Derivations of Applied Mathematics by Black T. H.

By Black T. H.

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7 z is an independent variable and a is an indeterminate constant. There is another way to view the power operation, however. One can view it as the exponential operation az , where the variable z is in the exponent and the constant a is in the base. 1 The logarithm The exponential operation follows the same laws the power operation follows, but because the variable of interest is now in the exponent rather than the base, the inverse operation is not the root but rather the logarithm: loga (az ) = z.

ROTATION 49 ˆx + y ˆ y + ˆzz is a vector. If x, y and z are complex, then 8 x |v|2 = |x|2 + |y|2 + |z|2 = x∗ x + y ∗ y + z ∗ z = [ (x)]2 + [ (x)]2 + [ (y)]2 + [ (y)]2 + [ (z)]2 + [ (z)]2 . 3) A point is sometimes identified by the vector expressing its distance and direction from the origin of the coordinate system. That is, the point ˆx + y ˆ y. However, in the general (x, y) can be identified with the vector x case vectors are not associated with any particular origin; they represent distances and directions, not fixed positions.

43). 11 Functions This book is not the place for a gentle introduction to the concept of the function. Briefly, however, a function is a mapping from one number (or vector of several numbers) to another. For example, f (x) = x 2 − 1 is a function which maps 1 to 0 and −3 to 8, among others. When discussing functions, one often speaks of domains and ranges. The domain of a function is the set of numbers one can put into it. The range of a function is the corresponding set of numbers one can get out of it.

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