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By I. M. Gelfand, E. G. Glagoleva, A. A. Kirillov, Mathematics

ISBN-10: 0817635335

ISBN-13: 9780817635336

The necessity for greater arithmetic schooling on the highschool and university degrees hasn't ever been extra obvious than within the Nineteen Nineties. As early because the Nineteen Sixties, I. M. Gel'fand and his colleagues within the USSR suggestion not easy approximately this comparable query and built a mode for proposing uncomplicated arithmetic in a transparent and easy shape that engaged the interest and highbrow curiosity of hundreds of thousands of highschool and faculty scholars. those comparable principles, this comparable content material, unchanged by way of over thirty years of expertise and mathematical improvement, are on hand within the current books to any scholar who's prepared to learn, to be motivated and to benefit. "The approach to Coordinates" is a manner of moving geometric pictures into formulation, a mode for describing photographs by means of numbers and letters denoting constants and variables. it's primary to the research of calculus and different mathematical issues. academics of arithmetic will locate right here a clean knowing of the topic and a invaluable route to the learning of scholars in mathematical ideas and talents.

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Sample text

For example, the xy-, xz-, and XI/­ planes pass through the x-axis. We shaH therefore say that the x-axis is the intersection of these planes. 57 ~ The six coordinate axes contain only one point in commOn. This Îs the point (0, 0, 0, 0) ­ the origin. 1 Question. What set of points is the intersection of the xy-plane and the yz-plane? of the xy-plane and the zu-plane? We see that we obtain a picture fuUy analogous to the one in three-dimensional space. We can even try to make a schema tic diagram that will help to create some visual model for the disposition of the co­ ordinate planes and axes of four-dimensional space.

It is c1ear that in this way one obtains every point satisfying equation (**). Thus, the set of points determined by equation (**) is obtained in the following manner: Take the circle with its center at the origin and radius 1 lying in the xy-plane, and through each point of this circle con­ struct a straight line parallel to the z-axis. We thus obtain a cylindrical surface (Fig. 25). 4. We have seen that a single equation generally defines a surface in space. But this is not always so. For example, the equation x 2 + y2 = 0 is satisfied only by the points of a tine - the z-axis - since it follows from the equation that x and y are equal to zero, and ail points for which these coordinates are equal to zero lie on the z-axis.

Let A, B, and C be points of four-dimensional space. We can define the angle ABC in the following way. As we are able to compute distances in four­ dimensional space, we can find d(A, B), d(B, C), and d(A, C), that is, the "Iengths of the sides" of the triangle ABC. We now construct in the ordinary two­ dimensional plane a triangle A' B'C' such that its sides A'B', B'C', and C'A' are equal to d(A, B), d(B, C), and d(C, A), respectively. Then the angle A'B'C' of 1 this triangle is defined to be the angle ABC in four­ dimensional space.

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The Method of Coordinates by I. M. Gelfand, E. G. Glagoleva, A. A. Kirillov, Mathematics

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