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.
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.
Def. a result of an "operation of deducing one function from another according to some fixed law" is called a derivative.
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.
as and go towards zero,
This ratio is called the derivative.
Partial derivatives edit
where z is held constant and
where x is held contstant.
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.
Notation: let the symbol be the gradient, i.e., derivatives for multivariable functions.
The graph at the top of this page shows a curve or curvature.
Area under a curve edit
Consider the curve in the graph at the top of the page. The x-direction is left and right, the y-direction is vertical.
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):
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 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 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."
"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 edit
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
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"
- Calculus can be described using set theory.
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