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Math Help - What is the state of development of Calculus in mathematics?

  1. #1
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    What is the state of development of Calculus in mathematics?

    In particular, I'm interested in knowing if Calculus is a developing math (i.e., there are mathematicians at the fringes of this area of study working to make breakthroughs).

    Also, is knowledge of calculus a component for higher theoretical mathematics?
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  2. #2
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    Re: What is the state of development of Calculus in mathematics?

    That question is not easily answered. Calculus, as it is taught to undergraduates, is not really a developing branch of mathematics. However, calculus has many broader definitions, and many of those are very much in development. By the very fact that you are posting this question, I am not certain you have the exposure to enough terminology that would allow me to answer this question. Many developing fields utilize calculus (as it is taught to undergraduates). Frequently, though, the mathematics in those fields is not under development, but rather development occurs with the applications of the mathematics (finding additional uses for the theorems).

    If you were to try to classify Calculus, it has roots in at least three of the major mathematical disciplines (algebra, analysis, and topology). Algebraically, one can define a calculus over a set where many of the operations are similar (but not equivalent to) the operations of differentiation and integration. In analysis, calculus is extended to any field possessing a complete metric, and modern approaches are extremely varied (typically some form of measure theory at their core). Point-set topology is fairly well understood, but "calculus" led to the study of differential geometry, differential manifolds, and more. In each of these disciplines, an accomplished mathematician would see the roots of their studies in undergraduate calculus, but to a student without the necessary training, these disciplines may appear to be completely foreign, even if that student completed and excelled at all undergraduate calculus classes.
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    Re: What is the state of development of Calculus in mathematics?

    Quote Originally Posted by Elusive1324 View Post
    In particular, I'm interested in knowing if Calculus is a developing math (i.e., there are mathematicians at the fringes of this area of study working to make breakthroughs).
    Also, is knowledge of calculus a component for higher theoretical mathematics?
    Even though the basic facts of calculus have been settled for several centuries, the teaching of calculus still evolving. So it is in the debates about what topics should be included under the rubric The Calculus and how should they be taught, where the action is now.

    In the 1960's Abraham Robinson found a way to put the theory infinitesimals on an absolutely logically consistent footing. He called Non-standard Analysis. Many of us had hoped that this development would lead to natural way of teaching calculus. In addition to several working groups there was even an entire textbook written and published. That text is now available as Elementary Calculus: An Infinitesimal Approach by Jerome Keisler whole book is a free down-load at.

    But by the mid 1980's the fast developments in Computer Algebra Systems (CAS) changed the whole direction of calculus education.
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