.
Gradient vector flow (GVF) is the process that spatially extends the edge map gradient vectors, yielding a new vector field that contains information about the location of object edges throughout the entire image domain. GVF is defined as a diffusion process operating on the components of the input vector field.
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 ...
Let X and Y be smooth time dependent vector fields and the flow of X.The following identity can be proved: | = (,) = (, ([,] + | =)) Also, we can define time dependent tensor fields in an analogous way, and prove this similar identity, assuming that is a smooth time dependent tensor field:
In physics, the Poynting vector (or Umov–Poynting vector) represents the directional energy flux (the energy transfer per unit area, per unit time) or power flow of an electromagnetic field. The SI unit of the Poynting vector is the watt per square metre (W/m 2 ); kg/s 3 in SI base units .
Intuitively, the behaviour of the flow at each point corresponds to the "direction" indicated by the vector field. It is a natural question to ask whether one may establish a similar correspondence between vector fields and more arbitrary Lie group actions on M {\displaystyle M} .
This equation says that the vector tangent to the curve at any point x(t) along the curve is precisely the vector F(x(t)), and so the curve x(t) is tangent at each point to the vector field F. If a given vector field is Lipschitz continuous, then the Picard–Lindelöf theorem implies that there exists a unique flow for small time.