dVdx

In mathematics and physics, dV/dx denotes the rate at which a velocity field changes with respect to a spatial coordinate x. If V is a scalar speed, dV/dx is the spatial derivative of that speed along x. If V is a vector field V(x, y, z, t), its derivative with respect to x is a vector of partial derivatives ∂V/∂x, with components such as dvx/dx, dvy/dx, and dvz/dx.

In fluid dynamics and continuum mechanics this derivative is part of the velocity gradient tensor ∇V, whose components are ∂Vi/∂xj. The single component dvx/dx describes how the x-component of velocity varies along the x direction; a positive value means the x-velocity increases with x. The full gradient tensor provides a local description of deformation and rotation: the symmetric part ∑ describes the rate of deformation or strain-rate tensor E = (∇V + ∇V^T)/2, while the antisymmetric part relates to rotation, linked to the vorticity ω = ∇ × V.

In motion analysis, the material (substantial) derivative a = DV/Dt = ∂V/∂t + (V · ∇)V gives the acceleration of a moving fluid particle. Its x-component involves ∂Vx/∂t and terms like Vx ∂Vx/∂x, Vy ∂Vx/∂y, and Vz ∂Vx/∂z, among others.

Examples help intuition: for a simple field V = (a x, b y, c z), dvx/dx = a, while in uniform flow dvx/dx = 0. In non-Cartesian coordinates, the derivative generalizes to covariant forms that account for curvature and metric factors.

See also: gradient, velocity field, velocity gradient tensor, strain rate, vorticity, Navier–Stokes equations.