What's your math background?
As I'm finding out, the quality of undergrad math courses can vary widely.
Was your calculus and linear algebra very proof oriented (i.e., did you DO a lot of proofs)? Are you familiar with the proof techniques yourself? Can you wrap your head around abstract constructs such as sigma algebras and infinite sequences?
I think a proper understanding of stochastic calculus requires a good analytical knowledge of probability. That in turn requires a good grounding in analytical proof techniques.
As of now, I'm still taking calculus 3 which is about sequences and series and multi-variable calculus. We were tested on proofs for calculs 1 and 2 but it wasn't the central thing. I've done linear algebra 1 wherein we focused more on properties and I'm currently taking linear algebra 2 wherein we are discussing about abstract vector spaces. It's heavy on proofs.
---------- Post added at 05:08 AM ---------- Previous post was at 03:19 AM ----------
Hi guys, this is more or less the material I have learned so far:
Calculus 1
This is the first course on calculus in a sequence of four Courses. The objective is to introduce basic notions of calculus and analytic geometry, including differentiation.
- Real numbers, functions, their inverses and graphs
- Transcendental functions: trigonometric and inverse trigonometric, logarithm and exponential, hyperbolic
- Limits of functions, continuity at a point, continuity on an interval
- Differentiability, derivatives of functions, chain rule, implicit differentiation, derivatives of higher order
- Local maxima and local minima, Rolle's Theorem and Mean Value Theorem, points of inflection, first-derivative and second-derivative tests, Netwon's Method
- Antidifferentiation
Calculus 2
This is the second course in the calculus sequence. The objective is to study integration and related topics.
- Indefinite and definite integrals, Mean Value Theorem for integrals, Fundamental Theorems of Calculus, area of plane regions
- Parametric equations, polar coordinates
- Volumes of solids, length of arcs, other applications of the definite integral
- Techniques of integration
- Elementary differential equations
Foundation of Mathematics
This course introduces fundamental ideas and techniques used in many different areas of mathematics.
- Elementary logic, mathematical statements, quantified statements
- Sets, operations on sets, Cartesian products, properties of sets
- Natural numbers, integers, rational numbers, real numbers, complex numbers
- Relations, equivalence relations, equivalence classes
- Functions, injective and surjective functions, inverse functions, composition of functions
- Mathematical proofs, mathematical induction.
Intro to Scientific Computing
This is an introductory Course on scientific programming using Fortran and C/
C++, intended primarily for students in physical and mathematical sciences. The objective is to equip the students with basic programming skills, including the use of existing libraries, useful in the study of physical and mathematical sciences.
- Fundamental concepts of programming
- Brief overview of scientific programming languages (Fortran, C/C++)
- Basic data types, functions, classes, templates, STL (container classes, algorithms), memory management
- Compilation process, use of existing C/C++/Fortran libraries
- Algorithmic problem solving and design process, program development, coding and debugging, fundamental programming constructs, data structures, recursions, simple file processing, algorithmic complexity
- Case studies in physical and mathematical sciences
Linear Algebra 1
This is the first of two courses on linear algebra. The main objective is to introduce basic notions in linear algebra that are often used in other areas of mathematics and applications.
- Systems of linear equations, Gaussian elimination
- Matrices, inverses, determinants
- Vectors, dot product, cross product
- Vector spaces, subspaces, linear independence, basis, dimension, row and column spaces, rank
Discrete Math
This Course introduces basic notions in discrete mathematics and number theory commonly used in mathematics and computer science.
- Counting, permutations and combinations, binomial theorem
- Inclusion-exclusion principle
- Boolean algebra
- Recursion
- Graphs, paths and circuits, isomorphisms, trees, spanning trees
- Division algorithm, greatest common divisor, Euclidean algorithm, fundamental theorem of arithmetic, modulo arithmetic
- Diophantine equations ax+by=c
As evidenced by the above, I don't have an advanced background and I'm lost as where to start so that I could understand Brownian motion, etc. I'm taking Introduction to Probability and Statistics 1 now but we have just started with the course.
Thanks very much
