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This is just a notational question, to see if there is a better way to write the following quantities:

  • $ v_1 = (f_{y} + f_{z}, f_{x}+f_{z}, f_{x} + f_{y} )$
  • $ v_2 = (f_{yy} + f_{zz}, f_{xx}+f_{zz}, f_{xx} + f_{yy} )$
  • $ t_1 = f_{xy}f_xf_y + f_{xz}f_xf_z + f_{yz}f_yf_z $

where $f_i$ is the partial derivative with respect to $i$, for $f:\mathbb{R}^3\rightarrow \mathbb{R}$. So, $v_1(x,y,z),v_2(x,y,z)\in\mathbb{R}^3$ are vector fields, while $t_1(x,y,z)\in\mathbb{R}$ is a scalar field.

How can I write these with common operators on $f$ (e.g. $\nabla,\times,\cdot,\Delta,\nabla\cdot,\nabla\times,\otimes$ and so on) and/or are there some operators I am missing?

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  • $\begingroup$ Question, What kinds of objects are these things? $\endgroup$ – Triatticus Mar 16 '17 at 22:41
  • $\begingroup$ @Triatticus They pop out as part of some larger expressions, from some differential geometry calculations $\endgroup$ – user3658307 Mar 16 '17 at 23:07
  • $\begingroup$ I was wondering if these were vectors, scalars, maybe higher order tensors $\endgroup$ – Triatticus Mar 16 '17 at 23:14
  • $\begingroup$ @Triatticus Oh, sorry. I have edited to clarify a bit. Is that alright? $\endgroup$ – user3658307 Mar 16 '17 at 23:20
  • $\begingroup$ Ah now its much more clear what we are dealing with $\endgroup$ – Triatticus Mar 17 '17 at 0:08

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