Brouwer′s Fixed-Point Theorem and a Generalization of the Formula for Change of Variables in Multiple Integrals

The most elementary one-dimensional change of variable formula for definite integrals does not require that the map involved be one-to-one. The Jacobian, i.e., the derivative, does not appear here in absolute value. Likewise the usual Jacobian change of variable formula for multiple integrals can be stated for maps that are not one-to-one as long as the boundaries are preserved diffeomorphically. From this an extremely simple proof is derived for the non-existence of a smooth retract of an embedded compact n-dimensional manifold with boundary onto its boundary.