Calculus for Life Sciences MAT 1332 C Winter 2010 Jing Li Department of Mathematics and Statistics University of Ottawa March 17, 2010 Jing Li (UofO) MAT 1332 C March 17, 2010 1 / 27
Outline Introductory Example 1 Functions of two or more independent variables 2 The level set 3 Jing Li (UofO) MAT 1332 C March 17, 2010 2 / 27
Introductory Example Introductory Example To survive in cold temperatures, humans must maintain a sufficiently high metabolic rate, or regulate heat loss by covering their skin with insulating material. There is a functional relationship that gives the lowest temperature for survival ( T e ) as a function of metabolic heat production ( M ) and whole-body thermal conductance ( g Hb ). The metabolic heat production depends on the type of activity; some values for humans are summarized in the following table: M in Wm − 2 Activity Sleeping 50 Working at a desk 95 Level walking at 2.5 mph 180 Level walking at 3.5 mph with a 40-lb pack 350 Jing Li (UofO) MAT 1332 C March 17, 2010 3 / 27
Introductory Example The whole body thermal conductance g Hb describes how quickly heat is lost. The value of g Hb depends on the type of protection; for instance, g Hb = 0 . 45 mol m − 2 s − 1 without clothing, g Hb = 0 . 14 mol m − 2 s − 1 for a wool suit, g Hb = 0 . 04 mol m − 2 s − 1 for warm sleeping bag. That is, the smaller g Hb , the better protection from the cold the material provides. Jing Li (UofO) MAT 1332 C March 17, 2010 4 / 27
Introductory Example The relationship between T e , M and g Hb is given by [see Campbell(1986)] T e = 36 − ( 0 . 9 M − 12 )( g Hb + 0 . 95 ) 27 . 8 g Hb where M is measured Wm − 2 ; g Hb is measured in mol m − 2 s − 1 T e is measured in degree Celsius. The temperature T e is a function of two variables, namely M and g Hb ; to meet the required temperature, we can change either M (by starting to move when we get cold) or g Hb (by putting on more clothes when we get cold). Jing Li (UofO) MAT 1332 C March 17, 2010 5 / 27
Introductory Example We can plot T e as a function of M for different values g Hb , 40 30 g Hb =0.45 20 10 0 e g Hb =0.14 T −10 −20 −30 −40 −50 0 50 100 150 200 250 300 350 M Figure: The graph of T e as a function of M for various values of g Hb . Jing Li (UofO) MAT 1332 C March 17, 2010 6 / 27
Introductory Example plot T e as a function of g Hb for different values of M , 100 0 M=50 M=180 −100 e −200 T −300 −400 −500 0 0.1 0.2 0.3 0.4 0.5 g Hb Figure: The graph of T e as a function of g Hb for various values of M . Jing Li (UofO) MAT 1332 C March 17, 2010 7 / 27
Introductory Example The goal of this chapter: to discuss functions of two or more independent variables, develop the theory of differential calculus for such functions, and discuss a number of application. Jing Li (UofO) MAT 1332 C March 17, 2010 8 / 27
Functions of two or more independent variables For the functions of one variable, for example, let f : [ 0 , 4 ] − → R √ x − → x 3.2 2.4 1.6 0.8 -1 -0.5 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 -0.8 -1.6 Figure: The graph of f ( x ) = √ x . The function y = f ( x ) has the domain, [ 0 , 4 ] ; the range, [ 0 , 2 ] Jing Li (UofO) MAT 1332 C March 17, 2010 9 / 27
Functions of two or more independent variables Now, we consider functions for which the domain consists of pairs of real numbers ( x , y ) with x , y ∈ R , or more generally, of n -tuples of real numbers ( x 1 , x 2 , . . . , x n ) with x 1 , x 2 , . . . , x n ∈ R . We also call n -tuples points. We use the notation R n to denote the set of all n -tuples ( x 1 , x 2 , . . . , x n ) with x 1 , x 2 , . . . , x n ∈ R , R n = { ( x 1 , x 2 , . . . , x n ) : x 1 , x 2 , . . . , x n ∈ R } Note: For n = 1, R 1 = R , which is the set of all real numbers. For n = 2, R 2 is the set of all points in the plane. We will use notation ( x , y ) for points in R 2 . For n = 3, R 3 is the set of all points in the space. We will use notation ( x , y , z ) for points in R 3 . n -tuples are ordered; for instance, ( 2 , 3 ) � = ( 3 , 2 ) Jing Li (UofO) MAT 1332 C March 17, 2010 10 / 27
Functions of two or more independent variables The functions of several variables Definition Suppose D ⊂ R n . A real-valued function f on D assigns a real number to each element in D and we write f : D − → R ( x 1 , x 2 , . . . , x n ) − → f ( x 1 , x 2 , . . . , x n ) The set D is the domain of the function f; The set { w ∈ R : f ( x 1 , x 2 , . . . , x n ) = w for some ( x 1 , x 2 , . . . , x n ) ∈ D } is the range of the function f. Jing Li (UofO) MAT 1332 C March 17, 2010 11 / 27
