Calculus

Front 1

Quadrature

Back 1

A process for determining area; sometimes achieved by seeking the dimensions of squarea and circles of the same area.

Front 2

The Fundamental Theorem

Back 2

dA/dx = y

Discovered by Isaac Newton and Gottfried Wilhelm Leibniz, the theorem states that for any curve, the accumulation under the curve is the product of the derivative of the function defining the curve and time.

Front 3

The Backward Problem

Back 3

The holy grail of calculus, the backward problem sought to find the area given a curve y. This problem is the basis of integral calculus.

Front 4

Arithmetica Infinitorum

Back 4

A book by John Wallis on infinity and quadrature that enthralled Isaac Newton. This work helped Newton formalize the infinity principle into aj algorithm.

Front 5

Infinity Principle

Back 5

When calculating a complicated area, reimagine it as an infinite series of simpler areas.

Front 6

Power Series

Back 6

Developed by Newton, the power series enhances the infinity principle by replacing series of calculable geometric areas with a series of powers of x. By chaining powers together in series, Newton was able to express any curve, area, or decimal, backwards and forwards.

Front 7

Madhava of Sangamagrama

Back 7

Founder of the Kerala School of Astronomy and Mathematics. He is credited with discovering the power series for trigometric functions.

Front 8

Unit Fraction

Back 8

A fraction with only a 1 in the numerator.

Front 9

Huygens Infinite Sum

Back 9

Can be solved by Liebniz's

S=(1/1) - (1/N+1)

Front 10

Fourier's Series of Triangle Waves

Back 10

Triangle Wave = sin(x)-(1/9)sin(3x)+(1/25)sin(5x)-(1/49)sin(7x)...

Front 11

d/dx(d/dx(sin(×)))

Back 11

= -sin(x)

Front 12

The Three Body Problem

Back 12

Need to define. Essentially it is the partial differential, nonlinear, equation to compute the forces that 3 bodies exert on each other.

Front 13

Ordinary Differential

Back 13

A differential equation with a single independent variable.

Front 14

Partial Differential

Back 14

A differential equation with two or more independent variables.

Front 15

Fundamental Theory of Algebra

Back 15

All polynomial equations with real or complex coefficients in a single variable have a solution in complex numbers.

Front 16

Complex Numbers

Back 16

A complex number is a number that can be expressed in the form "a + bi" where a and b are real numbers and i is the solution to the equation i^2 = -1.

Front 17

Nonlinear Differential

Back 17

A differential in which the result of integration is not the sum of all parts. The whole is greater than the sum of the parts.

Front 18

Peano's Primitives

Back 18

Peano set forth 3 primitive ideas for the substantiating the throry that all traditional pure mathematics can be derived from the natural numbers.

1) 0

2) Number

3) Successor

Front 19

Peano's 5 Propositions

Back 19

The primitive assumptions behind his 3 primitive ideas and what they mean to the whole of pure mathematics based on the set of natural numbers.

1) 0 is a number

2) The succesor of any number is a number

3) no two numbers have the same successor

4) 0 is not the successor of any number

5) any property which belongs to 0, and also to the successor of every number which has the property, belongs to all the numbers.