# Vector Calculus Notation for “Cross” Quantities

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?

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