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The difference between "multivariable" and "vector" calculus, as stated on Yale's website, is that multivariable would only go through 2 or 3 dimensions, and so would rely heavily on geometric analysis. Vector calculus on the other hand generalizes to n dimensions, which requires linear algebra as a prerequisite.

I was wondering if n-dimensions would be necessary to conduct condensed matter physics research.

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    $\begingroup$ The more math, and the deeper the math, you know, the better able to attack problems you will be. Linear algebra and vector calculus will be immensely useful in any hard science you wish to go into (and at this point, given where you are in math, you have no idea if condensed matter physics is for you). $\endgroup$
    – Jon Custer
    Commented Aug 28, 2015 at 19:05
  • $\begingroup$ This is really asking for opinions, which is something we don't do here. $\endgroup$
    – Kyle Kanos
    Commented Aug 29, 2015 at 0:30
  • $\begingroup$ This is not at all an opinion. You can guarantee an answer to whether you would ever need more than three dimensions in condensed matter research, without any leeway. The fact that it's a question that doesn't need any math to solve doesn't mean that it is an opinion. $\endgroup$
    – Striker
    Commented Aug 29, 2015 at 14:05

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In the long run I don't think it matters much which of the two you study now. If you truly understand calculus in 2-3 dimensions, you won't have too much trouble generalizing your understanding to $N$ dimensions. On the other hand, if you want to do research in condensed matter, you will need linear algebra anyway, so there's no harm in picking up that topic as well. Given that linear algebra and $N$-variable calculus are pretty much the bread and butter of physics, you can't possibly do yourself any disservice by trying to gain as deep an understanding of those topics as possible if physics research is your goal (provided you think you can keep up with the course material and workload).

As to whether you might encounter more than 3 dimensions (variables) in condensed matter research, the answer is a resounding 'yes'. For instance, if you're studying a lattice of particles, you're likely to have at least one variable (almost always more) for every particle on the lattice. Even a 10 particle system in 1 spatial dimension would have 20 variables (10 position, 10 velocity) coupled by 20 equations (well, 10 similar copies of 2 basic equations). If you want to get an idea of what sort of problems you might encounter in condensed matter, you might look up the Ising model. The basic 1D Ising model actually doesn't require calculus, only linear algebra, but one can easily think of similar systems with different interactions that would add calculus to the required toolbox.

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  • $\begingroup$ Shouldn't you allow for entangled degrees of freedom for your 10 particle example? $\endgroup$ Commented Aug 28, 2015 at 21:34
  • $\begingroup$ @NorbertSchuch In my example I have 20 equations (which you can assume are independent) and 20 unknowns. In my hasty mental setup of the problem, I had 10 equations in mind to encode the dynamics, and 10 to encode constraints on the system (e.g. entangled degrees of freedom). A "better" solution might encode constraints by combining variables, but I think my example is sound. $\endgroup$
    – Kyle Oman
    Commented Aug 28, 2015 at 21:45
  • $\begingroup$ Already one particle has an infinite number of degrees of freedom (as its wavefunction lives in an infinite dimensional Hilbert space). $\endgroup$ Commented Aug 28, 2015 at 22:14
  • $\begingroup$ I agree, but however, there is a difference between the dimension of the hilbert space of a system, which is ususal infinite dimensional, and the dimension of the space where it "lives" in. That is usually 3+1 dimensional. $\endgroup$ Commented Aug 28, 2015 at 22:26
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The fractional dimensional space approach (FDSA) can be adopted to introduce flexibility in examining optoelectronic properties in anisotropic systems (quantum dot, wells etc). Here the material are fitted to models that utilize a variable dimension, (alpha) which has provided good agreement with experimental results in many works. This is because the pseudo-two-dimensionality attribute of quasi-particles such as excitons can simplify the evaluation of electro-optical properties. By mapping of the anisotropic exciton in the real space into an isotropic environment parameterized by a single quantity, the problem is greatly simplified. The usefulness of alpha can be appreciated by the fact that it is independent of the physical mechanisms associated with confinement effects arising from the anisotropic structures. A specific optical spectrum can then be attributed to different systems as long as these possess the same dimensionality.

In most works, the dimensionality parameter is seen to vary between 1 and 3, but it is possible that alpha can exceed 3.

Good refs:

X.-F. He, Phys. Rev. B 43, 2063 (1991)

H. Mathieu, P. Lefebvre, and P. Christol, Phys. Rev. B 46, 4092 (1992).

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