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Different Types of Calculus: Traditional to Unusual

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The traditional calculus you come across in AP- or single-variable Calculus is also known as infinitesimal calculus. “Calculus” originally meant accounting or reckoning, originating with the Latin name for a small counting pebble.

different types of calculus

Infinitesimal calculus uses small additive changes (called delta-epsilon) for calculations.


Infinitesimal calculus is a formal system of calculation, involving two major branches: derivatives (the slope of a tangent line at a point) and integrals (a way to find the area under the curve). However, there are dozens of different types of calculus including vector calculus for three-dimensional (or higher) curves, surfaces and solid bodies and exterior calculus , a high-dimensional calculus which is particularly well suited for working in complex analysis.

Some of the other different types of calculus make small changes to the way the calculations are performed, like switching small additive changes to small multiplicative changes. Other types of “calculus” are far removed from the traditional calculus of Newton and Leibniz, spreading over to fields like mathematical logic.

Different Types of Calculus #1: Non-Newtonian Calculus

Unlike traditional calculus, which is linear with additive operators, Non-Newtonian calculus is non-linear and lacks additive operators. For example, exponential calculus (also called geometric calculus) has multiplicative operators. These operators can only be applied to positive functions, so only positive functions are valid in this particular calculus.

Grossman & Katz [1] outlined several branches, including multiplicative calculus, which uses multiplicative (instead of additive) operators. One sub branch is bigeometric calculus, where ratios measure changes in arguments and values and products are used for accumulations. Other lesser known branches of non-Newtonian calculus include: Anageometric, Biquadratic, and Harmonic calculus.

Other Different Types of Calculus

Some branches of calculus are used for very narrow purposes. For example:

  1. Propositional calculus (also called sentential calculus) is a types of symbolic logic—a way to analyze truth relationships via statements like and, or, not and if.
    propositional calculus

    A truth table used in propositional calculus.



    This calculi is the foundational of most logical-mathematical theories; Predicate Calculus is a more complex version, allowing functions, relations, and quantifiers [2].
  2. Ricci calculus, also called the calculus of congruences of curves [1], is the study of tensors and tensor fields.
  3. Stochastic calculus, used in probability and statistics, deals with stochastic (random) processes or equations that involve statistical noise.
  4. Finite calculus (also called calculus of finite differences) is calculus for discrete values. It useful for many areas like marginal economic analysis, growth and decay, and modeling human behavior.
  5. Lambda calculus is made up a transformation rule and a function definition scheme. It is used extensively in higher-order logic and computer programming, where it forms the underpinnings of many computer programs (like LISP).
  6. Umbral calculus (also called Blissard Calculus or Symbolic Calculus) is a modern way to do algebra on polynomials. Generally speaking, it’s a way to discover and prove combinatorial identities, but it can also be viewed as a theory of polynomials that count combinatorial objects [3].

To muddy the waters a little, there is another branch of calculus called nonstandard calculus. Despite the name, it doesn’t mean “non-traditional calculus”; It refers to a small branch of calculus that applies quite technical (and difficult to understand for many) nonstandard analysis tools to infinitesimal calculus.

umbral

References

[1] Grossman M., Katz R., (1972), Non-Newtonian Calculus, Lee Press, Piegon Cove, Massachusetts.
[2] Goldmakher, L. (2020). Propositional and Predicate Calculus.
[3] Ray, N. Universal Constructions in Umbral Calculus. Retrieved May 4, 2021 from: http://www.ma.man.ac.uk/~nige/ucuc.pdfH.

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Stephanie Glen. "Different Types of Calculus: Traditional to Unusual" From CalculusHowTo.com: Calculus for the rest of us! https://www.calculushowto.com/different-types-of-calculus/
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