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In three dimensions area is a vector because two dimensions have a direction relative to the third. If the world had four spatial dimensions then area would be a tensor?

And what form then the laws of physics which imply the concept of area, as electromagnetism or the definition of pressure, would have?

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    $\begingroup$ It's not the concept of area that changes, it's the relation between vectors and pseudovectors that changes. $\endgroup$
    – NDewolf
    Mar 9, 2021 at 11:10
  • $\begingroup$ I didn't want to put it in answer since it does not address your more important question of "how the laws of physics change" $\endgroup$
    – NDewolf
    Mar 9, 2021 at 13:18
  • $\begingroup$ Related: physics.stackexchange.com/q/130098/50583, physics.stackexchange.com/q/313091/50583 $\endgroup$
    – ACuriousMind
    Mar 9, 2021 at 16:39
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    $\begingroup$ Regarding physics, you would need to distinguish laws that use 2 dimensions (e.g. based on rotations) and laws that use n-1 dimensions (e.g. based on enclosing). $\endgroup$
    – M.S.
    Mar 11, 2021 at 16:44
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    $\begingroup$ I'd say that a more accurate heuristic is that area is a rank-2 tensor in any number of dimensions, but in three dimensions (only) it can also be throught of as a vector. $\endgroup$
    – tparker
    Mar 13, 2021 at 4:02

1 Answer 1

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Roughly speaking, Yes!

In 3 spatial dimensions a 2D thing (an area) uses 2 of the 3 available dimensions. So even though it isn't really a vector (an area should have twice the units of a vector) an area can be described by picking out the single direction that is orthogonal to it.

In 4D an area has a 2D space orthogonal to it, so one needs a 2-index thing (tensor) to specify its orthogonal space, or you could equally use 2-indicies to specify its actual (own) space.

For more details this wiki-page is very clear: https://en.wikipedia.org/wiki/Exterior_algebra

If you have some number of dimensions then a thing (eg. A volume) can be denoted either using either the number of dimensions it has, or the number it is missing. Not only is this why a 2D area in 3D can be specified by a 1D vector (3-2=1), it is also why a Volume in a 3D space can be represented by a scalar (3-3=0). In 4D 3-volume is a vector quantity.

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