Engineering Calculus Notes 426

# Engineering Calculus Notes 426 - 2 f 2 x 1,x 2 f 3 x 1,x 2...

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414 CHAPTER 4. MAPPINGS AND TRANSFORMATIONS If we expand the superscript notation by thinking of numbers as “1-vectors” ( R 1 = R ), then this definition and notation embrace all of the kinds of functions we have considered earlier. The term transformation is sometimes used when the domain and target live in the same dimension ( m = n ). In Chapter 3 we identified the input to a function of several variables as a vector, while in Chapter 2 we identified the output of a vector-valued function F as a list of functions f i , giving the coordinates of the output. In the present context, when we express the output as a list, we write down the coordinate column of the output vector: for example, a mapping F : R 2 R 3 from the plane to space could be expressed (using vector notation for the input) as [ F ( −→ x )] = f 1 ( −→ x ) f 2 ( −→ x ) f 3 ( −→ x ) or, writing the input as a list of numerical variables, [ F ( x 1 ,x 2 ,x 3 )] = f 1 ( x 1 ,x 2 ) f 2 ( x 1 ,x 2 ) f
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Unformatted text preview: 2 ) f 2 ( x 1 ,x 2 ) f 3 ( x 1 ,x 2 ) . Often we shall be sloppy and simply write F ( −→ x ) = f 1 ( x 1 ,x 2 ) f 2 ( x 1 ,x 2 ) f 3 ( x 1 ,x 2 ) . We cannot draw (or imagine drawing) the “graph” of a mapping F : R n → R m if m + n > 3, but we can try to picture its action by looking at the images of various sets. ±or example, one can view a change of coordinates in the plane or space as a mapping: speciFcally, the calculation that gives the rectangular coordinates of a point in terms of its polar coordinates is the map P : R 2 → R 2 from the ( r,θ )-plane to the ( x,y )-plane given by P ( r,θ ) = b r cos θ r sin θ B . We get a picture of how it acts by noting that it takes horizontal ( resp . vertical) lines to rays from ( resp . circles around) the origin (±igure 4.1 )....
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