Well, if you're serious about applied mathematics-and serious in that you don't just want "reciepe" books,rather applications that build on the meaty theory background you have-then you should avoid such texts and try and locate books that don't avoid theory,but merely downplay it. Those are the "real" applied mathematics textbooks.
You definitely need to strengthen your background in differential equations, you're pretty much dead in the water without that in your background. The best beginning textbook I know on differential equations is George Simmon's Differential Equations with Applications and Historical Notes. It is one of the most beautiful, richest textbooks you've ever find on any subject-it covers all the basics of ordinary and partial differential equations using only basic calculus and a context of not only physical applications,but incredibly detailed and scholarly notes on the historical founders of the subject. It's a must have for any mathematical library. The one flaw the book has is that it doesn't explicitly use linear algebra,but rather old fashioned linear equation notation. But by all means,don't let that chase you away from one of the best books there is on any subject. It really is the book to start with.
Another subject you'll need to be very comfortable with in terms of applications is linear algebra and there's no better book from an applied standpoint for this then Gilbert Strang's classic Linear Algebra With Applications. Be careful with this because there are several versions of this book and the one you want is this one-the other one is much gentler and less substantial. This is the one with the real red meat by one of the great applied mathematicians of our era-with a host of applications you won't find anywhere else,both standard and exotic. While you're at it, check out Strang's calculus text online at his website. It may be the best basic calculus text there is bar none-with more applications then just about any other text in it's wieght class. And best of all, it's free!
For financial mathematics,you really need a good background in probability theory. It sounds like you have a good working knowledge of the brute theory, but that'll only take you so far. You need to back up and learn some of the basics first, particularly the applications. One of the best books that currently exists on this subject for the beginner, which contains a host of applications, is Grinstead and Snell's Introduction to Probability. It provides not only a comprehensive undergraduate course in the subject,but it provides many applications, particularly to finance and the social sciences. Best of all,it's available online for free for download.
Lastly. a very good addition to your training will be advanced calculus with applications. Not "applied advanced calculus",which is mostly receipes in metric spaces. I'm talking about a relatively recent and very important trend in elementary analysis textbooks where real-life applications are included alongside a careful presentation of the theory of calculus. This is where the artificial separation of pure and applied mathematics-which is there for purely profiteering reasons in academia in my opinion-is taken down and both are discussed in an advanced calculus course in equal measure and import. There aren't many of these books yet, sadly-but the ones that do exist are excellent.The best one that currently exists is Jeffery Cooper's Working Analysis,in which a thorough course in advanced calculus of one and several variables is peppered with applications to physics, biology, numerical analysis, economics and so much more. This is a terrific text that most students of mathematics should have in thier library. A bit less advanced but also quite good is Donald Estep's Practical Analysis of One Variable. This is really an honors calculus book masquerading as an analysis book, but Estep writes beautifully and presents one variable analysis in complete unity with literally hundreds of applications, many fascinating insights and an interestingly original presentation. This one is well worth having as well,particularly for you since it's emphasis is on differential equations.Lastly and more sophisticated then either of the previous texts is Real Analysis and Applications: Theory in Practice by Kenneth R. Davidson and Allan P. Donsig.This isn't really a full blown analysis course but rather a supplementary text following up a standard intermediate analysis course based on baby Rudin or Charles Chapman Pugh's book. That being said,it has an amazing set of applications, including Fourier analysis and wavelet theory. It's definitely worth a look.
That should get you started. I'm sure the others will recommend other good texts as well and if I think of any others,I'll edit this post and add them. Good luck!
Rudin's book is too abstract in some sense because it requires some knowledge or sense of metric topology. Although Rudin explains the basic theory, but I don't think this is not appropriate to beginner.
Spivak's calculus is "calculus". Although it is quite tough, it is not a book for undergraduate analysis.
I recommend three books:
- Ross, Elementary Analysis: The Theory of Calculus
- Marsden and Hoffman, Elementary Classical Analysis
- Apostol, Mathematical Analysis
Ross helps reader to understand one dimensional real analysis. It gives quite good examples and appropriate exercise problem. But the book doesn't cover multivariable things.
Marsden and Hoffman gives a tons of examples and interesting exercise problems. Although it is quite a challenge to reader, the book gives many pictures and good explanation to the subject. Although some part is based on higher dimensional setting, it is quite readable. I strongly recommend this book. If you read this book, you have to aware that the definition of compactness in this book is `sequential compactness'.
Finally, the book of Apostol gives almost full details to the proof. It covers many topics. Maybe this book is more appropriate to person who want to know more advanced topics.
I heard that one of my friends says Pugh's Real Mathematical Analysis is good, but I didn't read that book.
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