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Define to be the linear transformation such that is angle preserving, the are also pairwise orthogonal.

So, this condition suffices to make be angle preserving. Then impliesi. Similarly, if is norm preserving, then the polarization identity together with the linearity of T give:. A linear transformation is called norm preserving if1.

Calculo En Variedades/ Calculus of Variations : Michael Spivak :

Further, maps each to a scalar multiple of itself; so is a map of the type in part b. If for some realthen substituting back into the equality shows that must be non-positive or must be 0. So since the integrand is always spivqk it must be that its discriminant is negative. I claim that equality holds precisely when one vector is a non-positive multiple of the other.

In fact, suppose that for eachthere is an with. Then part a gives the inequality of Theorem 2. Clearly, any of this form is angle preserving as the composition of two angle preserving linear transformations is angle preserving. The case where is treated similarly. Trabalho do Professor Shing Tung Yau. So T is If for some realthen substituting shows that the inequality is equivalent to and clearly equality holds if a is non. The linear transformation is angle preserving if is and for1.

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Further, giventhere is a withand so. Note, however, that the equality condition does not follow from a. For the converse, suppose that is angle preserving.

No, you could, for example, vary at discrete points without changing the values of the integrals. Since is a linear map of a finite dimensional vector space into itself, it follows that is also onto.

Spivak – Calculus – of – Manifolds – Solutions (2) – Soluções do Livro do Spivak

Further, is norm preserving since. The first assertion is varisdades triangle inequality. This is a consequence of the analogous assertion of the next problem.

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.

Michael David Spivak

Show that is norm preserving if and only if is inner product preserving. If there is a basis of and numbers such thatprove that is angle preserving if and only if all are equal.

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In particular, has an inverse. Equality holds precisely when one is a nonnegative multiple of the other. On the other hand, ifthen the result follows The inequality follows from Theorem Parte 1 de If and are continuous, then the assertion is true.

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

The transformation is by Enn Rule because the determinant of its matrix is 1. If and in are both non-zero, then the angle between anddenotedis defined to be which makes sense by Theorem 2. Geometrically, if, and are the vertices of a triangle, then the inequality says that the length of a side is no larger than cakculo sum of the lengths of the other two sides.

Let be norm preserving. Then the inequality holds true in an open neighborhood of since and are.