Editing Substantive derivative

The '''substantive derivative''' is a derivative taken along a path moving with velocity v, and is often used in fluid mechanics and [[classical mechanics]]. It describes the time rate of change of some quantity (such as [[heat]] or momentum) by following it, while moving with a – space and time dependent – velocity field.

The material derivative of a scalar field <math>\phi( x, t )</math> is:
:<math>    \frac{D\varphi}{Dt} = \frac{\partial \varphi}{\partial t} + \mathbf{v}\cdot\nabla \varphi,
</math>
where <math> \nabla \varphi</math> is the gradient of the scalar.

For a vector field <math>u( x, t )</math> it is defined as:
:<math>    \frac{D\mathbf{u}}{Dt} = \frac{\partial \mathbf{u}}{\partial t} + \mathbf{v}\cdot\nabla \mathbf{u},
</math>
where <math>\nabla \mathbf{u}</math> is the covariant derivative of a vector.

In case of the material derivative of a vector field, the term <math>\mathbf{v} \cdot \nabla \mathbf{u}</math> can both be interpreted as <math>\mathbf{v} \cdot (\nabla \mathbf{u})</math>, involving the tensor derivative of u, or as <math>(\mathbf{v} \cdot \nabla) \mathbf{u}</math>, leading to the same result.

==Alternative names==
There are many other names for this operator, including:
*'''material derivative'''
*'''convective derivative'''
*'''advective derivative'''
*'''substantive derivative'''
*'''substantial derivative'''
*'''Lagrangian derivative'''
*'''Stokes derivative'''
*'''particle derivative'''
*'''hydrodynamic derivative'''
*'''derivative following the motion'''
*'''total derivative'''

The notation varies likewise, with some authors retaining the usual ''d'' instead of ''D''.

==References==
<references/>
[[Category: mathematics]]
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	The '''substantive derivative''' is a derivative taken along a path moving with velocity v, and is often used in fluid mechanics and [[classical mechanics]]. It describes the time rate of change of some quantity (such as [[heat]] or momentum) by following it, while moving with a – space and time dependent – velocity field~~. Note that the familiar <math>d</math> now becomes <math>D</math>~~.		The '''substantive derivative''' is a derivative taken along a path moving with velocity v, and is often used in fluid mechanics and [[classical mechanics]]. It describes the time rate of change of some quantity (such as [[heat]] or momentum) by following it, while moving with a – space and time dependent – velocity field.

	The material derivative of a scalar field <math>\phi( x, t )</math> is:		The material derivative of a scalar field <math>\phi( x, t )</math> is: