Calculus

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

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.

Variations

Def. "a partial change in the form, position, state, or qualities of a thing"[2] or a "related but distinct thing"[2] is called a variation.

Differences

Def. "[s]ignificant change in or effect on a situation or state"[3] or a "result of a subtraction; sometimes the absolute value of this result"[3] is called a difference.

Notation: let the symbol \Delta represent change in.

Notation: let the symbol d represent an infinitesimal change in.

Notation: let the symbol \partial represent an infinitesimal change in one of more than one.

Derivatives

Def. a result of an "operation of deducing one function from another according to some fixed law"[4] is called a derivative.

Let

y = f(x)

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

\Delta x is a small finite change in x which when put into the function f(x) produces a \Delta y.

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

\Delta y = f(x + \Delta x) - f(x)

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

For example,

f(x) = x^2
f(x + \Delta x) = (x + \Delta x)^2 = x^2 + 2x\Delta x + (\Delta x)^2
\Delta y = (x^2 + 2x\Delta x + (\Delta x)^2) - x^2
\Delta y/\Delta x = (2x\Delta x + (\Delta x)^2)/\Delta x
\Delta y/\Delta x = 2x + \Delta x

as \Delta x and\Delta y go towards zero,

dy/dx = 2x + dx = limit_{\Delta x\to 0}{f(x+\Delta x)-f(x)\over \Delta x} = 2x.

This ratio is called the derivative.

Partial derivatives

Let

y = f(x,z)

then

\partial y = \partial f(x,z) = \partial f(x,z) \partial x + \partial f(x,z) \partial z
\partial y/ \partial x = \partial f(x,z)

where z is held constant and

\partial y / \partial z = \partial f(x,z)

where x is held contstant.

Gradients

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

\nabla f(x,z) = \partial y = \partial f(x,z) = \partial f(x,z) \partial x + \partial f(x,z) \partial z.

Area under a curve

For

\Delta x * \Delta y = [f(x + \Delta x) - f(x)] * \Delta x

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

\Delta x * \Delta y + f(x) * \Delta x = f(x + \Delta x) * \Delta x.

Any particular individual rectangle for a sum of rectangular areas is

f(x_i + \Delta x_i) * \Delta x_i.

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

\sum_{i=0}^{n} f(x_i + \Delta x_i) * \Delta x_i.

Integrals

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 \int represent the integral.

limit_{\Delta x\to 0}\sum_{i=0}^{n} f(x_i + \Delta x_i) * \Delta x_i = \int f(x)dx.

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

\int_a^b f(x) \; dx

when i = 0 the integral is evaluated at a and i = n the integral is evaluated at b. 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

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

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

DM = \int_0^\infty n_e\, ds

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]

EM = \int_0^\infty n_e^2 ds.

Research

Hypothesis:

  1. Calculus can be described using set theory.

Control groups

This is an image of a Lewis rat. Credit: Charles River Laboratories.

The findings demonstrate a statistically systematic change from the status quo or the control group.

“In the design of experiments, treatments [or special properties or characteristics] are applied to [or observed in] experimental units in the treatment group(s).[10] In comparative experiments, members of the complementary group, the control group, receive either no treatment or a standard treatment.[11]"[12]

Proof of concept

Def. a “short and/or incomplete realization of a certain method or idea to demonstrate its feasibility"[13] is called a proof of concept.

Def. evidence that demonstrates that a concept is possible is called proof of concept.

The proof-of-concept structure consists of

  1. background,
  2. procedures,
  3. findings, and
  4. interpretation.[14]

See also

References

  1. ↑ "mathematics, In: Wiktionary". San Francisco, California: Wikimedia Foundation, Inc. January 13, 2013. Retrieved 2013-01-31.
  2. 1 2 87.113.182.130 (14 April 2011). "variation, In: Wiktionary". San Francisco, California: Wikimedia Foundation, Inc. Retrieved 2015-06-25.
  3. 1 2 "difference, In: Wiktionary". San Francisco, California: Wikimedia Foundation, Inc. 28 May 2015. Retrieved 2015-06-25.
  4. ↑ Poccil (13 January 2015). "derivation, 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). n_e^2. The Astrophysical Journal 372 (05): L17-20. doi:10.1086/186013. http://adsabs.harvard.edu/full/1991ApJ...372L..17R. Retrieved 2013-12-17.
  10. ↑ Klaus Hinkelmann, Oscar Kempthorne (2008). Design and Analysis of Experiments, Volume I: Introduction to Experimental Design (2nd ed.). Wiley. ISBN 978-0-471-72756-9. http://books.google.com/?id=T3wWj2kVYZgC&printsec=frontcover.
  11. ↑ R. A. Bailey (2008). Design of comparative experiments. Cambridge University Press. ISBN 978-0-521-68357-9. http://www.cambridge.org/uk/catalogue/catalogue.asp?isbn=9780521683579.
  12. ↑ "Treatment and control groups, In: Wikipedia". San Francisco, California: Wikimedia Foundation, Inc. May 18, 2012. Retrieved 2012-05-31.
  13. ↑ "proof of concept, In: Wiktionary". San Francisco, California: Wikimedia Foundation, Inc. November 10, 2012. Retrieved 2013-01-13.
  14. ↑ Ginger Lehrman and Ian B Hogue, Sarah Palmer, Cheryl Jennings, Celsa A Spina, Ann Wiegand, Alan L Landay, Robert W Coombs, Douglas D Richman, John W Mellors, John M Coffin, Ronald J Bosch, David M Margolis (August 13, 2005). "Depletion of latent HIV-1 infection in vivo: a proof-of-concept study". Lancet 366 (9485): 549-55. doi:10.1016/S0140-6736(05)67098-5. http://www.ncbi.nlm.nih.gov/pmc/articles/PMC1894952/. Retrieved 2012-05-09.

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