If you do not understand the assignment, don't hesitate to ask. Follow the directions. Work neatly and accurately. Show your complete work, not just the answer. This will help you and your teacher when you are checking through for errors. Always check back to be sure you have done all simple arithmetic correctly. Do the work promptly before you have forgotten all the instructions.
If you get stuck, don't just give up! Look back at the book and your notes for ideas related to the problem. If your work on a problem seems to be completely confused, it sometimes helps to discard your paper entirely and start a fresh. If you still can't clear your thinking, ask the teacher about the problems as soon as possible. Help someone else, if you can. There is no better way to learn a topic than by trying to teach it!
Also, it is often helpful to call upon a classmate when you do not understand a problem. Often, they are able to explain the concept to you as well if not better than the teacher.
Here's what you can do: Analyze the error to see if you can find what you did wrong. If it is a careless error and you really knew how to do the work correctly, make a note of it, and if you find that you keep making careless errors frequently, start working more carefully. If you can't find where your error is, ask the teacher or a classmate to help you. Keep a page in your notebook entitled, "Warning: Errors to Avoid. In the minute or two before the class gets started, think over what you have been working on recently.
Have all necessary equipment: book, pencils or pens, notebook, homework assignment. Take down the assignment promptly and accurately. This takes an effort if you are the kind whose mind tends to wander. Ask questions when you do not understand. Listen to the questions and answers of others in the class. Some ODEs can be solved analytically , that is, with a closed-form solution, using elementary functions. However, the solution to many ODEs can only be written as a series or integral relationship.
ODEs can be solved "numerically", on the computer, using approximate methods. A large part of quantitative finance involves numerically solving differential equations in this manner. There is no lack of study materials available for ODEs as they are a staple of the first year undergraduate mathematics program.
I used the book written by my lecturer at University, and I found it to be approachable for a first year undergraduate see Robinson, below. Geometry is one of the most fundamental areas of mathematics.
It is absolutely essential for many areas of deeper mathematics, including those related to quantitative finance. Many undergraduate courses introduce Euclidean geometry to students in their first year, and it is also an appropriate place to start for the autodidact.
The primary setting is often Euclidean Geometry in three-dimensions, namely the geometry of "everyday life". You will learn a lot about constructing proofs from studying geometry, particularly with regards to projective geometry in the plane and geometry of the sphere. In highschool or at GCSE! Euclidean Geometry eventually leads on to more general geometries such as Spherical Geometry or Hyperbolic Geometry, where familiar results from Euclidean Geometry are shown not to hold.
In addition, and perhaps more relevant to the quant, having a good understanding of trigonometry is essential for later courses such as Fourier Analysis, which plays a substantial role in signals analysis and time series analysis.
Geometry is a tricky subject to introduce as it is extremely broad and covers such a diverse area of mathematics. However, I have found the following book, part of the Springer Undergraduate Mathematics Series, to be very helpful:. Groups are one of the most important algebraic structures found in mathematics. They provide the basis for studying more complex structures such as rings, fields, vector spaces which we mentioned above in Linear Algebra.
They are also strongly related to the idea of mathematical symmetry. While it might be considered that groups are more of a "pure mathematics" topic, and thus are less applied, this is actually not the case. Groups find applications in chemistry crystallisation , physics symmetry and conservation laws as well as in cryptography. However, are they relevant to the quantitative analyst?
This is a tricky question to answer. While it isn't clear how a direct study of groups and symmetry might be applied on a day-to-day basis in the world of a quant, the study of groups does form the basis of many more advanced mathematical topics, particularly advanced Linear Algebra. For the autodidact who is short on time, I would state that it is worth studying them at an introductory level in order to "be aware of their existence", as many advanced quantitative techniques will indirectly refer to them.
Note however that one of the most successful quant hedge funds in history, Renaissance Technologies, was founded by Jim Simons, a notable mathematician who carried out a substantial amount of work on manifolds which requires a solid understanding of group theory. Read into this what you will!
There is no shortage of elementary textbooks on group theory. Since it it such a common topic for first year undergraduates, many authors have tried to write introductory books.
