# 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.

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]

# References

1. "mathematics, In: Wiktionary". San Francisco, California: Wikimedia Foundation, Inc. January 13, 2013. Retrieved 2013-01-31.
2. 87.113.182.130 (14 April 2011). "variation, In: Wiktionary". San Francisco, California: Wikimedia Foundation, Inc. Retrieved 2015-06-25.
3. "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. 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.
11. R. A. Bailey (2008). Design of comparative experiments. Cambridge University Press. ISBN 978-0-521-68357-9.
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. Retrieved 2012-05-09.