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

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