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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>.
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| The material derivative of a scalar field <math>\phi( x, t )</math> is:
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| :<math> \frac{D\varphi}{Dt} = \frac{\partial \varphi}{\partial t} + \mathbf{v}\cdot\nabla \varphi,
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| </math>
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| where <math> \nabla \varphi</math> is the gradient of the scalar.
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| For a vector field <math>u( x, t )</math> it is defined as:
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| :<math> \frac{D\mathbf{u}}{Dt} = \frac{\partial \mathbf{u}}{\partial t} + \mathbf{v}\cdot\nabla \mathbf{u},
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| </math>
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| where <math>\nabla \mathbf{u}</math> is the covariant derivative of a vector.
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| 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.
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| ==Alternative names==
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| 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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| *'''derivative following the motion''' | | *'''derivative following the motion''' |
| *'''total derivative''' | | *'''total derivative''' |
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| The notation varies likewise, with some authors retaining the usual ''d'' instead of ''D''.
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| ==References==
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| <references/>
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| [[Category: mathematics]]
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