Most important theorems

[Killjoy]

What are people's opinions on which are the most important theorems in mathematics? By important I mean those with the most significant consequences both in pure and applied mathematics (not just obscure results that happen to be interesting to a few individuals). I would suggest the the Pythagoras Theorem should be up there. What else?

[ajb]

The fundamental theorem of calculus and the mean value theorem. (these two are very similar in fact)

 

 

Also the fundamental theorem of algebra.

Edited May 1, 2009 by ajb

[Kyle]

I think Taylor's theorem and whatever theorem introduced Fourier Series (not sure what it would be called) are pretty important.

[Bignose]

The fundamental theorem of calculus

 

This one gets my vote by and far. Not a whole lot gets done without this being true. I guess that's why it's been named "fundamental" eh?

[Killjoy]

The fundamental theorem of calculus

 

This one gets my vote by and far. Not a whole lot gets done without this being true. I guess that's why it's been named "fundamental" eh?

 

I agree that the fundamental theorem of calculus should be included. There is also a generalisation called Stoke's Theorem which, while perhaps a little advanced to be called "fundamental", actually incorporates several concepts in physics and mathematics - including the fundamental theorem (and Green's theorem, and the divergence theorem), is used to derive equations such as the wave equation and the diffusion equation, and is of central importance in differential geometry, general relativity, and many other fields.

[Bignose]

Firstly, welcome to the forum

 

Secondly, The guy the theorem is named after is Stokes, not Stoke, so it would be properly written Stokes'

 

Thirdly, the link is broken (extra http://) in it

 

Fourthly, Stokes' Theorem isn't the same as the fundamental theorem. Stokes' Theorem is how to turn surface integrals into line integrals. The Gauss-Green-Ostrogradsky divergence theorem (has many names depending on the reference you are looking at) is how to turn volume integrals into surface integrals. Divergence is how to turn 3-D volume into 2-D surface and Stokes' is the 2-D to 1-D equivalent. Neither one covers the same ground as the fundamental theorem which proves the truly fundamental property that differentiation and integration are intricately related and are inverse operations (to an arbitrary constant).

[Killjoy]

Firstly, welcome to the forum

 

Secondly, The guy the theorem is named after is Stokes, not Stoke, so it would be properly written Stokes'

 

Thirdly, the link is broken (extra http://) in it

 

Fourthly, Stokes' Theorem isn't the same as the fundamental theorem. Stokes' Theorem is how to turn surface integrals into line integrals. The Gauss-Green-Ostrogradsky divergence theorem (has many names depending on the reference you are looking at) is how to turn volume integrals into surface integrals. Divergence is how to turn 3-D volume into 2-D surface and Stokes' is the 2-D to 1-D equivalent. Neither one covers the same ground as the fundamental theorem which proves the truly fundamental property that differentiation and integration are intricately related and are inverse operations (to an arbitrary constant).

 

WRT point 1: Thank you

 

WRT points 2&3: Sorry - my bad

 

WRT point 4: The name "Stokes' Theorem" sometimes refers to a specific case of a more general theorem which, rather confusingly, is also called "Stokes' Theorem". This more general theorem relates what happens in an N-dimensional hypervolume to what happens on the orientable (N-1)-dimensional hypersurface of that hypervolume, all embedded in an M dimensional manifold. Depending on the values of M and N, and what kinds of functions are being considered you get the various different theorems mentioned, and many others. If N=1 you recover the fundamental theorem of calculus.

I'll try to find a better reference and post it.