Material derivative

The material derivative is the rate of change of a field following a moving point in a continuum. It combines change at a fixed location with change caused by motion through spatial gradients.

Local derivative: The time rate of change of a field measured at a fixed spatial location. It is the fixed-location contribution to the material derivative.
Continuum mechanics: The study of matter modeled as continuously distributed rather than as separate particles. The material derivative describes rates for fields in this continuum framework.
Navier–Stokes equations: The equations governing momentum conservation in a viscous fluid. Their acceleration term is the material derivative of velocity.
Partial derivative: The derivative of a multivariable function with respect to one variable while holding others fixed. It captures only fixed-position change, not transport through spatial variation.
Convective derivative: The part of a material derivative caused by motion through spatial variations in a field. It isolates the transport contribution added to local change.
Scalar field: A function assigning a scalar value to each point in space and time. Temperature and pressure are scalar fields whose rates can be followed.
Fluid acceleration: The rate of change of a fluid particle's velocity as it moves. It is the material derivative when the field is velocity.
Total derivative: The derivative of a quantity that depends on a variable both directly and through other variables. It supplies the chain-rule framework for the material derivative.
Velocity field: A vector field assigning a velocity to each point in a region of space and time. It determines how a moving point travels through the field.
Vector field: A function assigning a vector to each point in space and time. The same derivative applies componentwise to vector quantities such as velocity.

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