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Material Derivative

The material derivative of a scalar field or vector field F(x,t) is the rate of change measured while following a moving material element. If the element follows a path x(t) with velocity v, then

(DF)/(Dt)=d/(dt)F(x(t),t)=(partialF)/(partialt)+v·del F.

The first term is the local rate of change at a fixed point, while the second is the change caused by motion through spatial variation in F. The material derivative is also called the convective derivative. Applied to a velocity field, it gives the convective acceleration

(Dv)/(Dt)=(partialv)/(partialt)+(v·del )v,

where the second summand is the convective term.

See also

Convective Acceleration, Convective Derivative, Convective Term, Euler's Equations of Inviscid Motion, Navier-Stokes Equations, Velocity

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References

Batchelor, G. K. An Introduction to Fluid Dynamics. Cambridge, England: Cambridge University Press, p. 73, 1977.Quanta Magazine. "Why Navier-Stokes Pushes Math and Physics to the Edge." Sep. 17, 2026. https://www.youtube.com/watch?v=PsWR1rQEWYo.Tritton, D. J. "The Substantive Derivative." §5.5 in Physical Fluid Dynamics, 2nd ed. Oxford, England: Clarendon Press, pp. 53-55, 1989.

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Material Derivative

Cite this as:

Weisstein, Eric W. "Material Derivative." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/MaterialDerivative.html

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