Higher Mathematics Pdf
Yoana Terrano
Further Mathematics is the title given to a number of advanced secondary mathematics courses. The term "Higher and Further Mathematics", and the term "Advanced Level Mathematics", may also refer to any of several advanced mathematics courses at many institutions.
In the United Kingdom, Further Mathematics describes a course studied in addition to the standard mathematics AS-Level and A-Level courses.[1] In the state of Victoria in Australia, it describes a course delivered as part of the Victorian Certificate of Education (see Australia (Victoria) for a more detailed explanation). Globally, it describes a course studied in addition to GCE AS-Level and A-Level Mathematics, or one which is delivered as part of the International Baccalaureate Diploma.In other words, more mathematics can also be referred to as part of advanced mathematics, or advanced level math.
With regard to Mathematics degrees, most universities do not require Further Mathematics, and may incorporate foundation math modules or offer "catch-up" classes covering any additional content. Exceptions are the University of Warwick,[2] the University of Cambridge which requires Further Mathematics to at least AS level; University College London requires or recommends an A2 in Further Maths for its maths courses; Imperial College requires an A in A level Further Maths, while other universities may recommend it or may promise lower offers in return. Some schools and colleges may not offer Further mathematics, but online resources are available.[3]Although the subject has about 60% of its cohort obtaining "A" grades,[4] students choosing the subject are assumed to be more proficient in mathematics, and there is much more overlap of topics compared to base mathematics courses at A level.
Some medicine courses do not count maths and further maths as separate subjects for the purposes of making offers.[5] This is due to the overlap in content, and the potentially narrow education a candidate with maths, further maths and just one other subject may have.
There are numerous sources of support for both teachers and students. The AMSP (formerly FMSP) is a government-funded organisation that offers professional development, enrichment activities and is a source of additional materials via its website. Registering with AMSP gives access to Integral, another source of both teaching and learning materials hosted by Mathematics Education Innovation (MEI). Underground Mathematics is another resource in active development which reflects the emphasis on problem solving and reasoning in the UK curriculum. A collection of tasks for post-16 mathematics can be also found on the NRICH site.
Further Mathematics is available as a second and higher mathematics course at A Level (now H2), in addition to the Mathematics course at A Level. Students can pursue this subject if they have A2 and better in 'O' Level Mathematics and Additional Mathematics, depending on the school.[7] Some topics covered in this course include mathematical induction, complex number, polar curve and conic sections, differential equations, recurrence relations, matrices and linear spaces, numerical methods, random variables and hypothesis testing and confidence intervals.[8]
Further Mathematics, as studied within the International Baccalaureate Diploma Programme, was a Higher Level (HL) course that could be taken in conjunction with Mathematics HL or on its own. It consisted of studying all four of the options in Mathematics HL, plus two additional topics.
I graduated with an M.S. in Computer science about a decade ago, standard curriculum that I believe is still somewhat taught (Calc, Multivariate Calc, Dif Eq, Linear Algebra, Discrete Math, etc.). I work as a software engineer (they give us a title of Computer Scientist for some reason) for an Contract R&D (gov stuff).
I have found my math skills withering over the years, probably for lack of use of particular fields. For the past couple of years, I am constantly reading research papers (computer science related) for background when developing a new algorithm. What I notice is that I will often get stuck on some mathematical notation or methodology that I am unfamiliar with, when trying to understand the paper. I have been attributing this to my withering math skills, and having to do with fields I never studied in school (or deeply enough).
I try to go back and review what I need to understand the paper, but this leads to a seeming unending link of I need to know this before I can understand that, etc.. With sometimes unsatisfying results.
I was wondering what people have experienced as the best way to learn higher math (advanced calculus, advanced prob and stats, tensor calculus, advanced linear algebra, etc.) as well as refreshing what they were taught in school MANY years ago.
