Engineering Calculus Notes 266

Engineering Calculus Notes 266 - functions the level sets...

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254 CHAPTER 3. REAL-VALUED FUNCTIONS: DIFFERENTIATION Level Curves and Implicit Diferentiation For a function of two variables, there is another way to think about the level set, which in this case is called a level curve : the graph of f ( x,y ) is the locus of the equation z = f ( x,y ) which is a surface in space, and L ( f,c ) is found by intersecting this surface with the horizontal plane z = c , and then projecting the resulting curve onto the xy -plane. Of course, this is a “generic” picture: if for example the function itself happens to be constant, then its level set is the xy -plane for one value, and the empty set for all others. We can cook up other examples for which the level set is quite exotic. However, for many
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Unformatted text preview: functions, the level sets really are curves. For example (see Figure 3.1 : • The level curves of a non-constant a±ne function are parallel straight lines. • The level curves of the function f ( x,y ) = x 2 + y 2 are concentric circles centered at the origin for c > 0, just the origin for c = 0, and the empty set for c < 0. • For the function f ( x,y ) = x 2 4 + y 2 the level sets L ( f,c ) for c > 0 are the ellipses centered at the origin x 2 4 c 2 + y 2 c 2 = 1 which all have the same eccentricity. For c = 0, we again get just the origin, and for c < 0 the empty set. • The level curves of the function f ( x,y ) = x 2 − y 2 are hyperbolas:...
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