w4-C - Math 20C Multivariable Calculus Lecture 8 1 Slide 1...

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Unformatted text preview: Math 20C Multivariable Calculus Lecture 8 1 Slide 1 & $ % Scalar functions of 2, 3 variables Graph and level curves/surfaces. (Sec. 14.1) Limits and continuity. (Sec. 14.2) x f(x,y) z y Slide 2 & $ % Scalar functions of 2 variables is denoted as f ( x, y ) Definition 1 A scalar function f of two variables ( x, y ) is a rule that assigns to each ordered pair ( x, y ) D IR 2 a unique real number, denoted by f ( x, y ) , that is, f : D IR 2 R IR. Examples: f ( x, y ) = x 2 + y 2 , g ( x, y ) = x- y. Math 20C Multivariable Calculus Lecture 8 2 Slide 3 & $ % Compare f ( x, y ) with r ( t ) Vector valued functions, r : IR IR 2 t h x ( t ) , y ( t ) i Scalar function of two variables, f : IR 2 IR ( x, y ) f ( x, y ) . Slide 4 & $ % The graph of f ( x, y ) is a surface in IR 3 Definition 2 The graph of a function f ( x, y ) is the set of all points ( x, y, z ) in IR 3 of the form ( x, y, f ( x, y )) . z f(x ,y ) = x + y x y 2 2 Math 20C Multivariable Calculus Lecture 8 3 Slide 5 & $ % The domain of a function may not be the whole plane Consider f ( x, y ) = x- y . y x D={(x,y) : x > y } y = x Slide 6 & $ % Curves of constant f ( x, y ) are called level curves Definition 3 The level curves of f ( x, y ) are the curves in in the domain of f , D IR 2 , solutions of the equation f ( x, y ) = k, for k R , a real constant in the range of f . z f ( x , y ) = x + y x y 2 2 Math 20C Multivariable Calculus Lecture 8 4 Slide 7 & $ % Scalar functions of 3 variables are f ( x, y, z ) Definition 4 A scalar function f of three variables ( x, y, z ) is a rule that assigns to each ordered triple ( x, y, z ) D IR 3 a unique real number, denoted by f ( x, y, z ) , that is, f : D IR 3 R IR. Example: f ( x, y, z ) = x 2 + y 2 + z 2 . The graph a function f ( x, y, z ) requires four space dimensions. We cannot picture such graph Slide 8 & $ % Level curves can be generalized from f ( x, y ) to f ( x, y, z ) . In this case they are called level surfaces R z x y R 2 = f ( x, y, z ) = x 2 + y 2 + z 2 . Math 20C Multivariable Calculus Lecture 8 5 Slide 9 & $ % The function f ( x, y ) has the number L as limiting value at the point ( x , y ) roughly means: x f(x,y) z y f(x ,y ) (x ,y ) that for all points ( x, y ) near ( x , y ) the value of f ( x, y ) differs little from L Slide 10 & $ % The definition of limit requires the notion of distance in the plane Definition 5 Given a function f ( x, y ) : D IR 2 IR and a point ( x , y ) IR 2 , we write lim ( x,y ) ( x ,y ) f ( x, y ) = L, if and only if for all...
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This note was uploaded on 04/30/2008 for the course MATH 20C taught by Professor Helton during the Spring '08 term at UCSD.

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w4-C - Math 20C Multivariable Calculus Lecture 8 1 Slide 1...

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