I have tried looking course work on MITs website, to see what graduate math students are being taught. I procure those books and notes, and try to go through the class syllabus myself. But I guess its the lack of rigor, that is failing me the most (school imposed a strict rigor), so I end up just glossing over things when I should be trying to deeply understand thee material (trying to get at the meat of what I am trying to understand, for the task at hand). But over-all this seems ultimately flawed and I only come out with partial understanding.
I want to try to follow a method that would eventually get my math skills on par with a computer science PhD graduate level of understanding of the involved math (say with a focus on computer vision, AI, ML, and computer graphics). What I have been doing over the years is not working for me.
So the path is: Read good books to acquire knowledge and maturity $\Longrightarrow$ At the same time, use Internet to find lecture notes, graphics and videos that can give you less formal and intuitive explanations $\Longrightarrow$ Have your own ideas and your own understandings on what you learned, and at the same time are more skilled at tackling math problems.
which is super-detailed, zero-gap. A real gem. But warning: while the book is extremely helpful, you should not indulge yourself in the comfortable proofs and go through the material all way long without thinking. Always ask: what theory I have learned, what proof methods I have mastered, and can I remember and reproduce the whole machinery on my own?
Since last year, I've been interested in higher mathematics and don't really know where to start and don't know what knowledge I still need to obtain, before being able to understand the concepts. Currently, I'm in a trigonometry class (I understand it well), but I have studied other subjects by reading textbooks and taking free courses, when possible. If I were to sum up my knowledge, up until trigonometry, I have a good understanding of the topics. I'm also good with symbollic logic. After that point, I have a fragmented knowledge of the topic. I want to study logic, set theoy, abstract algebra, etc. and topology (the most seemingly interesting subject to me). Where do I start? What subject should I start with? What lower mathematics should I learn first?
One way to go would be to try to accelerate yourself through the standard Trigonometry, Geometry, Pre-Calculus, Calculus curriculum. This is a good way to go, but won't be the most direct way to the more abstract topics you've listed. If this is of interest, perhaps "A Hitchhiker's Guide to Calculus" by Spivak would be a good place to start.
A good place to start might be number theory. You already have some feelings for numbers, and it'll be a great way to introduce yourself to proving things rigorously. For number theory, this link might be of interest: Best book ever on Number Theory.
Alternatively, you can never start learning to prove things rigorously (essentially thinking clearly) too early. The text "Foundations of Higher Mathematics" by Fletcher and Patty might be a good place to start.
My 2cents: Winning Ways by Berlekamp, Conway and Guy is a beautiful but sneaky introduction to advanced mathematics. It's definitely a lot of fun and is great training for a mathematician. Just take it very slow!
I think people should have a background in analysis and algebra before attempting to learn serious set theory and logic. Those subjects tend to be a bit too abstract to be the place where you first learn to reason mathematically. (I'm not talking about the rudiments of set theory that are a prerequisite for analysis and algebra.)
I think in your shoes I would start learning calculus with Spivak's book Calculus. That is an excellent introduction to mathematics in general, not just calculus. It has a solutions manual, which you should use mainly to check your own solutions.
If you find that book too difficult at first, you could try reading the high-school level books in the Gelfand series (The Method of Coordinates, Functions and Graphs, Algebra and Trigonometry). They have problems that are probably more challenging than what you're doing in school. You could also have a look at some of the books in the New Mathematical Library. They are written on various topics and directed at bright high-schoolers. The books by Niven in particular are good. After a while learning from these kinds of books, Spivak's ought to seem easier.
If you do want to jump straight into abstract algebra, that might be difficult, but you could try Algebra by Artin. You could also read Stark's Introduction to Number Theory. Learning topology in a meaningful way is probably impossible at this stage.
byPatrick Keef and David GuichardDepartment of Mathematics
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This HTML version was produced using a script written by David Farmerand adapted by David Guichard. It is generated from the TeX source;the output of TeX should be treated as the definitive version: _math/higher_math.pdf. Otherversions are available at _math/.
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