## Sharipov R.'s Course of analytical geometry PDF By Sharipov R.

ISBN-10: 5747725742

ISBN-13: 9785747725744

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8. 7 can enter as subexpressions into simple and multiple sums which will be external sums with respect to them. Then some of their free indices or all of their free indices can turn into summation indices. Those of free indices that remain free are included into the list of free indices of the whole expression. 7 is applied. Taking into account this rue, each free index of the ultimate expression should enter it exactly once, while each summation index should enter it exactly twice — once as an upper index and once as a lower index.

1 the term “ordered system of vectors” is used. This term means a system of vectors in which some ordering of vectors is fixed: e1 is the first vector, e2 is the second vector. If we exchange the vectors e1 and e2 and take e2 for the first vector, while e1 for the second vector, that would be an˜1 , e ˜2 different from other basis e the basis e1 , e2 : ˜ 1 = e2 , e ˜ 2 = e1 . e Let e1 , e2 be a basis on a plane α and let x be some vector § 17. BASES ON A PLANE. 47 lying on this place. Let’s choose some arbitrary point O ∈ α and let’s build the geometric realizations of the three vectors e1 , e2 , and x with the initial point O: −−→ e1 = OA , −−→ e2 = OB , −−→ x = OC .

Conversely, changing a new basis for an old one is understood as an inverse transition. ˜1 , e ˜2 , e ˜3 be two bases in the space E, Let e1 , e2 , e3 and e ˜1 , e ˜2 , e ˜3 is a new basis. where e1 , e2 , e3 is an old basis and e In the direct transition procedure vectors of a new basis are expanded in an old basis, i. e. 1) ˜3 = S31 e1 + S32 e2 + S33 e3 . 1) are called the direct transition formulas. 1) are the coordinates 54 CHAPTER I. VECTOR ALGEBRA. ˜1 expanded in the old basis. 2, they are enumerated by an upper index.