From Math 2220 Class 2 Point Set Terminology Dr. Allen Back Aug. - PowerPoint PPT Presentation

From Math 2220 Class 2 V1 Limits Surface Pictures From Math 2220 Class 2 Point Set Terminology Dr. Allen Back Aug. 29, 2014

Limits From Math 2220 Class 2 V1 Intuition: Limits ( x , y ) → ( a , b ) f ( x , y ) = L lim Surface Pictures Point Set means that as ( x , y ) gets close to ( a , b ), the function values Terminology get close to L .

Limits From Math 2220 Class 2 Intuition: V1 ( x , y ) → ( a , b ) f ( x , y ) = L lim Limits Surface means that as ( x , y ) gets close to ( a , b ), the function values Pictures get close to L . Point Set Terminology As in the 1 variable case, the value of f ( a , b ) (or its undefinedness) does not affect the existence of the limit. If the limit coincides with the value of f ( a , b ), we say f is continuous at (a,b).

Limits Theorems (see pages 95 and 98) say that rational operations From Math 2220 Class 2 (as long as you don’t divide by zero) preserve limits and V1 continuity. Limits For example: Surface Theorem: If Pictures lim ( x , y ) → ( a , b ) f ( x , y ) = L 1 Point Set Terminology exists and lim ( x , y ) → ( a , b ) g ( x , y ) = L 2 � = 0 exists, then f ( x , y ) g ( x , y ) = L 1 lim ( x , y ) → ( a , b ) L 2 (And in particular the limit of the quotients exists.)

Limits From Math 2220 Class 2 V1 Limits Or Theorem: If f ( x , y ) and g ( x , y ) are continuous on a domain Surface U with g ( x , y ) always nonzero, then f ( x , y ) Pictures g ( x , y ) is also continuous Point Set Terminology on U .

Limits From Math 2220 Class 2 V1 Limits xy + 3 Show that f ( x , y ) = 2 − cos y is a continuous function. (i.e. Surface Pictures continuous at every p = ( x 0 , y 0 ) ∈ R 2 . ) Point Set Terminology

Limits From Math 2220 Class 2 V1 Limits What about the corresponding statements (re Surface Pictures continuity/limits) for compositions? Point Set Terminology

Limits From Math 2220 Class 2 V1 What about the corresponding statements (re Limits continuity/limits) for compositions? Surface Pictures The composition of continuous functions is continuous, but the Point Set case of limits of compositions is more complicated. Can you Terminology construct an example to show this?

Limits From Math 2220 Class 2 V1 In 1 variable, when we look at e.g. Limits sin x Surface lim Pictures x x → 0 Point Set Terminology the domain is 1 dimensional and there are just two ways to approach 0; from the left or from the right.

Limits In 1 variable, when we look at e.g. From Math 2220 Class 2 sin x V1 lim x x → 0 Limits the domain is 1 dimensional and there are just two ways to Surface Pictures approach 0; from the left or from the right. Point Set But in two or more variables Terminology p → p 0 f ( p ) lim there are (infinitely) many ways (directions, or curves, or . . . ) to approach p 0 . This is the big difference in higher dimensions.

Limits From Math 2220 Class 2 V1 Does the following limit exist? If so its value? Limits Surface xy Pictures lim x 2 + y 2 Point Set ( x , y ) → (0 , 0) Terminology

Limits From Math 2220 Class 2 V1 Does the following limit exist? If so its value? Limits xy lim Surface x 2 + y 2 Pictures ( x , y ) → (0 , 0) Point Set Terminology Approach 1: Think about behavior along the axes; along lines y = kx in the domain.

Limits From Math 2220 Class 2 Does the following limit exist? If so its value? V1 xy Limits lim x 2 + y 2 ( x , y ) → (0 , 0) Surface Pictures Point Set Terminology Approach 1: Think about behavior along the axes; along lines y = kx in the domain. 0 along the x-axis, but 1 2 along the line y = x shows non-existence.

Limits From Math 2220 Class 2 Does the following limit exist? If so its value? V1 xy Limits lim x 2 + y 2 Surface ( x , y ) → (0 , 0) Pictures Point Set Terminology The point is that limits, if they exist are unique. So finding two different paths through ( a , b ) along which the function approaches different values shows non-existence.

Limits From Math 2220 Class 2 Does the following limit exist? If so its value? V1 xy Limits lim x 2 + y 2 Surface ( x , y ) → (0 , 0) Pictures Point Set Terminology Polar coordinates are another approach. The function simplifies to sin 2 θ for r � = 0 which definitely depends on θ ; so we have 2 non-existence.

