Telecommunications Sensors

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By D. P. Howson (Auth.)

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24. Prove A . (Β x C) = Β . (C χ A) = C . (Α χ Β) 25. Prove ( A x B ) x ( C x D ) = ( À x B . D ) C - ( A x B , C)D 2 26. Prove (Α χ Β). (A x C) = A ( B . C) - (A. B)(A. C) 27. ^ + A ^ dt d A dt η A/ ι d and dt A dB — À x B = τ xB-t-Ax — dt dt dt 28. Calculate the value of j* J * x(x + y) dy dx over a surface defined by the following four lines in the x, y plane, χ = 0, χ = 1, y = 0, y = 2 . *2>'2(*) 29. Change the variables in |* j / ( x , y) dx dy to polar coordinates. In χσι(χ) what conditions might it be an advantage to use polar rather than Cartesian coordinates to evaluate a double integral?

If it is found by some means that n f(x) = Σαη(χ — a) for χ ^ a, it is straightforward to show that this series is the Taylor series, f o r / ' ( x ) = 0 + al + 2a2(x — a) + ... and eventually f(x) = 0 + . . + + an. n\ + a n +i . (n + 1)! (x - a) + . . 26 M A T H E M A T I C S FOR E L E C T R O N I C TECHNOLOGY so that (n) f{a) = n\an showing that an has the correct value to be the Taylor series coefficient for (x - af. The use of the Taylor series to give the value of a function when χ is slightly perturbed from an equilibrium position is of great importance.

J Hence not only are the terms in the expansion 1 of Zl the same as those of A, but their signs are the same, proving that the interchange of rows and columns in a determinant leaves the value unchanged. Hence in following work whenever rows are mentioned, columns may be read, and vice versa. 2,2 The Laplace Development It has been shown that an «th-order determinant may be expanded with reference to its first column in terms of its minors (each with(« — 1) rows), and the same is true for its first row.

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