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 $\phi (x,t)$ is:

${\frac {D\varphi }{Dt}}={\frac {\partial \varphi }{\partial t}}+\mathbf {v} \cdot \nabla \varphi ,$

where $\nabla \varphi$ is the gradient of the scalar.

For a vector field Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle u(x,t)} it is defined as:

Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle {\frac {D\mathbf {u} }{Dt}}={\frac {\partial \mathbf {u} }{\partial t}}+\mathbf {v} \cdot \nabla \mathbf {u} ,}

where Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle \nabla \mathbf {u} } is the covariant derivative of a vector.

In case of the material derivative of a vector field, the term $\mathbf {v} \cdot \nabla \mathbf {u}$ can both be interpreted as Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle \mathbf {v} \cdot (\nabla \mathbf {u} )} , involving the tensor derivative of u, or as Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle (\mathbf {v} \cdot \nabla )\mathbf {u} } , 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