Limits From Math 2220 Class 2 V1 Does the following limit exist? Limits Surface x 2 − y 2 Pictures lim Point Set x − y ( x , y ) → (1 , 1) Terminology

Limits From Math 2220 Class 2 V1 Does the following limit exist? If so its value? Limits Surface xy Pictures lim x − y Point Set ( x , y ) → (0 , 0) Terminology

Limits From Math 2220 Class 2 V1 Does the following limit exist? If so its value? Limits Surface 3 2 y x Pictures lim x 2 + y 2 Point Set ( x , y ) → (0 , 0) Terminology

Limits From Math 2220 Class 2 V1 Does the following limit exist? If so its value? Limits Surface sin ( x 2 ) + y 2 Pictures lim x 2 + y 2 Point Set ( x , y ) → (0 , 0) Terminology

Plane z = x + y From Math 2220 Class 2 V1 Limits Sketch the level curves of f ( x , y ) = x + y for Surface c = − 2 , − 1 , 0 , 1 , 2 . Think about vertical sections of the graph Pictures obtained by intersecting with the xz and yz planes. What does Point Set Terminology the graph look like?

Paraboloid z = x 2 + 4 y 2 From Math 2220 Class 2 V1 Limits Surface Pictures Point Set Terminology

Contour Curves of f ( x , y ) = x 2 + 4 y 2 From Math 2220 Class 2 V1 Limits Surface Pictures Point Set Terminology

Hyperbolic Cylinder x 2 − z 2 = − 4 A level surface of f ( x , y , z ) = x 2 − z 2 . From Math 2220 Class 2 V1 Limits Surface Pictures Point Set Terminology

Hyperboloid x 2 + y 2 − z 2 = 4 A level surface of f ( x , y , z ) = x 2 + y 2 − z 2 . From Math 2220 Class 2 V1 Limits Surface Pictures Point Set Terminology

Hyperboloid with Plane Section y = 0 From Math 2220 Class 2 V1 Limits Surface Pictures Point Set Terminology

2 Sheeted Hyperboloid x 2 + y 2 − z 2 = − 4 A level surface of f ( x , y , z ) = x 2 + y 2 − z 2 . From Math 2220 Class 2 The xz plane y = 0 is shown. V1 Limits Surface Pictures Point Set Terminology

Saddles From Math 2220 Class 2 V1 Limits Surface Pictures A fundamental example is the saddle Point Set Terminology z = x 2 − y 2 .

Saddles A fundamental example is the saddle From Math 2220 Class 2 z = x 2 − y 2 . V1 Limits Surface Pictures Point Set Terminology

Saddles From Math 2220 Class 2 V1 A fundamental example is the saddle Limits Surface z = x 2 − y 2 . Pictures Point Set Terminology z = x 2 − y 2 .

Saddles From Math 2220 Class 2 V1 Limits Surface Pictures Point Set z = x 2 − y 2 . Terminology The saddle’s basic property is that along some directions (e.g. a y = 0 section) it is an upward parabola ( z = x 2 ) while in other directions (e.g. an ( x = 0) section) it is a downward parabola z = − y 2 .

Saddles From Math 2220 Class 2 V1 z = x 2 − y 2 . Limits Surface Pictures The saddle’s basic property is that along some directions (e.g. Point Set a y = 0 section) it is an upward parabola ( z = x 2 ) while in Terminology other directions (e.g. an ( x = 0) section) it is a downward parabola z = − y 2 .

Saddles From Math 2220 Class 2 V1 Limits Surface Pictures Point Set Terminology z = x 2 − y 2 . In fact there are also vertical sections (e.g. along y = x ) where z = 0 .

Saddles From Math 2220 Class 2 V1 Limits Surface Pictures Point Set Terminology In fact there are also vertical sections (e.g. along y = x ) where z = 0 . The general pattern is that along a line y = kx we generally get a parabola, but how wide or narrow as well as which way it points depends on the value of k .

Saddles From Math 2220 Class 2 Note that z = xy has the same saddle shape as z = x 2 − y 2 ; V1 upward along y = x and downward along y = − x . Limits Surface Pictures Point Set Terminology

Saddles From Math 2220 Class 2 V1 Limits Surface Pictures Point Set Terminology

Saddles From Math 2220 Class 2 V1 Limits Surface Pictures Point Set Terminology The algebra � x + y � � x − y � = x 2 − y 2 2 √ √ 2 2 may be interpreted as saying z = 2 xy is a 45 degree rotated z = x 2 − y 2 .

Swallowtail z = x 3 − xy Combines all cubic curves z = x 3 − cx in the plane at once! From Math 2220 Class 2 V1 Limits Surface Pictures Point Set Terminology

Point Set Topology Terms From Math 2220 Class 2 V1 Limits D r ( p 0 ) : open disk (or ball) of radius r about p 0 ; Surface Pictures { p : � p − p 0 � < r } . Point Set Terminology

Point Set Topology Terms From Math 2220 Class 2 V1 Limits D r ( p 0 ) : open disk (or ball) of radius r about p 0 ; Surface { p : � p − p 0 � < r } . Pictures Interior Point p of a Set A: Some ball D r ( p ) with r > 0 lies Point Set Terminology entirely in A .