# Fundamental Theorems

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##### Sure, there are lots of topics in mathematics. And for each topic, there is a number of
theorems. But only one theorem in each subject area earns the title of

# THE FUNDAMENTAL THEOREM OF *X*

### where *X* is the particular subject in question.

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So here we go...**

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The Fundamental Theorem of Arithmetic:

Any positive integer *n* can be represented in exactly one way as a product of primes *p*_{i}:

*n* = *p*_{1}*p*_{2}*p*_{3}...
*p*_{k}

The Fundamental Theorem of Algebra:

Every Polynomial equation having Complex Coefficients and degree *n* > 0
has at least one Complex Root.

The five postulates of (Euclidean) Geometry:

*(note: we do not list a fundamental theorem here. Rather, we note that all theorems
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must follow from a set of statements simply assumed to be true. Since all other theorems

follow from these postulates, we acknowledge said postulates as the "fundamentals".)

1. A straight line segment can be drawn joining any two points.

2. Any straight line segment can be extended indefinitely in a straight Line.

3. Given any straight line segment, a circle can be drawn having the segment as radius

and one endpoint as center.

4. All right angles are congruent.

5. Given a line and a point not on that line, there exists exactly one line through the

given point parallel to the given line. (*The parallel postulate*)

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*(note: in *Hyperbolic Geometry* we replace the parallel postulate by the following:
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5. Given a line and a point not on that line, there exist an infinite number of

lines through the given point parallel to the given line.)

*(note: in *Spherical Geometry* (where lines are defined to be great circles on a sphere)
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we may replace the parallel postulate by the following:

5. All lines intersect in exactly two points.* -- i.e. there are no parallel lines.)
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The Fundamental Theorem of Calculus:

If *f* is a continuous function, then

and A'(*x*) = f(*x*)

Let *f* be continuous on [*a*,*b*] and let *F* be an antiderivative
for *f*. Then

The Fundamental Theorem(s) of Multivariable Calculus:

(Note: technically, all these theorems are a special case of an *n*-dimensional Stokes' Theorem.)

Green's Theorem: Let C be a simple, closed path, orientated counterclockwise, that encloses
Q. Then

Stokes' Theorem: If the surface S is bounded by the curve C, then

Divergence Theorem (Gauss): If the surface S encloses the region R, then