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

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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

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

From Math 2220 Class 2

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

Limits

Intuition: lim

(x,y)→(a,b) f (x, y) = L

means that as (x, y) gets close to (a, b), the function values get close to L.

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

Limits

Intuition: lim

(x,y)→(a,b) f (x, y) = L

means that as (x, y) gets close to (a, b), the function values get close to L. 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).

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

Limits

Theorems (see pages 95 and 98) say that rational operations (as long as you don’t divide by zero) preserve limits and continuity. For example: Theorem: If lim(x,y)→(a,b)f (x, y) = L1 exists and lim(x,y)→(a,b)g(x, y) = L2 = 0 exists, then lim(x,y)→(a,b) f (x, y) g(x, y) = L1 L2 (And in particular the limit of the quotients exists.)

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

Limits

Or Theorem: If f (x, y) and g(x, y) are continuous on a domain U with g(x, y) always nonzero, then f (x, y) g(x, y) is also continuous

  • n U.
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From Math 2220 Class 2 V1 Limits Surface Pictures Point Set Terminology

Limits

Show that f (x, y) = xy + 3 2 − cos y is a continuous function. (i.e. continuous at every p = (x0, y0) ∈ R2.)

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

Limits

What about the corresponding statements (re continuity/limits) for compositions?

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

Limits

What about the corresponding statements (re continuity/limits) for compositions? The composition of continuous functions is continuous, but the case of limits of compositions is more complicated. Can you construct an example to show this?

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

Limits

In 1 variable, when we look at e.g. lim

x→0

sin x x the domain is 1 dimensional and there are just two ways to approach 0; from the left or from the right.

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

Limits

In 1 variable, when we look at e.g. lim

x→0

sin x x the domain is 1 dimensional and there are just two ways to approach 0; from the left or from the right. But in two or more variables lim

p→p0 f (p)

there are (infinitely) many ways (directions, or curves, or . . . ) to approach p0. This is the big difference in higher dimensions.

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

Limits

Does the following limit exist? If so its value? lim

(x,y)→(0,0)

xy x2 + y2

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

Limits

Does the following limit exist? If so its value? lim

(x,y)→(0,0)

xy x2 + y2 Approach 1: Think about behavior along the axes; along lines y = kx in the domain.

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

Limits

Does the following limit exist? If so its value? lim

(x,y)→(0,0)

xy x2 + y2 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.

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

Limits

Does the following limit exist? If so its value? lim

(x,y)→(0,0)

xy x2 + y2 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.

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

Limits

Does the following limit exist? If so its value? lim

(x,y)→(0,0)

xy x2 + y2 Polar coordinates are another approach. The function simplifies to sin 2θ 2 for r = 0 which definitely depends on θ; so we have non-existence.

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

Limits

Does the following limit exist? lim

(x,y)→(1,1)

x2 − y2 x − y

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

Limits

Does the following limit exist? If so its value? lim

(x,y)→(0,0)

xy x − y

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

Limits

Does the following limit exist? If so its value? lim

(x,y)→(0,0)

x

3 2 y

x2 + y2

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

Limits

Does the following limit exist? If so its value? lim

(x,y)→(0,0)

sin (x2) + y2 x2 + y2

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

Plane z = x + y

Sketch the level curves of f (x, y) = x + y for c = −2, −1, 0, 1, 2. Think about vertical sections of the graph

  • btained by intersecting with the xz and yz planes. What does

the graph look like?

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

Paraboloid z = x2 + 4y 2

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

Contour Curves of f (x, y) = x2 + 4y 2

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

Hyperbolic Cylinder x2 − z2 = −4

A level surface of f (x, y, z) = x2 − z2.

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

Hyperboloid x2 + y 2 − z2 = 4

A level surface of f (x, y, z) = x2 + y2 − z2.

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

Hyperboloid with Plane Section y = 0

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

2 Sheeted Hyperboloid x2 + y 2 − z2 = −4

A level surface of f (x, y, z) = x2 + y2 − z2. The xz plane y = 0 is shown.

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

Saddles

A fundamental example is the saddle z = x2 − y2.

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

Saddles

A fundamental example is the saddle z = x2 − y2.

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

Saddles

A fundamental example is the saddle z = x2 − y2. z = x2 − y2.

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

Saddles

z = x2 − y2. The saddle’s basic property is that along some directions (e.g. a y = 0 section) it is an upward parabola (z = x2) while in

  • ther directions (e.g. an (x = 0) section) it is a downward

parabola z = −y2.

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

Saddles

z = x2 − y2. The saddle’s basic property is that along some directions (e.g. a y = 0 section) it is an upward parabola (z = x2) while in

  • ther directions (e.g. an (x = 0) section) it is a downward

parabola z = −y2.

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

Saddles

z = x2 − y2. In fact there are also vertical sections (e.g. along y = x) where z = 0.

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

Saddles

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.

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

Saddles

Note that z = xy has the same saddle shape as z = x2 − y2; upward along y = x and downward along y = −x.

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

Saddles

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

Saddles

The algebra 2 x + y √ 2 x − y √ 2

  • = x2 − y2

may be interpreted as saying z = 2xy is a 45 degree rotated z = x2 − y2.

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

Swallowtail z = x3 − xy

Combines all cubic curves z = x3 − cx in the plane at once!

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

Point Set Topology Terms

Dr(p0) : open disk (or ball) of radius r about p0; {p : p − p0 < r}.

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

Point Set Topology Terms

Dr(p0) : open disk (or ball) of radius r about p0; {p : p − p0 < r}. Interior Point p of a Set A: Some ball Dr(p) with r > 0 lies entirely in A.

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

Point Set Topology Terms

Dr(p0) : open disk (or ball) of radius r about p0; {p : p − p0 < r}. Interior Point p of a Set A: Some ball Dr(p) with r > 0 lies entirely in A. Open Set U: All points are interior points.

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

Point Set Topology Terms

Dr(p0) : open disk (or ball) of radius r about p0; {p : p − p0 < r}. Interior Point p of a Set A: Some ball Dr(p) with r > 0 lies entirely in A. Open Set U: All points are interior points. Neighborhood U of a point p0 : Any open set containing p0.

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

Point Set Topology Terms

Boundary Point p of a set A: every ball Dr(p) around p contains both points of A and points not in A.

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

Point Set Topology Terms

Boundary Point p of a set A: every ball Dr(p) around p contains both points of A and points not in A. Closed Set: contains all its boundary points. (Or complement is open.)

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

Point Set Topology Terms

The precise definition of lim

p→p0 f (p) = L

is that for any ǫ > 0 (and points p in the domain of f ) there is a δ > 0 so that p ∈ Dδ(p0) and p = p0 ⇒ f (p) ∈ Dǫ(L). Don’t worry about the textbook’s “eventually in” terminology; also we won’t work much with this formal definition.

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

Point Set Topology Terms

The precise definition of lim

p→p0 f (p) = L

is that for any ǫ > 0 (and points p in the domain of f ) there is a δ > 0 so that p ∈ Dδ(p0) and p = p0 ⇒ f (p) ∈ Dǫ(L). Don’t worry about the textbook’s “eventually in” terminology; also we won’t work much with this formal definition. I wrote f (p) ∈ Dǫ(L) rather than |f (p) − L| < ǫ to allow for the possibility that L ∈ Rm rather than L simply being a scalar.