Scalar functions of several variables (Sect. 14.1)

Transcription

1 Scalar functions of several variables (Sect. 14.1) Functions of several variables. On open, closed sets. Functions of two variables: Graph of the function. Level curves, contour curves. Functions of three variables. Graphs, level surfaces. f(,) Scalar functions of several variables A scalar function of n variables is a function f : D R n R R, where n N, the set D is called the domain of the function, and the set R is called the range of the function. Remark: Comparison between f : R 2 R with r : R R 2. f(,) r(t) t

2 Functions of several variables Eample An eample of a scalar-valued function of two variables, T : R 2 R is the temperature T of a plane surface, sa a table. Each point (, ) on the table is associated with a number, its temperature T (, ). An eample of a scalar-valued function of three variables, T : R 3 R is the temperature T of this room. Each point (,, ) in the room is associated with a number, its temperature T (,, ). Another eample of a scalar function of three variables is the speed of the air in the room. An eample of a vector-valued function of three variables, v : R 3 R 3, is the velocit of the air in the room. Scalar functions of several variables Eample Find the maimum domain D and range R sets where the function f : D R 2 R R given b f (, ) = is defined. The function f (, ) = is defined for all points (, ) R 2, therefore, D = R 2. f(,) = R Since f (, ) = for all (, ) D, then the range space is R = [0, ). D

3 Scalar functions of several variables Eample Find the maimum domain D and range R sets where the function f : D R 2 R R given b f (, ) = is defined. The function f is defined for points (, ) R 2 such that 0. So, D = {(, ) R 2 : }. = D={(,) : > } Since f (, ) = 0 for all (, ) D, the range space is R = [0, ). R D Scalar functions of several variables Eample Find the maimum domain D and range R sets where the function f : D R 2 R R given b f (, ) = 1/ is defined. The function f is defined for points (, ) R 2 such that > 0. So, D = {(, ) R 2 : > }. = D={(,) : > } Since f (, ) = 1/ 0 for all (, ) D, the range space is R = (0, ). R D

4 Scalar functions of several variables (Sect. 14.1) Functions of several variables. On open, closed sets. Functions of two variables: Graph of the function. Level curves, contour curves. Functions of three variables. Graphs, level surfaces. On open and closed sets in R n Remark: We first generalie from R 3 to R n the definition of a ball of radius r centered at ˆP c. An open ball of radius r > 0 centered at ˆP c = (ˆ 1,, ˆ n ) is the set in R n, with n N, given b B r (ˆP c ) = {( 1,, n ) R n : ( 1 ˆ 1 ) ( n ˆ n ) 2 < r 2 }. Remark: An open ball B r (ˆP c ) contains the points inside a sphere of radius r centered at ˆP c without the points of the sphere.

5 On open and closed sets in R n A point P S R n, with n N, is called an interior point iff there is a ball B r (P) S. A point P S R n, with n N, is called a boundar point iff ever ball B r (P) contains points in S and points outside S. The boundar of a set S is the set of all boundar points of S. R 2 Boundar point S Interior point Boundar On open and closed sets in R n A set S R n, with n N, is called open iff ever point in S is an interior point. The set S is called closed iff S contains its boundar. A set S is called bounded iff S is contained in ball, otherwise S is called unbounded. closed and unbounded open and bounded bounded closed and bounded

6 On open and closed sets in R n Eample Find and describe the maimum domain of the function f (, ) = ln( 2 ). The maimum domain of f is D D = {(, ) R 2 : > 2 }. D is an open, unbounded set. Scalar functions of several variables (Sect. 14.1) Functions of several variables. On open, closed sets. Functions of two variables: Graph of the function. Level curves, contour curves. Functions of three variables. Graphs, level surfaces.

7 The graph of a function of two variables is a surface in R 3 The graph of a function f : D R 2 R is the set of all points (,, ) in R 3 of the form (,, f (, )). The graph of a function f is also called the surface = f (, ). Eample Draw the graph of f (, ) = The graph of f is the surface = This is a paraboloid along the ais. f(,) = Scalar functions of several variables (Sect. 14.1) Functions of several variables. On open, closed sets. Functions of two variables: Graph of the function. Level curves, contour curves. Functions of three variables. Graphs, level surfaces.

8 Level curves, contour curves The contour curves of a function f : D R 2 R R are the curves in R 3 given b the equation f (, ) = k, = k, (, ) D, k R. The level curves of the function f are the curves in the domain D R 2 given b the equation f (, ) = k, (, ) D, k R. Remark: Contour curves are the intersection of the graph of f with horiontal planes = k. Remark: Level curves are the vertical translation of contour curves to the function domain. Level curves, contour curves. Eample Find and draw few level curves and contour curves for the function f (, ) = f(,) = The level curves are solutions of the equation = k with k 0. These curves are circles of radius k in D = R 2. contour curves The contour curves are the circles {(,, ) : = k, = k}. D level curves

9 Level curves, contour curves Eample Find the maimum domain, range of, and graph the function 1 f (, ) = Since the denominator never vanishes, hence D = R 2. 1 Since 0 < , the + 2 range of f is R = (0, 1]. 1 f(,) The contour curves are circles on horiontal planes in (0, 1]. Level curves, contour curves N Eample Given the topographic map in the figure, which wa do ou choose to the summit? W S E 1000 From the east side

10 Scalar functions of several variables (Sect. 14.1) Functions of several variables. On open, closed sets. Functions of two variables: Graph of the function. Level curves, contour curves. Functions of three variables. Graphs, level surfaces. Scalar functions of three variables The graph of a scalar function of three variables, f : D R 3 R R, is the set of points in R 4 of the form (,,, f (,, )) for ever (,, ) D. Remark: The graph a function f : D R 3 R requires four space dimensions. We cannot picture such graph. The level surfaces of a function f : D R 3 R R are the surfaces in the domain D R 3 of f solutions of the equation f (,, ) = k, where k R is a constant in the range of f.

11 Scalar functions of three variables Eample Draw one level surface of the function f : D R 3 R R 1 f (,, ) = The domain of f is D = R 3, the range is R = (0, ). For k > 0 the level surfaces f (,, ) = k are = 1 k, spheres radius R = 1 k. R