What area of math can I begin learning before I have completed

What are your objectives? You can do most of basic pire math without calculus. You can learn discrete mathematics, linear algebra, abstract algebra, number theory all without calculus. In fact you don't see any calculus in those courses except maybe as examples.
Realistically you can also learn complex analysis without calculus (up until you're differentiating and integrating but it sounds like you're way passed that point). Hell, you can do basic Analysis without calculus.

>>7981327
Linear algebra, general topology.
If you defend yourself well enough, you can start with analysis too.
Actually, calculus is very applied math, and you aren't probably facing those monster integrals again in your life, unless you do something really specific.

>>7981337
I have no real objectives. I am a first year student in mathematical physics, so I guess anything that can help with physics is ideal. Really though, I am just learning for fun and a challenge, since my current courses are easy, and they won't let me take the advanced math courses since I am in the science department, instead of the math department.
All of my maths courses so far have been calculus, and linear algebra (which is easy, but pure agony).

>>7981379
Start looking into a book on ordinary differential equations. They show up all the time in physics.

Don't discount linear algebra. Linear transforms can be very useful. Also, this sets the stage for tensors, which I like to call "hyper" vectors.

>>7981404
I can't discount linear algebra even if I wanted to; I am required to complete two more courses on it before I graduate.
Do you have any recommendations on books for differential equations?

>>7981409
from the wiki
http://www.cargalmathbooks.com/#Differential%20Equations
The Simmons book is good for outlining the people/stories involved in the discovery of these equations, and has plenty of physics examples in it.

Go read through Courant's "What is Mathematics?" which has devotes a chapter each to most fields of undergrad math. It should help to clarify what your interests are, which you can then pursue further.

>>7981337

If I recall there's a point when learning linear algebra that integration becomes somewhat important. Non euclidean inner products can be defined with integration sometimes (if I'm recalling it correctly). Something to do with function spaces and orthogonality. The best example I can think of is with fourier series and proving with an integral inner product that each of the sine/cosine terms in those series is linearly independent.

But I'm an MecE so what the fuck do I know

>>7981327
You can do linear algebra and introductory vector stuff. For linear algebra you can probably go through row reduction, modeling and solving simple systems of equations, and as far as eigenvalues/eigenvectors. For introductory vector stuff (my school called it vector geometry) you can easily go through what vectors are, dot products, cross products, and even as far as euler's identity (e^(i*pi) = -1) and the imaginary axis (for graphing complex numbers as vectors where the x axis is the real part of the number and the y axis is the imaginary part - if you're good with polar coordinates you're ready for this, if not you can do everything up to it).

Honestly, if you've done derivatives and integrals already you can start multivariate calculus because you don't really need the sequences/series part, it's mostly just double/triple integrals and partial derivatives.

>>7982924
Oh I just got out of a test on this stuff. OP probably won't need to worry about it until like junior or senior year in a math major. For now he only really needs to focus on the mechanics, stuff like basic matrices, how to use matrices/vectors to model Ax=b, how to row reduce, then eigenvalues/eigenvectors. If he wants to go really far for his level he can look at diagonalizable matrices/similar matrices, change of basis, and maybe markov chains. He won't need vector spaces, norm spaces, or inner product spaces for a long time.

Group/ring/field theory can be doable once you know basic linear algebra