Reference request: an elementary proof of Brouwer fixed-point theorem. One of the elementary way to prove of the Brouwer fixed-point theorem is,  making it follow from the (smooth) Non-Retraction theorem. The latter is then proven by contradiction by means of a simple computation on the "oriented area" of smooth mappings $g:B\subset \mathbb {R}^n\rightarrow\mathbb {R}^n$
$$\int_B \operatorname{det} D g(x) dx$$
and only involves a differentiation under the sign of integral with respect to the parameter of deformation (I mentioned this proof in this wiki-article) . Due to this fact, I sometimes like to use it in elementary courses as a meaningful application of differential calculus and Lebesgue integration (on the other hand, the geometrical ideas behind remain a bit hidden, but that is an other story). 
However, a slight annoyance to me now is, that I can't remember where I read this proof the first time, several years ago. I would be very glad to learn a reference, and (if it is known) the name of the inventor of this nice proof.  
 A: There is a completely elementary and very elegant  proof of the Brower fixed point theorem  based on a beautiful combinatorial result called Sperner lemma. For details I recommend  Section 2.3, page 72 of the beautiful book

V.V. Prasolov: Elements of
  Combinatorial and Differential
  Topology,  Graduate Studies in
  Mathematics, vol. 74, Amer. Math.
  Soc., 2006

The proof    is constructive and it leads to an algorithm for generating a sequence of points converging to a fixed point of the map.   Prasolov  attributes this approach  to

B. Kuratowski, C. Knaster, C.
  Mazurkiewicz: Ein Beweis des Fixpunktsatz fur n-dimensionale Simplexe,  Funt. Math. 14(1929),
  132-137.

Note This does not really answer your question.
A: In these notes by Tony Carbery, it is mentioned that a proof along these lines appears in the book Differential Forms and Applications by do Carmo, where it is attributed to E. Lima.
A: Could one of these two be what you're looking for?


*

*J. Milnor, Analytic proofs of the “hairy ball theorem” and the Brouwer fixed-point theorem, Amer. Math. Monthly 85 (1978), no. 7, 521–524. MR MR505523 (80m:55001)

*C. A. Rogers, A less strange version of Milnor’s proof of Brouwer’s fixed-point theorem, Amer. Math. Monthly 87 (1980), no. 7, 525–527. MR MR600910 (82b:55004)

A: According to Lax in his expository paper Change of Variables in Multiple Integrals, Hadamard's original proof of the Brouwer fixed-point theorem from 1910 is based on the determinant formula for change of variables in integrals. The reference he gives is 


*

*J. Hadamard, Sur quelques applications de l'indice Kronecker, pp. 437-477, in J. Tannery, Introduction a la theorie des functions d'une variable, vol. 2, Paris, 1910.

A: Here are two further references of proofs of the fixed point theorem that rely on evaluating determinants:
MR0117523 Dunford, Nelson ;  Schwartz, Jacob T.  Linear Operators. I. General Theory.
With the assistance of W. G. Bade and R. G. Bartle. Pure and Applied
 Mathematics, Vol. 7 Interscience Publishers, Inc., New York; Interscience Publishers, Ltd.,
 London  1958 xiv+858 pp. The proof is on page 467.
MR0610487  Kannai, Yakar . An elementary proof of the no-retraction theorem.
 Amer. Math. Monthly  88  (1981),  no. 4, 264--268.
A: There is an interesting essay on Brouwer's Fixed Point theorem, including a contructive proof, at Kevin Brown's MathPages site http://www.mathpages.com/home/kmath262/kmath262.htm
Looking at the home page, I see he has now written a string of books. If his articles are any indication, these books are doubtless excellent and well worth buying.
