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Calculus 2 - Course Notes 3

# Calculus 2 - Course Notes 3 - MAT 1332 Frithjof Lutscher 63...

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MAT 1332: Frithjof Lutscher 63 Functions of several variables I - Introduction Introductory example We can measure the rate of food uptake of a single individual as a function of temperature. We will probably find some optimal temperature T opt , where the uptake rate is highest. At lower temperatures, it is too cold, at higher temperatures, it is too hot for the organism to function properly. If we denote the rate by r and temperature by T then we might try to model this situation with the function r ( T ) = r max exp( ( T T opt ) 2 ) . We can also measure the uptake rate at a constant temperature but in the presence of other individuals. Typically, we see the uptake rate decrease in the presence of others due to competition for food. We have seen such functions before, for example r ( N ) = r max N 1 + N , where N is the number of individuals around. Now we want to vary T and N independently. We could simply multiply the two expressions and get r ( N, T ) = r max N 1 + N exp( ( T T opt ) 2 ) . This function now depends on the two variables T and N. While it is easy to plot the two functions of a single variable above, it is much harder to get a good impression of the function of two variables, see Figure 5. Not only is it more diﬃcult to visualize functions of two and more variables, it is also harder to analyze them. The goal of this chapter is to define concepts such as level sets and derivatives for these functions. Definition: The set R n is the set of all n -tuples ( x 1 , x 2 , . . . , x n ) where all x i are real numbers. So R 1 = R , the real numbers; R 2 is the set of points ( x 1 , x 2 ) in the plane; R 3 are the points in space. We also use the notation ( x, y ) and ( x, y, z ) for points in R 2 and R 3 , respectively. A real-valued function on some subset D R n is a function f : D R that assigns a real number to each element in D. The set D is called the domain of definition of f. The graph of a function f : D R of two variables is the set G = { ( x, y, z ) R 3 : ( x, y ) D, z = f ( x, y ) } . In particular, the graph is a subset of three-dimensional space, and as such, it is not so easy to visualize. Graphs of functions of more variables are defined analogously, but as they are subsets of spaces of dimension 4 and higher, they cannot be visualized in a similar manner. 63

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MAT 1332: Frithjof Lutscher 64 1 1.5 2 2.5 3 1 2 3 4 5 0 2 4 6 8 10 0 1 2 3 4 5 1 2 3 0 5 10 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 Figure 5: Uptake rate as a function of temperature alone (top left), as a function of population density alone (bottom left), and as a function of both independent variables (right). Examples 1. n = 2 , D = R 2 , f ( x 1 , x 2 ) = x 2 1 + x 2 2 . Then for example f (0 , 0) = 0 , f (1 , 0) = 1 , f (1 , 1) = 2 , f (2 , 4) = 20 . If we fix x 2 = 0 the we have a function of a single variable f ( x 1 , 0) = x 2 1 . Similarly, if we fix x 1 . For a visualization of this function, see Figure 6.
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Calculus 2 - Course Notes 3 - MAT 1332 Frithjof Lutscher 63...

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