Calculus/Definitions

MathematicsEdit

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.

DifferencesEdit

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.

ChangesEdit

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.

DerivativesEdit

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 derivativesEdit

Let

 

then

 
 

where z is held constant and

 

where x is held contstant.

AreasEdit

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.

GradientsEdit

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

 

CurvaturesEdit

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

VariationsEdit

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 curveEdit

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):

 

IntegralsEdit

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 calculusEdit

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 integralsEdit

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]

 

HypothesesEdit

  1. Calculus can be described using set theory.
  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. September 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.