Objective time derivative

An objective time derivative transforms covariantly under time-dependent rigid changes of observer, so constitutive predictions do not depend on the observer's rotation.

The upper-convected derivative of a contravariant tensor is $\stackrel{▽}{\mathbf{C}}=D\mathbf{C}/Dt-(\nabla \mathbf{u})\mathbf{C}-\mathbf{C}(\nabla \mathbf{u}{)}^T$.

The lower-convected derivative of a covariant tensor is $\stackrel{△}{\mathbf{C}}=D\mathbf{C}/Dt+(\nabla \mathbf{u}{)}^T\mathbf{C}+\mathbf{C}\nabla \mathbf{u}$.

The Jaumann derivative is the corotational objective derivative $\stackrel{\circ }{\mathbf{C}}=D\mathbf{C}/Dt-\mathbf{\Omega }\mathbf{C}+\mathbf{C}\mathbf{\Omega }$, where $\mathbf{\Omega }$ is the spin tensor.