On pg 2, the fluid velocity in the Navier-Stokes system of equations is noted as:

$v(x,t) \equiv (v^1, v^2, \ldots, v^N)^t$,

where I am assuming that the velocity vector field is time-dependent. The same notation appears for the gradient operator, $\nabla$:

$\nabla = \left( \frac{\partial}{\partial x_1}, \frac{\partial}{\partial x_2}, \ldots, \frac{\partial}{\partial x_N} \right)^t$,

yet, when introduced just after the above, the Laplace operator, $\Delta$, has no such time-dependence notation:

$\Delta = \sum_{j=1}^N \frac{\partial^2}{\partial {x_j}^2}$.

Can anyone please explain this inconsistency in notation? Thank you!

  • $\begingroup$ The $t$ superscript doesn't mean "time-dependent". It means "transpose". Which is good, because the gradient operator shouldn't be time-dependent. $\endgroup$
    – mjqxxxx
    Dec 28, 2013 at 3:48
  • $\begingroup$ Oh I see. So it's just like a column vector? $\endgroup$ Dec 28, 2013 at 4:04

1 Answer 1


(Answer to remove from unanswered list)

In this book, when $t$ appears as a superscript, it means the transpose most of the time. Thus $$ v(x,t) = (v^1,v^2,\dots,v^N)^t = \begin{pmatrix}v^1(x,t)\\v^2(x,t)\\ \vdots\\ v^N(x,t)\end{pmatrix} , \text{ and} \\ \nabla = \left( \frac{\partial}{\partial x_1}, \frac{\partial}{\partial x_2}, \ldots, \frac{\partial}{\partial x_N} \right)^t = \begin{pmatrix}\frac{\partial}{\partial x_1}\\ \frac{\partial}{\partial x_2} \\ \vdots \\ \frac{\partial}{\partial x_N}\end{pmatrix}.$$

It is unfortunate that $t$ is also used to denote the time variable, but it is what it is.


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