Cálculo en Variedades – Michael – Download as PDF File .pdf), Text File .txt) or read online. Cálculo en Variedades – Michael Spivak – Download as PDF File .pdf) or read online. Title, Cálculo en variedades. Author, Michael Spivak. Edition, illustrated, reprint. Publisher, Reverte, ISBN, ,

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If for some realthen substituting back into the equality shows that must be non-positive or must be 0. If for some realthen substituting shows that the inequality is equivalent to and clearly equality holds if a is non. Trabalho do Professor Shing Tung Yau. Let be an orthogonal basis of.

### Calculo En Variedades/ Calculus of Variations : Michael Spivak :

This completes the characterization. If and are continuous, then the assertion is true. Further, giventhere is a withand so. Define to be the variedzdes transformation such that is angle preserving, the are also pairwise orthogonal. In fact, suppose that for eachthere is an with. Since is a linear map of a finite dimensional vector space into itself, it follows that is also onto.

Then the inequality holds true in an open neighborhood of since and are.

If and in are both non-zero, then the angle between anddenotedis defined to be which makes sense by Theorem 2. If there is a basis of and numbers such thatprove that is angle preserving if and only if all are equal.

## Spivak – Calculus – of – Manifolds – Solutions (2)

Let be norm preserving. To correct the situation, add the condition that the be pairwise orthogonal, because all the cross terms are zero. Geometrically, if, and are the vertices of a triangle, then the inequality says that the length of a side is no larger than the spuvak of the lengths of the other two sides. I claim that equality holds precisely when one vector is a non-positive multiple of the other.

Further, maps each to a scalar multiple variedaded itself; so is a map of the type in part b. Then impliesi.

The case where is treated similarly. The first assertion is the triangle inequality. So, this condition suffices to make be angle preserving. The linear transformation is angle preserving if is and for1.

### Calculo En Variedades/ Calculus of Variations: Michael Spivak: Books –

No, you could, for example, vary at discrete points without changing the spiak of the integrals. So T is This is a consequence of the analogous assertion of the next problem.

Similarly, if is norm preserving, then the polarization identity together with the linearity of T give:. The transformation is by Cramer’s Rule because the determinant of its matrix is 1. On the other hand, ifthen the result follows The inequality follows from Theorem The angle preserving are precisely those which can be expressed in the form where U is angle preserving of the kind in part bV is norm preserving, and the operation is functional composition.

Then part a gives the inequality of Theorem 2.

## Michael David Spivak

Note, however, that the equality condition does not follow from a. Prove that if is norm preserving, then is angle preserving.

Equality holds precisely when one is a nonnegative multiple of the other. Further, is norm preserving since. So since the integrand is always non-negative it must be that its discriminant is negative.