Would condensed matter physicists need more than three dimensional calculus? 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. 
 A: 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.
A: 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).
