What is material time derivative?

In continuum mechanics, the material derivative describes the time rate of change of some physical quantity (like heat or momentum) of a material element that is subjected to a space-and-time-dependent macroscopic velocity field.

What is the difference between covariant derivative and Lie derivative?

Hopefully this illustrates the big differences between the two derivatives: the covariant derivative should be used to measure whether a tensor is parallel transported, while the Lie derivative measures whether a tensor is invariant under diffeomorphisms in the direction of the vector ξa.

Is material derivative Lagrangian or Eulerian?

A material derivative is the time derivative – rate of change – of a property following a fluid particle ‘p’. The material derivative is a Lagrangian concept. By expressing the material derivative in terms of Eulerian quantities we will be able to apply the conservation laws in the Eulerian reference frame.

Is material derivative total derivative?

The material derivative computes the time rate of change of any quantity such as temperature or velocity (which gives acceleration) for a portion of a material moving with a velocity, v . They include total derivative, convective derivative, substantial derivative, substantive derivative, and still others.

Is Lie derivative covariant?

On the other hand the Lie derivative satisfies LfXY=fLXY−(Xf)Y, so it cannot be a covariant derivative.

What does the material derivative do?

The material derivative computes the time rate of change of any quantity such as temperature or velocity (which gives acceleration) for a portion of a material moving with a velocity, v . If the material is a fluid, then the movement is simply the flow field.

What is a total time derivative?

In mathematics, the total derivative of a function f at a point is the best linear approximation near this point of the function with respect to its arguments. Unlike partial derivatives, the total derivative approximates the function with respect to all of its arguments, not just a single one.

How do you take a derivative of a lie?

1 Answer

1. The Lie derivative of a smooth function f:M→R with respect to a tangent vector X∈Tp(M) at a point p is just the directional derivative of f with respect to X at p.
2. =x2(2x)+2y2(0)
3. In general, if X is a smooth vector field on M, then a smooth function f:M→R results in a smooth function X(f):M→R.