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]

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[edit]

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[edit]

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[edit]

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[edit]

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[edit]

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[edit]

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[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 \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[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

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[edit]

Hypothesis:

  1. Calculus can be described using set theory.

Control groups[edit]

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[edit]

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[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 87.113.182.130 (14 April 2011). "variation, In: Wiktionary". San Francisco, California: Wikimedia Foundation, Inc. Retrieved 2015-06-25. 
  3. 3.0 3.1 "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). "Line Integrals of ne and n_e^2 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. 
  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. 

External links[edit]

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