Definition
The material derivatives of a scalar field φ( x, t ) and a vector field u( x, t ) are defined respectively as:
where the distinction is that is the gradient of a scalar, while is the covariant derivative of a vector. In case of the material derivative of a vector field, the term v•∇u can both be interpreted as v•(∇u) involving the tensor derivative of u, or as (v•∇)u, leading to the same result.
Confusingly, the term convective derivative is both used for the whole material derivative Dφ/Dt or Du/Dt, and for only the spatial rate of change part, v•∇φ or v•∇u respectively. For that case, the convective derivative only equals D/Dt for time independent flows.
These derivatives are physical in nature and describe the transport of a scalar or vector quantity in a velocity field v( x, t ). The effect of the time independent terms in the definitions are for the scalar and vector case respectively known as advection and convection.
Read more about this topic: Material Derivative
Famous quotes containing the word definition:
“Mothers often are too easily intimidated by their children’s negative reactions...When the child cries or is unhappy, the mother reads this as meaning that she is a failure. This is why it is so important for a mother to know...that the process of growing up involves by definition things that her child is not going to like. Her job is not to create a bed of roses, but to help him learn how to pick his way through the thorns.”
—Elaine Heffner (20th century)
“... we all know the wag’s definition of a philanthropist: a man whose charity increases directly as the square of the distance.”
—George Eliot [Mary Ann (or Marian)
“The man who knows governments most completely is he who troubles himself least about a definition which shall give their essence. Enjoying an intimate acquaintance with all their particularities in turn, he would naturally regard an abstract conception in which these were unified as a thing more misleading than enlightening.”
—William James (1842–1910)