I know that the operator $\nabla$ denotes $$ \nabla = \begin{pmatrix} \frac{\partial}{\partial x} \\ \frac{\partial}{\partial y} \end{pmatrix} $$

Is there some kind of similar notation to denote

$$ A = \begin{pmatrix} \frac{\partial}{\partial u_x} \\ \frac{\partial}{\partial u_y} \end{pmatrix} $$

where $u_x = \frac{\partial}{\partial x} u$, likewise $u_y$. Here $u$ denotes some differentiable function in $\Omega \subset \mathbb{R}^2$.

  • $\begingroup$ Like the gradient on the fibre of the tangent bundle? $\endgroup$ – Troy Woo Aug 1 '18 at 9:28
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    $\begingroup$ I'm not exactly sure what you're asking, but the directional derivative along a vector field $\mathbf v = (v_x, v_y)$ is usually denoted $(\mathbf v \cdot \nabla)$, which means $v_x \frac{\partial}{\partial x} + v_y \frac{\partial}{\partial y}$. This is used to define the material derivative in continuum mechanics. $\endgroup$ – Rahul Aug 1 '18 at 9:30
  • $\begingroup$ @Rahul, No, I'm looking for a shorthand for $$A = \frac{\partial}{\partial x} \frac{\partial}{\partial v_x} + \frac{\partial}{\partial y} \frac{\partial}{\partial v_y}$$ $\endgroup$ – user8469759 Aug 1 '18 at 9:35

It is fairly common to use the notation $\nabla_{\mathbf{u}}$ for gradient with respect to the velocity field. An alternative notation is $\partial/\partial{\mathbf{u}}$ so that the usual gradient is written as $\partial/\partial{\mathbf{x}}$.

  • $\begingroup$ If I applied $\nabla_{\bf{u}} f$ wouldn't I get $\langle \nabla f, \bf{u} \rangle $? $\endgroup$ – user8469759 Aug 1 '18 at 9:41
  • $\begingroup$ No, you should get $\hat{e}_x\partial f/\partial u_x + \hat{e}_y\partial f/\partial u_y$ in two dimensions. $\endgroup$ – Amey Joshi Aug 1 '18 at 9:43
  • $\begingroup$ But couldn't $\nabla_\bf{u}$ be confused with the directional derivative? $\endgroup$ – user8469759 Aug 1 '18 at 9:45
  • $\begingroup$ I did mean the directional derivative. I gave an analog of the nabla operator. If it is not a directional derivative with respect to a velocity field, why would want to write it as a vector. Can you possible write an expression in a long form that you want to write in a compact notation? $\endgroup$ – Amey Joshi Aug 1 '18 at 9:49
  • $\begingroup$ Euler Lagrange equations are an example. I want ti write to gradient wrt a different basis I think (instead of the canonical basis I'd use the velocity field I think). $\endgroup$ – user8469759 Aug 1 '18 at 9:52

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