Substantive derivative - Revision history

Dduque: /* Alternative names */

2010-05-21T10:45:07Z

Alternative names

← Older revision		Revision as of 12:45, 21 May 2010
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	*'''derivative following the motion'''		*'''derivative following the motion'''
	*'''total derivative'''		*'''total derivative'''

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

	==References==		==References==
	<references/>		<references/>
	[[Category: mathematics]]		[[Category: mathematics]]

Carl McBride: Added a note on nomenclature: d vs D

2010-05-19T12:11:35Z

Added a note on nomenclature: d vs D

← Older revision		Revision as of 14:11, 19 May 2010
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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.		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 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:

Carl McBride: Added two internal links and changed category

2010-05-19T12:05:34Z

Added two internal links and changed category

← Older revision		Revision as of 14:05, 19 May 2010
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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.		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:
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	</math>		</math>
	where <math> \nabla \varphi</math> is the gradient of the scalar.		where <math> \nabla \varphi</math> is the gradient of the scalar.


	For a vector field <math>u( x, t )</math> it is defined as:		For a vector field <math>u( x, t )</math> it is defined as:
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	where <math>\nabla \mathbf{u}</math> is the covariant derivative of a vector.		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.~~[7]~~		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.

	==~~Names~~==		==Alternative names==
	There are many other names for this operator, including:		There are many other names for this operator, including:
	*'''material derivative'''		*'''material derivative'''
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	==References==		==References==
	<references/>		<references/>
	[[Category: ~~classical mechanics~~]]		[[Category: mathematics]]

Dduque: Formulas added

2010-05-19T11:12:27Z

Formulas added

@@ Line 1: / Line 1: @@
 There are many other names for this operator, including:
 *'''material derivative'''
@@ Line 11: / Line 27: @@
 *'''derivative following the motion'''
 *'''total derivative'''

Dduque: new page. Some names

2010-05-19T11:07:08Z

new page. Some names

New page

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'''