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In continuum mechanics, why does local derivative and convective derivative only apply to spatial descriptions?

Is it because in the material description, the motion focuses on the particles, and not on fixed spots in space, and in each particle's motion the time and space are simultaneously changing, so we can't perform derivative w.r.t time and space separately?

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Yes, your second paragraph is a pretty good natural-language explanation.

The way I'd have put it is as follows:

A partial derivative is a derivative with respect to (something), calculated on the assumption that (something else) is held constant. Specifically, the "local derivative" is defined as the derivative with respect to time, calculated on the assumption that the location in space under observation is held constant; the "material derivative" is defined as the derivative with respect to time, calculated on the assumption that the identity of the parcel of fluid under observation is held constant; and the "convective derivative" is defined in some texts as being the same thing as the material derivative, and in other texts as being the difference between the material derivative and the local derivative. The material derivative of velocity is the same thing as the "acceleration" in Newton's second law, so when one writes Newton's second law for a fluid (which is what the Navier-Stokes momentum equation is) in terms of the material derivative of velocity, the "mass times acceleration" side of the equation appears as a simple, single term; whereas when one writes Newton's second law for a fluid in terms of the local derivative of velocity, one has to add something on to the local derivative of velocity to obtain the acceleration, hence ending up with two terms on the "mass times acceleration" side of the equation.

(A little linguistic note of caution: the verb "derive" does not mean "compute a derivative".)

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