How to fill my mathematical gaps?

To do the story short, I became interested in mathematics in a serious way like two years ago, I'm currently in graduate school, but the problem is that my mathematical background is not as good as other students, since in my undergraduate I didn't take math seriously, I was studying music on my own and my major was biology.

On the other hand, my old university was not really serious (the classes were not that hard and we didn't cover what i was suppose to learn), the only exception was my old adviser that basically he taught me mathematics for the last two years, but one person cannot cover all the material that I needed to learn.

Now, I fell that I have a lot of gaps in a few areas of mathematics in which I consider really important. Here is the list of the mathematical knowledge that i feel that i have little to none.

1. Multivariable Calculus - I saw the introductory stuff but in general, I dont know a lot about the topic.
2. Advanced Calculus- I have zero experience in advanced calculus, since my advanced calculus was different, I can say that I learned more of differential equations that the calculus with proofs.
3. Understanding Proofs- I take discrete mathematics but I find difficult how to construct proofs in general.
4. Linear Algebra- I think we stop when we find eigenvalues and eigenvectors, after that I know zero.
5. PDE- I saw very little about the topic

Know this are the courses that I have some experience with:

1. Advanced Algebra- im currently taking a year of algebra using Dummit and Foote, which was really hard for me since I took only one semester in advanced algebra and was really easy.
2. I have Topology (one semester) and mathematical modeling (one semester).
3. In general my strongest background is differential equations in which I think I know a lot so far.

My plan for fall and next spring are the following classes:

fall:

• Partial Differential Equations

• Real Analysis

• Thesis

Spring

• Complex Variables

• Real Analysis 2

• Thesis

I want to know what books are better to start filling my gaps in mathematics, especially i want to focus in learning diffusion which I need in my thesis, how to understand proofs to tackle real variables and advanced calculus, maybe some PDE that I really needed too. I maybe want to learn some dynamical system as well. My plan is to use the summer to learn the most I can.

I am currently studying my master degree in applied mathematics, especially working in nonlinear system of differential equations applied to biology. Right Now i would like to understand the structures on how to creates proofs since is going to help me to understand Real Variables.

Thanks

I will give some suggestions and you tell me what you think about them. Then i can suggest others.

First of all, some revision:

Topology------ Introduction to metric and topological spaces by Sutherland. Very good book with an excellent motivation for compactness (with an example that appears in analysis)

Real Analysis and Advanced Calculus---David Brannan A first curse in real Analysis The book is very easy tpo read and guide you in the proofs. You learn proofs reading good ones and trying to do some of them by your own. If that book is very easy or when you finish with it, then you can try Bartle and Sherbert Introduction to Real Analysis.

Multivariate Calculus, Advanced Calculus and Analysis all in one---- Mathematical Analysis 1 and 2 by Zorich. I haven't used this book but i have been told that it is very good. And it deals with all the material.

Linear Algebra ---- Algebra Done right Axler.

Then you can jump to PDES or Dynamical Systems.