Engineering Calculus Notes 426

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

Info icon This preview shows page 1. Sign up to view the full content.

View Full Document Right Arrow Icon
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
Image of page 1
This is the end of the preview. Sign up to access the rest of the document.

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 )....
View Full Document

{[ snackBarMessage ]}

What students are saying

  • Left Quote Icon

    As a current student on this bumpy collegiate pathway, I stumbled upon Course Hero, where I can find study resources for nearly all my courses, get online help from tutors 24/7, and even share my old projects, papers, and lecture notes with other students.

    Student Picture

    Kiran Temple University Fox School of Business ‘17, Course Hero Intern

  • Left Quote Icon

    I cannot even describe how much Course Hero helped me this summer. It’s truly become something I can always rely on and help me. In the end, I was not only able to survive summer classes, but I was able to thrive thanks to Course Hero.

    Student Picture

    Dana University of Pennsylvania ‘17, Course Hero Intern

  • Left Quote Icon

    The ability to access any university’s resources through Course Hero proved invaluable in my case. I was behind on Tulane coursework and actually used UCLA’s materials to help me move forward and get everything together on time.

    Student Picture

    Jill Tulane University ‘16, Course Hero Intern