Calculus/Definitions

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" is called mathematics.

Differences
Here's a theoretical definition:

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

Notation: let the symbol $$\Delta$$ represent difference in.

Notation: let the symbol $$d$$ represent an infinitesimal difference in.

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

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

Derivatives
Def. a result of an "operation of deducing one function from another according to some fixed law" 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,


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

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.

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


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

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

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

Area under a curve
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


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

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" is called an integral.

Notation: let the symbol $$\int $$ represent the integral.


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

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."

"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)."

Line integrals
Def. an "integral the domain of whose integrand is a curve" 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"


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

Hypotheses

 * 1) Calculus can be described using set theory.