Dr. Allen Back. Aug. 27, 2014

Transcription

1 Dr. Allen Back Aug. 27, 2014

2 Math 2220 Preliminaries (2+ classes) Differentiation (12 classes) Multiple Integrals (9 classes) Vector Integrals (15 classes)

3 Math 2220 Preliminaries (2+ classes) Differentiation (12 classes) Multiple Integrals (9 classes) Vector Integrals (15 classes) Preliminaries: functions of several variables limits and continuity

4 Math 2220 Differentiation: partial derivatives chain rule linear approximation curves tangent planes, gradients extreme values Lagrange multipliers inverse functions

5 Math 2220 Multiple Integrals: double triple polar/cylindrical/spherical coordinates general change of variables

6 Math 2220 Vector Integrals: line/surface integrals conservative fields Green/Stokes/Gauss integral theorems differential forms

7 (Briefly) Grading: 20% each prelim, 40% final, 20% homework/quizzes Evening Prelims: Tu Sep 30 and Tu Nov 4. (Please send about any conflicts by Fri Sep 12.) No Late Homework: But Grading Notes Files for excuses and explanations. Homework Quizzes: Not intended to be pressing. Primarily to confirm people are not just copying solutions from the internet, etc.

8 , Graphs Domain U R n : The places where f (x 1, x 2,..., x n ) is defined.

9 , Graphs Domain U R n : The places where f (x 1, x 2,..., x n ) is defined. Range of a Function: The values actually assumed.

10 , Graphs The graph of a function f : U R 2 R is actually a set; namely {(x, y, f (x, y)) : (x, y) U}

11 , Graphs Level Curve of f (x, y): A subset of the plane R 2 where f takes on some value.

12 , Graphs Level Curve of f (x, y): A subset of the plane R 2 where f takes on some value. Level of f (x, y, z): A subset of space R 3 where f takes on some value.

13 , Graphs Horizontal sections of the graph of f (x, y) are intersections of the graph with z = f (x, y); Marsden refers to them as level curves lifted up to the graph.

14 , Graphs (Vertical) sections are intersections of the graph with coordinate planes like the xz or yz plane, as well, or any other vertical plane (e.g. y = kx for some constant k or y y 0 = k(x x 0 ).)

15 Paraboloid z = x 2 + 4y 2

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

17 Hyperbolic Cylinder x 2 z 2 = 4 A level surface of f (x, y, z) = x 2 z 2.

18 Hyperboloid x 2 + y 2 z 2 = 4 A level surface of f (x, y, z) = x 2 + y 2 z 2.

19 Hyperboloid with Plane Section y = 0

20 2 Sheeted Hyperboloid x 2 + y 2 z 2 = 4 A level surface of f (x, y, z) = x 2 + y 2 z 2. The xz plane y = 0 is shown.

21 Swallowtail z = x 3 xy Combines all cubic curves z = x 3 cx in the plane at once!

22 Intuition: lim f (x, y) = L (x,y) (a,b) means that as (x, y) gets close to (a, b), the function values get close to L.

23 Intuition: lim f (x, y) = L (x,y) (a,b) 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).

24 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) = L 1 exists and exists, then lim (x,y) (a,b) g(x, y) = L 2 0 lim (x,y) (a,b) f (x, y) g(x, y) = L 1 L 2 (And in particular the limit of the quotients exists.)

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

26 What about the corresponding statements for compositions?

27 Does the following limit exist? If so its value? lim (x,y) (0,0) xy x 2 + y 2

28 Does the following limit exist? If so its value? lim (x,y) (0,0) xy x 2 + y 2 Approach 1: Think about behavior along the axes; along lines y = kx in the domain.

29 Does the following limit exist? If so its value? lim (x,y) (0,0) xy x 2 + y 2 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.

30 Does the following limit exist? If so its value? lim (x,y) (0,0) xy x 2 + y 2 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.

31 Does the following limit exist? If so its value? lim (x,y) (0,0) xy x 2 + y 2 Polar coordinates are another approach. The function simplifies sin 2θ to for r 0 which definitely depends on θ; so we have 2 non-existence.

32 Does the following limit exist? lim (x,y) (1,1) x 2 y 2 x y

33 Does the following limit exist? If so its value? lim (x,y) (0,0) xy x y

34 Does the following limit exist? If so its value? lim (x,y) (0,0) x 3 2 y x 2 + y 2

35 Does the following limit exist? If so its value? lim (x,y) (0,0) sin (x 2 ) + y 2 x 2 + y 2

36 Show that f (x, y) = is continuous for all (x, y). x 2 cos (xy)

37 Topology Terms D r (p 0 ) : open disk (or ball) of radius r about p 0 ; {p : p p 0 < r}.

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

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

40 Topology Terms D r (p 0 ) : open disk (or ball) of radius r about p 0 ; {p : p p 0 < r}. Interior Point p of a Set A: Some ball D r (p) with r > 0 lies entirely in A. Open Set U: All points are interior points. Boundary Point p of a set A: every ball D r (p) around p contains both points of A and points not in A.

41 Topology Terms D r (p 0 ) : open disk (or ball) of radius r about p 0 ; {p : p p 0 < r}. Interior Point p of a Set A: Some ball D r (p) with r > 0 lies entirely in A. Open Set U: All points are interior points. Boundary Point p of a set A: every ball D r (p) around p contains both points of A and points not in A. Closed Set: contains all its boundary points. (Or complement is open.)

42 Topology Terms D r (p 0 ) : open disk (or ball) of radius r about p 0 ; {p : p p 0 < r}. Interior Point p of a Set A: Some ball D r (p) with r > 0 lies entirely in A. Open Set U: All points are interior points. Boundary Point p of a set A: every ball D r (p) around p contains both points of A and points not in A. Closed Set: contains all its boundary points. (Or complement is open.) Bounded Region (or Set): A set lying in some ball of finite radius.