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Vector Fields
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The velocity of a fluid at each point of space-time is a vector field (or flow). It is best to think of it in terms of the effect of fluid flow on some scalar field. A vector field is thus a first order partial differential operator, called the material derivative in fluid mechanics. The path of a speck of dust carried along (advected) by the fluid is the integral curve of the velocity field. Even simple vector fields can have quite complicated integral curves: a manifestation of chaos. Of special interest are incompressible (with zero divergence) and irrotational (with zero curl) flows. A fixed point of a vector field is a point at which it vanishes. The derivative of a vector field at a fixed point is a matrix (the Jacobi matrix) whose spectrum is independent of the choice of coordinates.
Title: Vector Fields
Description:
The velocity of a fluid at each point of space-time is a vector field (or flow).
It is best to think of it in terms of the effect of fluid flow on some scalar field.
A vector field is thus a first order partial differential operator, called the material derivative in fluid mechanics.
The path of a speck of dust carried along (advected) by the fluid is the integral curve of the velocity field.
Even simple vector fields can have quite complicated integral curves: a manifestation of chaos.
Of special interest are incompressible (with zero divergence) and irrotational (with zero curl) flows.
A fixed point of a vector field is a point at which it vanishes.
The derivative of a vector field at a fixed point is a matrix (the Jacobi matrix) whose spectrum is independent of the choice of coordinates.
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