Functions of two or more independent variables The functions of several variables Definition Suppose D ⊂ R n . A real-valued function f on D assigns a real number to each element in D and we write f : D − → R ( x 1 , x 2 , . . . , x n ) − → f ( x 1 , x 2 , . . . , x n ) The set D is the domain of the function f; The set { w ∈ R : f ( x 1 , x 2 , . . . , x n ) = w for some ( x 1 , x 2 , . . . , x n ) ∈ D } is the range of the function f. Remark: If a function f depends on just two independent variables, write f ( x , y ) . In the case of three variables, write f ( x , y , z ) . If f is a function of more than three independent variables, it is more convenient to use subscripts to label the variables, for example, f ( x 1 , x 2 , x 3 , x 4 ) . Jing Li (UofO) MAT 1332 C March 17, 2010 11 / 27
Functions of two or more independent variables Example 1. When n = 2, D = R 2 , f ( x , y ) = x 2 + y 2 , evaluate function f at points ( 0 , 0 ) , ( 1 , 0 ) , ( 0 , 1 ) , ( 2 , 4 ) . Solution: f ( 0 , 0 ) = 0 1 f ( 1 , 0 ) = 1 2 3 f ( 0 , 1 ) = 1 f ( 2 , 4 ) = 2 2 + 4 2 = 20 4 Jing Li (UofO) MAT 1332 C March 17, 2010 12 / 27
Functions of two or more independent variables Example 1. When n = 2, D = R 2 , f ( x , y ) = x 2 + y 2 , evaluate function f at points ( 0 , 0 ) , ( 1 , 0 ) , ( 0 , 1 ) , ( 2 , 4 ) . Solution: f ( 0 , 0 ) = 0 1 f ( 1 , 0 ) = 1 2 3 f ( 0 , 1 ) = 1 f ( 2 , 4 ) = 2 2 + 4 2 = 20 4 Fix y = 0, then we have a function of a single variable f ( x , 0 ) = x 2 ; Jing Li (UofO) MAT 1332 C March 17, 2010 12 / 27
Functions of two or more independent variables Example 1. When n = 2, D = R 2 , f ( x , y ) = x 2 + y 2 , evaluate function f at points ( 0 , 0 ) , ( 1 , 0 ) , ( 0 , 1 ) , ( 2 , 4 ) . Solution: f ( 0 , 0 ) = 0 1 f ( 1 , 0 ) = 1 2 3 f ( 0 , 1 ) = 1 f ( 2 , 4 ) = 2 2 + 4 2 = 20 4 Fix y = 0, then we have a function of a single variable f ( x , 0 ) = x 2 ; Fix x = 0, then we have a function of a single variable f ( 0 , y ) = y 2 . Jing Li (UofO) MAT 1332 C March 17, 2010 12 / 27
Functions of two or more independent variables function f ( x , y ) = x 2 + y 2 Figure: The graph of function f ( x , y ) = x 2 + y 2 . Jing Li (UofO) MAT 1332 C March 17, 2010 13 / 27
Functions of two or more independent variables Example 2. For n = 2, let D = { ( x , y ) ∈ R 2 | 0 ≤ x ≤ 1 , 0 ≤ y ≤ 1 } and f ( x , y ) = x + y Graph the domain of f in the x − y plane and determine the range of f . Solution: The domain of f is the set D : Jing Li (UofO) MAT 1332 C March 17, 2010 14 / 27
Functions of two or more independent variables Example 2. For n = 2, let D = { ( x , y ) ∈ R 2 | 0 ≤ x ≤ 1 , 0 ≤ y ≤ 1 } and f ( x , y ) = x + y Graph the domain of f in the x − y plane and determine the range of f . Solution: The domain of f is the set D : (1,1) 1 D 0.5 -1 -0.5 0 0.5 1 1.5 2 Figure: The domain of the function in Example 2. Jing Li (UofO) MAT 1332 C March 17, 2010 14 / 27
Functions of two or more independent variables f ( x , y ) = x + y , D = { ( x , y ) ∈ R 2 | 0 ≤ x ≤ 1 , 0 ≤ y ≤ 1 } To find the range of f , we need to determine what values f can take when we plug in points ( x , y ) from the domain D . Jing Li (UofO) MAT 1332 C March 17, 2010 15 / 27
Functions of two or more independent variables f ( x , y ) = x + y , D = { ( x , y ) ∈ R 2 | 0 ≤ x ≤ 1 , 0 ≤ y ≤ 1 } To find the range of f , we need to determine what values f can take when we plug in points ( x , y ) from the domain D . The function z = f ( x , y ) is smallest when ( x , y ) = ( 0 , 0 ) , namely, f ( 0 , 0 ) = 0; Jing Li (UofO) MAT 1332 C March 17, 2010 15 / 27
Functions of two or more independent variables f ( x , y ) = x + y , D = { ( x , y ) ∈ R 2 | 0 ≤ x ≤ 1 , 0 ≤ y ≤ 1 } To find the range of f , we need to determine what values f can take when we plug in points ( x , y ) from the domain D . The function z = f ( x , y ) is smallest when ( x , y ) = ( 0 , 0 ) , namely, f ( 0 , 0 ) = 0; The function z = f ( x , y ) is largest when ( x , y ) = ( 1 , 1 ) , namely, f ( 1 , 1 ) = 2; Jing Li (UofO) MAT 1332 C March 17, 2010 15 / 27
Functions of two or more independent variables f ( x , y ) = x + y , D = { ( x , y ) ∈ R 2 | 0 ≤ x ≤ 1 , 0 ≤ y ≤ 1 } To find the range of f , we need to determine what values f can take when we plug in points ( x , y ) from the domain D . The function z = f ( x , y ) is smallest when ( x , y ) = ( 0 , 0 ) , namely, f ( 0 , 0 ) = 0; The function z = f ( x , y ) is largest when ( x , y ) = ( 1 , 1 ) , namely, f ( 1 , 1 ) = 2; The range of f is the set { z : 0 ≤ z ≤ 2 } . Jing Li (UofO) MAT 1332 C March 17, 2010 15 / 27
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