I've found the following to be useful:. Along with Linear Algebra and Real Analysis Calculus , introductory Probability is the most important first year course for a quant to know. This applies for quantitative traders, quantitative analysts derivatives pricers , risk managers VaR, CVA etc and data scientists.
I cannot stress enough how important it is for a practising quant to have an intuitive grasp of probabilistic concepts. Time spent studying here will pay dividends over a quant career. Undergraduate introductory probability courses usually begin by discussing the laws of probability, including Bayes' Theorem, probability distributions, discrete random variables, expectation, covariance and continuous random variables.
These are all necessary topics for the quantitative analyst. As with Groups, there are no shortage of textbooks on Probability for the undergraduate student, nor MOOCs for that matter. I learnt probability primarily from Ross, below, as well as the Schaum's Guide I prefer to learn by doing! There is also a Coursera course on Probability, given by the University of Pennsylvania:.
What is "Mathematical Computing"? Broadly, it is carrying out mathematical analysis using computer programs. This is essentially the definition of a quant! Hence, it is absolutely essential that you gain a grounding in programming algorithms at the earliest possible stage. For the autodidact, such a course may seem a little unnecessary, as it is straightforward enough to learn how to program from the various sources on the internet, along with a large array of textbooks.
However, I will state that "learning how to program" and understanding how to take a mathematical algorithm and turn it into efficient computer code are completely different skillsets. I can see the difficulties in the first years that I tutor in that they are not able to coherently put all ideas together to form a proof. Let me tell you two ways of how I learned to write a proof. One: when reading a theorem, or anything from a book like say Atiyah - Macdonald copy down everything word for word.
When you reach a theorem, try and prove it by yourself. If you can't, take a sneak peak i. If you can't at all prove anything, copy down the proof word for word. The idea is that by doing this, the essential techniques in the proof, the key ideas get ingrained into your mind.
People often talk about methods of proof contradiction, contrapositive, etc but what you should know is that in each subject there are "methods", little tricks that one can use over and over again. By writing out line by line, word for word everything in a book, you learn these methods. As for books on the subject, you should choose a few and work intensely on them.
Since you are two years away from University, I suggest training in Algebra and Analysis first :. Work through these two books, and you should not only be able to handle the material well, but also have enough confidence in your ability to write a proof. When I was an undergraduate the entire 4-semester elementary calculus sequence, linear algebra I think , and one semester of a standard 2-semester advanced calculus course was required for computer science majors, along with some discrete math.
Perhaps this has changed. However, I'm currently tutoring a computer engineering student, a major I would expect to be a bit further from mathematics than computer science is he's a 2nd year undergraduate at a large public U.
That said, I think the answers and comments posted thus far overlooked your comment about not yet having studied any form of calculus or precalculus. If someone has not yet studied trigonometry beyond basic right triangle triangle trigonometry or conic sections with axes parallel to the coordinate axes or composition of functions or geometric series or polar coordinates or curve sketching of polynomials and rational functions, the advanced undergraduate level texts suggested thus far in the answers and comments would be entirely inappropriate.
Assuming that you really meant what you said, namely that you haven't yet studied any precalculus or calculus, I suggest looking at the following. I've listed these in approximate order of difficulty. There's two aspects to that.
One is the so-called "elementary" or "naive" set theory. This is something you'll need everywhere in mathematics, all the time, so getting up to speed on that before approaching anything else is a good idea. This is going to be stuff explaining what a union of sets is, what a function is, maybe how you can define different types of numbers using sets and such.
However, what mathematicians actually call set theory is a highly abstract topic that you will not be able to follow without having a rather solid idea of how mathematics of the "common" kind works, and it is in fact a field that many mathematicians don't know too much about beyond the acronym ZFC.
Basically just saying - don't try to start with some "introduction" to set theory that is actually of the second kind such as Jech. There are short talks from members of staff followed by a student view. The Faculty distributes a leaflet called Study Skills in Mathematics to help students generally fresh from school acquire the very different skills needed to make the best of our challenging mathematics course.
This is distributed free to all first year mathematicians, and may be downloaded in PDF form.
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