Difference between revisions of "Calculus"

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m (Remove redlink. If someone writes about this (it would be interesting!) by all means put the link back in.)
(What is an "Infinitesimal"?)
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''An infinitesimal is a framework for doing differential calculus'', nothing more.  It involves two very small quantities having a functional relationship to each other, such that they both decrease toward zero simultaneously.  Their ratio is the derivative.
 
''An infinitesimal is a framework for doing differential calculus'', nothing more.  It involves two very small quantities having a functional relationship to each other, such that they both decrease toward zero simultaneously.  Their ratio is the derivative.
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If one really wants to point one's finger at a specific thing and call it an infinitesimal, in the "epsilon/delta" formulation of [[Limit_(mathematics)|limits]] the quantities <math>|f(x) - Y|</math> and <math>|x - X|</math> are the infinitesimals.  These are, of course, not actual numbers, but expressions of dependency on <math>x</math> in the context of the derivative.
  
 
This concept was invented by Gottfried Leibniz with his "dx/dy" notation, and has been used ever since.  Dx and dy are the infinitesimals.  Isaac Newton's "fluxion" method of calculus did not involve this concept.
 
This concept was invented by Gottfried Leibniz with his "dx/dy" notation, and has been used ever since.  Dx and dy are the infinitesimals.  Isaac Newton's "fluxion" method of calculus did not involve this concept.

Revision as of 15:46, 7 August 2018

This article/section deals with mathematical concepts appropriate for late high school or early college.
The word "calculus" has historically had a much more general meaning than at present. It means "a method or 'trick' for calculating something". (The word means "little stone" in Latin, referring to the use of pebbles for counting.) Among the types of calculus are:
  • Propositional calculus—a notation for proving theorems about logical expressions.
  • Predicate calculus—a notation for proving theorems about functions in set theory.
  • Lambda calculus—a notation for proving theorems about recursive functions and computability in computer science.
  • Residue calculus—a trick for evaluating integrals of complex functions around a closed loop, by examining the singularities of the function inside the loop.
  • and, of course, Differential and Integral calculus—the common modern meaning of the term, and the subject of this article. These are sometimes collectively called infinitesimal calculus.

Calculus (that is, the "infinitesimal calculus", see above) is the mathematical subject that studies rates of change of functions. There are two main branches of calculus—differential calculus, and integral calculus. There are subfields of these: single variable calculus, exterior calculus, and multi variable calculus.

Calculus has numerous applications in many fields, such as engineering, physics, chemistry, biology, and economics. Many physical properties can be explained and/or modeled through the use of calculus.

Isaac Newton was one of the most famous inventors of calculus, however there are several others who developed calculus simultaneously including Gottfried Leibniz.[1] This controversy over the originator of calculus created rifts within the European mathematical community for years.

Integrals

For a more detailed treatment, see Integral.
Integration is primarily defined as the method to calculate the area in an xy-plane above the x-axis. There are two fundamentally different kinds of integrals.

Indefinite Integrals

For a more detailed treatment, see Indefinite integral.
The antiderivative of a function is often called the indefinite integral. It is called indefinite because the limits of integration are not specified. So, for example, the derivative of is . From this it follows that the antiderivative of could be . But note that the "7" in that formula was a red herring. Adding any constant to a function doesn't change its derivative, so the antiderivative of could have any constant added to it. This arbitrary constant is usually written C and is called the "constant of integration". The indefinite integral could be written:

,

The Fundamental Theorem of Calculus says that the area under the graph of between a and b is the difference in the values of between a and b. The constant of integration cancels out in the definite integral (see next section).

Definite Integrals

For a more detailed treatment, see Definite integral.
This kind of integral is called a definite integral, written with the limits :

The above is a simplified "intuitive" treatment of calculus and of this theorem. The actual "rigorous" proof, "rigorous" definitions of derivative and integral, and statement of the conditions under which the theorem is true, are beyond the scope of this article. Definite integrals are strongly connected to the Fundamental Theorem of Calculus.

Derivatives

For a more detailed treatment, see Derivative.
Derivatives are defined as the instantaneous rate of change of differentiable functions. Derivatives themselves can be functions that give the slope of the instantaneous rate of change of a function at any differentiable point. The slope of a tangent line is another way of defining a derivative. A tangent line touches a graph at only one point (locally). So its slope can be interpreted as the instantaneous rate of change of that graph.

What is an "Infinitesimal"?

The word "infinitesimal" is sometimes used in informal speech to mean "something extremely small", as in "The effect of the decline in soybean prices on the overall recession was infinitesimal." It's roughly a synonym for "inconsequential" in such a case.

But in calculus it has a somewhat clear, though not really precise meaning. It means some very small thing that is compared to another very small thing as they both decrease toward zero under some rule that connects them. It does not just mean "some very small number". It is not some kind of "opposite of infinity". Infinity is not a number either, but at least it behaves sort of like a number in that it can be compared to actual numbers. For any genuine number , the question "Is ?" always has an answer of "yes"; that's the definition. But (ignoring negative numbers) if an infinitesimal were some kind of opposite of infinity in the sense of being infinitely small instead of infinitely large, the answer to the question "Is infinitesimal?" could only have an answer of "yes" in all cases if that infinitesimal were exactly zero. Which would make it a useless concept.

An infinitesimal is a framework for doing differential calculus, nothing more. It involves two very small quantities having a functional relationship to each other, such that they both decrease toward zero simultaneously. Their ratio is the derivative.

If one really wants to point one's finger at a specific thing and call it an infinitesimal, in the "epsilon/delta" formulation of limits the quantities and are the infinitesimals. These are, of course, not actual numbers, but expressions of dependency on in the context of the derivative.

This concept was invented by Gottfried Leibniz with his "dx/dy" notation, and has been used ever since. Dx and dy are the infinitesimals. Isaac Newton's "fluxion" method of calculus did not involve this concept.

External links

See also

References

  1. Calculus, 5th edition, by James Stewart, Nelson Canada, ISBN 053439339X