Chapter 10.1 Limit and Continuity of multivariat

{(x, y)|0(xx0)2 (yy0)2 2} or
{(x, y)||xx0| , |yy0| , (x, y) (x0, y0)},
denoted as U °(A; ) or U °(A).
b. Relationship between point and point set:
2 2 (P , P ) ( x x ) ( y y ) . 1 2 1 2 1 2
y
P1(x1, y1)
P2(x2, y2)
x
O
Euclidean distance has the following properties:
(1) Positivity: (P,Q) ≥0; (P,Q)=0 P=Q; (2) Symmetry: (P,Q) = (Q, P); (3) Triangle inequality: (P,Q)≤ (P,S)+ (S,Q).
(iv) Accumulation point—If there exists point belong to any deleted neighborhood U °(A) of E, then A is called the accumulation point of E. Accumulation point itself may belong to E, or not belong to E.
Example 1. Assume D {(x, y)|1 x2 y2 4}, then the points, satisfy1 x2 y2 4, are the interior points of D; the points, satisfy x2 y2 4, are the boundary points of D, and they are all belong to D; the points, satisfy x2 y2 1, are also the boundary points of D, but they do not belong to D; DUD are the accumulation points of D. Open set—If every points of E belong to the interior of E(namely, int E E), then E is called an open set.
Basic Requirements
1.Master some concepts about the point sets in the plane, such as neighborhood, interiorpoint, boundary point, accumulation point, 0pen set, closed set, open domain, closed domain, bounded set, and unbounded set, etc. 2.To understand and describe the closed rectangle cover theorem on continuity of R2, finite cover theorem, accumulation principle, compactness theorem and Cauchy Convergence Criterion.
As any circular neighborhood of A can be contained in a square neighborhood of A, and vice versa, so it is often denoted circular neighborhood of A or square neighborhood of A indiscriminately as U(A; ) or U(A). Deleted neighborhood of A refers to:
When the Euclidean distance is introduced in R2, then it is called 2- dimensional Euclidean space, or 2dimensional metric space. The concept of neighborhood: We call the set of points {(x, y)|(xx0)2 (yy0)2 2} and {(x, y)| ||xx0 | , |yy0| } the -circular neighborhood with central A(x0, y0) and -square neighborhood with central A(x0, y0) , respectively.
The relationship between any point AR2 and any point set E R2: (i) Interior point—If there is a neighborhood U(A) of point A, such that U(A) E , then A is called the interior point of E; A set consists of all interior points of E is called interior of E, denoted as int E. (ii) Exterior point—If there is a neighborhood U(A) of point A, such that U(A)∩ E , then A is called the exterior point of E. (iii) Boundary point—If any neighborhood of the point A contains points in both E and Ec, then A is called the boundary point of E, namely, for any >0, formula U(A; )∩ E and U(A; )∩ Ec are always valid, where Ec R2 \ E. A set consists of all boundary points of E is called the boundary of E, denoted as E.
Examples of planar point sets 1.The first quadra: G = {(x, y)| x > 0, y>0}; 2. Lower half plane with X axis: H- = {(x, y)| y≤0};
3. Circle with central (x0, y0) and radius r:



§10.4 Taylor formula of two valued-function
§10.1 Multivariate Function
1. N-dimensional Euclidean space
2. The limit of points in R2 and continuity of R2 3. The concept of multivariate function
C ={(x, y)| (xx0)2 (yy0)2 = r 2}.
4. Disc with central (x0, y0) and radius r:
D = {(x, y)|(xx0)2 (yy0)2 < r 2};
5. Rectangle parallel to axis:
S={(x, y)|a≤ x ≤ b, c ≤ y ≤ d}=[a, b]×[c, d],
y b O
Point P(a,b)
number pair (a,b)
x a
Representation of points in plane: The points in plane is usually denoted by capital letters P, Q, X, Y etc. In order to emphasize its coordinates, they also can be expressed with P(x,y), Q(a,b),or Denoted as (x, y), (a, b) etc, directly.
C = {(x, y)| (xx0)2 (yy0)2 = r 2}.
Distance—The most basic concept
Euclidean Distance The Euclidean Distance between P1(x1, y1) and P2(x2, y2) in coordinate plane is defined by
(v) Isolated point—If AE, but not the accumulation point of E, namely, there exists a >0, such that U °(A; )∩E =, then A is called the isolated point of E.
3.Master the concept of two valued function and its limit.
a. Basic concept Coordinate Plane: The plane which is decised a coordinate system is called Coordinate Plane, denoted by R2. It makes 1-1 correspondence between the real number pair and the plane point.
Obviously, the isolated point must be a boundary point; The interior point and non-isolated boundary point must be the accumulation point; If the point is neither accumulation point nor isolated point, then it must be to the exterior point.