The expression on the left side is a material derivative.
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By expanding the material derivative, the equations become:
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Note that the Lagrangian rate of change is also known as the material derivative.
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The portion of the material derivative represented by the spatial derivatives is called the convective derivative.
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It turns out, however, that many physical concepts can be written concisely with the material derivative.
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Where \ mathbf v is the velocity, V the volume, and the derivatives of and are material derivatives.
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Mathematically, this constraint implies that the material derivative ( discussed below ) of the density must vanish to ensure incompressible flow.
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When the flow is incompressible, does not change for any fluid particle, and its material derivative vanishes : 0 } }.
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This equation expresses the material derivative of the field, that is, the derivative with respect to a coordinate system attached to the moving surface.
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"' Objectivity rates "'are modified material derivatives that allows to have an "'objective "'time differentiation.
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