Calculus

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This diagram shows an approximation to an area under a curve. Credit: Dubhe.

Calculus uses methods originally based on the summation of infinitesimal differences.

It includes the examination of changes in an expression by smaller and smaller differences.

Mathematics[edit]

Main source: Mathematics

Mathematics is about numbers (counting), quantity, and coordinates.

Def. "[a]n abstract representational system used in the study of numbers, shapes, structure and change and the relationships between these concepts"[1] is called mathematics.

Differences[edit]

Here's a theoretical definition:

Def. an abstract relation between identity and sameness is called a difference.

Notation: let the symbol represent difference in.

Notation: let the symbol represent an infinitesimal difference in.

Notation: let the symbol represent an infinitesimal difference in one of more than one.

Changes[edit]

Main sources: Abstractions/Changes and Changes

Def. "[s]ignificant change in or effect on a situation or state"[2] or a "result of a subtraction; sometimes the absolute value of this result"[2] is called a difference.

Derivatives[edit]

Def. a result of an "operation of deducing one function from another according to some fixed law"[3] is called a derivative.

Let

be a function where values of may be any real number and values resulting in are also any real number.

is a small finite change in which when put into the function produces a .

These small changes can be manipulated with the operations of arithmetic: addition (), subtraction (), multiplication (), and division ().

Dividing by and taking the limit as → 0, produces the slope of a line tangent to f(x) at the point x.

For example,

as and go towards zero,

This ratio is called the derivative.

Partial derivatives[edit]

Let

then

where z is held constant and

where x is held contstant.

Areas[edit]

Main sources: Spaces/Areas and Areas

In the figure on the right at the top of the page, an area is the difference in the x-direction times the difference in the y-direction.

This rectangle cornered at the origin of the curvature represents an area for the curve.

Gradients[edit]

Main sources: Spaces/Gradients and Gradients

Notation: let the symbol be the gradient, i.e., derivatives for multivariable functions.

Curvatures[edit]

Main sources: Spaces/Curvatures and Curvatures

The graph at the top of this page shows a curve or curvature.

Variations[edit]

Main sources: Spaces/Variations and Variations

Def. "a partial change in the form, position, state, or qualities of a thing"[4] or a "related but distinct thing"[4] is called a variation.

Area under a curve[edit]

Consider the curve in the graph at the top of the page. The x-direction is left and right, the y-direction is vertical.

For

the area under the curve shown in the diagram at right is the light purple rectangle plus the dark purple rectangle in the top figure

Any particular individual rectangle for a sum of rectangular areas is

The approximate area under the curve is the sum of all the individual (i) areas from i = 0 to as many as the area needed (n):

Integrals[edit]

Def. a "number, the limit of the sums computed in a process in which the domain of a function is divided into small subsets and a possibly nominal value of the function on each subset is multiplied by the measure of that subset, all these products then being summed"[5] is called an integral.

Notation: let the symbol represent the integral.

This can be within a finite interval [a,b]

when i = 0 the integral is evaluated at and i = n the integral is evaluated at . Or, an indefinite integral (without notation on the integral symbol) as n goes to infinity and i = 0 is the integral evaluated at x = 0.

Theoretical calculus[edit]

Def. a branch of mathematics that deals with the finding and properties ... of infinitesimal differences [or changes] is called a calculus.

"Calculus [focuses] on limits, functions, derivatives, integrals, and infinite series."[6]

"Although calculus (in the sense of analysis) is usually synonymous with infinitesimal calculus, not all historical formulations have relied on infinitesimals (infinitely small numbers that are nevertheless not zero)."[7]

Line integrals[edit]

Def. an "integral the domain of whose integrand is a curve"[8] is called a line integral.

"The pulsar dispersion measures [(DM)] provide directly the value of

along the line of sight to the pulsar, while the interstellar Hα intensity (at high Galactic latitudes where optical extinction is minimal) is proportional to the emission measure"[9]

Hypotheses[edit]

Main source: Hypotheses
  1. Calculus can be described using set theory.

See also[edit]

References[edit]

  1. "mathematics, In: Wiktionary". San Francisco, California: Wikimedia Foundation, Inc. January 13, 2013. Retrieved 2013-01-31. 
  2. 2.0 2.1 "difference, In: Wiktionary". San Francisco, California: Wikimedia Foundation, Inc. 28 May 2015. Retrieved 2015-06-25. 
  3. Poccil (13 January 2015). "derivation, In: Wiktionary". San Francisco, California: Wikimedia Foundation, Inc. Retrieved 2015-06-25. 
  4. 4.0 4.1 87.113.182.130 (14 April 2011). "variation, In: Wiktionary". San Francisco, California: Wikimedia Foundation, Inc. Retrieved 2015-06-25. 
  5. "integral, In: Wiktionary". San Francisco, California: Wikimedia Foundation, Inc. 30 May 2015. Retrieved 2015-06-25. 
  6. "Calculus, In: Wikipedia". San Francisco, California: Wikimedia Foundation, Inc. October 13, 2012. Retrieved 2012-10-14. 
  7. "infinitesimal calculus, In: Wiktionary". San Francisco, California: Wikimedia Foundation, Inc. Setember 19, 2012. Retrieved 2013-01-31. 
  8. "line integral, In: Wiktionary". San Francisco, California: Wikimedia Foundation, Inc. September 18, 2013. Retrieved 2013-12-17. 
  9. R. J. Reynolds (May 1, 1991). "Line Integrals of ne and at High Galactic Latitude". The Astrophysical Journal 372 (05): L17-20. doi:10.1086/186013. http://adsabs.harvard.edu/full/1991ApJ...372L..17R. Retrieved 2013-12-17. 

External links[edit]

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