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What IS calculus?

  1. Sep 1, 2007 #1
    I am starting university next month. I will be studying physics, maths, and chemistry, although I will be majoring in physics.

    Basically, I want to understand what calculus IS!!! We were never really taught what it is in high school math. Can someone here explain what calculus, as in differentiation and integration, is?

    Btw, in college maths, will you be taught the what and whys of all the principles of math? As in ^^^


    Thank you, all your help will be invaluable to me :-)
     
  2. jcsd
  3. Sep 1, 2007 #2

    mathwonk

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    it is a technique for solving non linear problems by approximating them everywere locally by linear ones (differentiation), and then integrating those approximations to obtain the global solution.

    e.g. to obtain the volume of a solid of revolution, we visualize the volume as growing, compute the derivative of the growing volume function to be the area of the leading face of the growing volume, (a disc), and then integrate that moving area function to obtain the volume.
     
    Last edited: Sep 2, 2007
  4. Sep 2, 2007 #3
    Differential calculus is essentially the study of instantaneous rates of change. For example, consider a particle moving in one dimension, who's equation of motion is [tex]x(t)[/tex] (where t is time). It's velocity, [tex]v(t)[/tex] is defined as the rate at which it's position changes with respect to time. In calculus terms, we can write [tex]v(t)=\frac{dx}{dt}[/tex]. The particle's acceleration can be defined as the rate at which its velocity changes with respect to time, so we can write [tex]a(t)=\frac{dv}{dt}[/tex]. Differential calculus is concerned with calculating these rates of changes (derivatives), and various applications.

    In integral calculus, we are essentially concerned with fining the area under a curve. If you have a graph [tex]f(x)[/tex], then the area under the curve, between points a and b is: [tex]\int_a^b{f(x)dx = F(b) - F(a)[/tex], where [tex]\frac{dF}{dx} = f(x)[/tex]. In integral calculus we are essentially concerned with calculating these anti-derivatives.

    This is essentially the basic of calculus, and what you will learn in your first year, but it can all be generalized to much more advanced applications.
     
  5. Sep 2, 2007 #4

    HallsofIvy

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    Whether you will be taught "the what and whys of all the principles of math" depends on how much math you take and which. Most Calculus courses, especially those intended for engineering majors (not to mention Business Admin) give rules and formulas with out a lot of explanation. A math major, in either "Mathematical Analysis" or "Advanced Calculus" will see much of the basic concepts of calculus. In "Abstract Algebra" or "Modern Algebra" you get more of the fundamentals. Not all college require or even offer "Topology" for undergraduates but that, I think, would complete the fundamentals of mathematics. In a sense, all mathematics is either "discrete" (Algebra) or "continuous" (Topology) and Analysis combines the two.
     
  6. Sep 2, 2007 #5

    matt grime

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    I would disagree with Hall's last sentence. Discrete mathematics frequently means 'not calculus/analysis'.
     
  7. Sep 2, 2007 #6

    mathwonk

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    i guess in my opinion, which is certainly arguable, the term calculus should refer to the connection between derivatives and integrals. I.e. there is to me no such thing as differential calculus, or integral calculus separately, regardless of the traditional textbook usages of those terms.

    the reason i say this is that doing integration essentially by taking limits of riemann sums, was known to archimedes, and finding tangent lines by linear approximation, was also known to descartes and fermat.

    but the power of calculus comes from noticing that one does not need to labor at taking lmits of sums in order to compute integrals, if one sees the connection between the moving area function and the height functioin, i.e. that the height function is the derivative of the area function.

    so i feel the problem of finding tangents is not of THAT much importance alone, and the more significant problems of finding areas and volumes and moments and arclengths, etc... are too hard to solve when considered alone as limits.

    so thats why i said calculus is the combination of the two techniques, i.e. the integration of a family of local linearizations, to obtain a global solution to a non linear problem.

    and this is why i think the invention of calculus is attributed not to archimedes or fermat, but to newton and leibniz, much later.

    i should admit i have changed my opinion on this matter several times over 40 years, and have only come to this view recently. but presumably "calculus" should refer to a method forcalcuklating thigns, ans essentially archimedes had no method for calculating his sums anywhere neara s effective as the modern calculus. he ahd a wonderful way of exopressing non linear problems as limits of linear ones, but he could only calculate those limits in the simpelst cases, and did not even know for e.g. the area under a cubic curve it seems.

    to oversimplify, the greeks were only masters of the "calculus" of conics and lines.
     
    Last edited: Sep 2, 2007
  8. Sep 2, 2007 #7
    Good point Mathwork. Early on, you tend to think of differential calculus and integral calculus separately, but later on you use them both as part of a greater framework.
     
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