Insert%20B - GAPHS, LEVEL SETS, SECTIONS This is a...

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GAPHS, LEVEL SETS, SECTIONS This is a geometrical look at functions of several variables. Many of you do much better in the course, if on the top of analytical development presented in the text, some geometrical view is given. For calculus of one variable it has been easy, the piece of paper is xy-plane and a graph of a function was the set of points ( x , ). Its different story in higher dimensions, ) ( x f y = ) ( x f although the ideas are similar. Let T h e s e t U is sometimes called a “natural” domain of the function f. R R U f n : (for example a natural domain of the function 2 2 1 1 ) , ( y x y x f = is ) } 1 : ) , {( 2 2 < + = y x y x U Def . The graph of f is the subset of R n+1 consisting of all the points )) ... , , ( , , .... , , ( 2 1 2 1 n n x x x f x x x for ( ) in U. n x x x ..., , , 2 1 graph f ={ )) ... , , ( , , .... , , ( 2 1 2 1 n n x x x f x x x 1 + n R : ( ) n x x x ..., , , 2 1 U } For n = 2 , the graph is a surface S in R 3 , S = { } U y x R y x f y x ) , ( : )) , ( , , ( 3 Def . The level set of value is the set of those points x R c U at which . c f = ) ( x L c = n R c f U = } ) ( : { x x Note that the level set is always in the domain space. If n = 2, we speak of a level curve (of value c ). If n = 3, we speak of a level surface. Example 1 . Describe the level curves of the quadratic function . 2 2 2 ) , ( , : y x y x R R f + a Solution : We may rewrite f as . The level sets of value c are . 2 2 y x z + = c y x = + 2 2 The level curve of value c is empty for c < 0, for c > 0 it is a circle of radius
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Very often this information is enough to sketch a graph of a function, in the same way as looking at the geographical map of mountains give an idea about their shape. Below are the level curves of the Example 1, raised to the graph: But in many cases the sections may help. Def . A section of the graph of f is the intersection of the graph and a “vertical” hyperplane P i in R n+1 S i = P i graph f In R 3 those hyperplanes are usually plains in space x = a , y = b or y = kx for some a , b , k R . Example 2
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This note was uploaded on 10/21/2010 for the course MATHEMATIC MAT237Y1 taught by Professor Romauldstanczak during the Fall '09 term at University of Toronto- Toronto.

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Insert%20B - GAPHS, LEVEL SETS, SECTIONS This is a...

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