Functions of several variables 14 1 Limits and Continuity 14 2 Math 241 Spring 2014 Jayadev S Athreya Spring 2014 Jayadev S Athreya Math 241 Spring 2014 Functions of several variables 14 1 Limits and Continuity 14 2 Functions from R2 to R Functions from R3 to R Inputs and outputs A function from R2 to R is a machine that takes in two inputs and spits out one output We write f R2 R Examples f x y 2x 3y f x y x 2 y 2 There are different geometric ways of looking at functions f Jayadev S Athreya Math 241 Spring 2014 Functions of several variables 14 1 Limits and Continuity 14 2 Functions from R2 to R Functions from R3 to R Graphs Like in one variable we can look at the graph For a function g R R the graph is a curve in R2 given by x y y f x For f R2 R it will plotted in 3 space and will be an surface sitting in R3 x y z z f x y 3D Plotter Jayadev S Athreya Math 241 Spring 2014 Functions of several variables 14 1 Limits and Continuity 14 2 Functions from R2 to R Functions from R3 to R Level sets The level sets of a function f R2 R are the sets of points in R2 where f has a constant value That is given c R the level set of level c is given by x y R2 f x y c Jayadev S Athreya Math 241 Spring 2014 Functions of several variables 14 1 Limits and Continuity 14 2 Functions from R2 to R Functions from R3 to R Circles Example f x y x 2 y 2 What are level sets of f For c 0 For c 0 For c 0 Jayadev S Athreya Math 241 Spring 2014 Functions of several variables 14 1 Limits and Continuity 14 2 Functions from R2 to R Functions from R3 to R Circles Example f x y x 2 y 2 What are level sets of f For c 0 Circle centered at 0 of radius c For c 0 the origin 0 0 For c 0 Empty Jayadev S Athreya Math 241 Spring 2014 Functions of several variables 14 1 Limits and Continuity 14 2 Functions from R2 to R Functions from R3 to R Lines Example f x y 2x 3y For any c R the level curve is the line 2x 3y c or y 1 c 2x 3 Jayadev S Athreya Math 241 Spring 2014 Functions of several variables 14 1 Limits and Continuity 14 2 Functions from R2 to R Functions from R3 to R Other examples Exponential f x y ex ey ex y Level sets are again lines since they are level sets of x y Polynomial f x y x 2 y Level sets are parabolas y c x 2 Jayadev S Athreya Math 241 Spring 2014 Functions of several variables 14 1 Limits and Continuity 14 2 Functions from R2 to R Functions from R3 to R Examples What if we want to consider functions with 3 inputs That is functions f R3 R f x y z x 2 y 2 z 2 f x y z x y z Now if we wanted to draw the graph we d have to work in 4 dimensions Can t draw this Jayadev S Athreya Math 241 Spring 2014 Functions of several variables 14 1 Limits and Continuity 14 2 Functions from R2 to R Functions from R3 to R Level surfaces But we can consider level surfaces places where the function is constant That is given c R consider the set of points x y z R3 f x y z c f x y z x 2 y 2 z 2 The level sets x y z f x y z c are c 0 Sphere of radius c centered at 0 c 0 The point 0 0 0 c 0 empty f x y z x y z For any c R the level set is the plane x y z c 0 which has normal vector n 1 1 1 Jayadev S Athreya Math 241 Spring 2014 Functions of several variables 14 1 Limits and Continuity 14 2 Functions from R2 to R Functions from R3 to R Quadric surfaces 12 6 Jayadev S Athreya Math 241 Spring 2014 Functions of several variables 14 1 Limits and Continuity 14 2 Functions from R2 to R Functions from R3 to R Quadric surfaces 12 6 Jayadev S Athreya Math 241 Spring 2014 Functions of several variables 14 1 Limits and Continuity 14 2 Functions from R2 to R Functions from R3 to R Quadric surfaces 12 6 Quadric Surfaces Tool More in section Jayadev S Athreya Math 241 Spring 2014 Functions of several variables 14 1 Limits and Continuity 14 2 One variable limits Higher dimensional limits Limits The most important concept of calculus is the limit Informally we say lim f x L x a if no matter how close you want f x to get to L you can find a small neighborhood of size so that x being within of a guarantees that f x is within of L All this relies on is a notion of distance We have to be careful in higher dimensions because you can approach a point in many different ways Jayadev S Athreya Math 241 Spring 2014 Functions of several variables 14 1 Limits and Continuity 14 2 One variable limits Higher dimensional limits One variable limits The sentence limx a f x L means the following For every 0 acceptable error There is 0 how close you need to be Such that 0 x a implies f x L Jayadev S Athreya Math 241 Spring 2014 Functions of several variables 14 1 Limits and Continuity 14 2 One variable limits Higher dimensional limits One variable limits reprise Given h R h 6 0 let E h f a h L The sentence limx a f x L means the following For every 0 acceptable error There is 0 how close you need to be Such that 0 h implies E h Jayadev S Athreya Math 241 Spring 2014 Functions of several variables 14 1 Limits and Continuity 14 2 One variable limits Higher dimensional limits One sided limits In one variable a limit exists if and only if the one sided limits exist and are equal Limit from the left limx a f x L if For every 0 acceptable error There is 0 how close you need to be Such that 0 a x implies f x L Limit from the right limx a f x L if For every 0 acceptable error There is 0 how close you need to be Such that 0 x a implies f x L Jayadev S Athreya Math 241 Spring 2014 Functions of several variables 14 1 Limits and Continuity 14 2 One variable limits Higher dimensional limits One sided limits reprise In one variable a limit exists if and only if the one sided limits …
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