1、Section 9.1Fermat2Space RnSince the domain of a multivariable function is a set in the n-dimensional space Rn,we need to begin by introducing some primary knowledge of points in the space Rn.The real vector,real vector space and real linear space.Euclidean space The length or norm of a vector Domain
2、s and Ranges 3The set Rn with the operations of addition of vectors and multiplication of a vector by a number is called an n dimensional real vector space n维向量空间,or n dimensional real linear spacen维线性空间.1,niiix yx y If x and y are two vectors in Rn,then the inner product of x and y isand Rn with th
3、is inner product is called n dimensional Euclideanspace 欧氏空间.The ith component of a vector x,xi,is also called the ith coordinate of A vector in n dimensional Euclidean space is also called a point.the point x.Space Rn4In particular,a point(or vector)in R2 may be denoted by(x,y),andA point(orR2 may
4、be regarded as the set of all points in the plane.vector)in R3 may be expressed by(x,y,z)and R3 is regarded as3 dimensional space.The length(or norm)of a vector x in the space Rn is defined by22212|x|x,x.nxxxThe distance 距离距离 between two points x and y is defined by 2221122(x,y)|xy|()()().nnxyxyxy S
5、pace Rn5Space RnDefinition(Neighbourhoods 邻域邻域)Let P0 be a point in Rn and 0 be a constant.A point set in Rn consisting of all points such that the distance between any point of the set and the point a is less that is called a neighbourhood of the point a and denoted by U(P0,),that is00(,)xR|x|,nU P
6、Pwhere is called the radius of the neighbourhood.Oxy 0P6Space RnThe neighbourhood U(P0,)but omitting the point P0,is called a deleted neighbourhood of a and is denoted by 000(,)(,).U PU PP Oxy 0P 02220(1,2)(,1)(,)R|121,PU Px yxy For example,2220(,1)(,)R|0121,U Px yxy Oxy127Space RnDefinition(Interio
7、r Points 内点内点 and Interior of a Set 内集内集).PS If there exists athen P is called(,),U PS and a pointLet S is a point setRnS points is called the interior of the set S,denoted by S0 or int S.(,)U P of P such thatneighbourhood an interior point of the set S and the set consisting of all interiorInterior
8、 pointS 22201(,)R|121,(1,2),(0.5,2)Sx yxyPP P0 is an interior point of the set S P1 is an interior point of the set S 8Space RnDefinition (Exterior Points 外点外点 and Exterior of a Set 外集外集)R.nP If there exists aand a pointLet S is a point setRnS is called the exterior of the set S,denoted by ext S.(,)
9、U P of P such that none of the points inneighbourhoodthen P is called an exterior(,)U P(,),cU PS belong to the set S,that is point of the set S.The set consisting of all the exterior points of S isExterior pointS9contains an point of the set S and also contains an point of the set Sc,Definition(Boun
10、dary Points 边界点边界点 and Boundary of a Set)(which may or may notLet S is a point setR.nS boundary points of S is called the boundary of the set S,denoted(,)U P belong to S),such that for any 0,the neighbourhood is called a boundary point of the set S.The set consisting of all the RnP A point.S bySpace
11、 RnSBoundary point 10Primary Knowledge of Point Sets in the Space RnSuppose22222(,)R|014.Sx yxyxy or Then by the definitions,the interior of S exteriorof S and boundary of S are respectively 22222(0,0)(,)R|14.Sx yxyxy or 0222(,)R|14,Sx yxy22222ext(,)R|014,Sx yxyxy or Oxy11APrimary Knowledge of Point
12、 Sets in the Space RnAccumulation point contains at least one point of the set S,then the point PDefinition(Accumulation Points 聚点聚点)Let S is a point setRnS (,)U P not belong to S).If for any 0,the deleted neighbourhoodis called an accumulation point or point of accumulation of the set S.RnP and a p
13、oint(which may or mayS12Space Rncomplementary set Sc of the set S is an open set,then S is called a closed set.Definition(Open Set 开集开集 and Closed Set 闭集闭集)If all the points of S are interior pointsR.nS 0,SS then S is called an open set in Rn;if theIn particular,the null set and the whole space Rn a
14、re both open andLet S is a point setof S,that is,close.xyOBoundary of unit diskxyOClosed unit diskxyOOpen unit disk13Space Rnis called the line segment connecting the points a and b in Rn.a(1)b|R,01tttt Suppose a and b are two different points in Rn.The point setSuppose S is an open set.If any two p
15、oint of S can be connected by a broken line consisting of a finite number of line segments which areall contained in the set S,then S iscalled a connected set（连通集）（连通集）.A connected open set is called anopen region or simply a region.A region together with its boundary is called a closed region.Defin
16、ition(Regions 区域区域)14SSpace Rnan unbounded set.Definition(Bounded Sets有界集有界集 and Unbounded Sets 无界集无界集)If there is a constant M 0,such that|x|M holdsR.nS x,S then S is called a bounded set;otherwise S is said to beSupposefor allSxyOOxy15The Concept of a Multivariable FunctionDefinition(Function of n
17、 variables)be a set.The mapping f:A R is called a function of nis called the independent variable,and w isLet12(,)nx xxA variables defined in the set A,denoted byis called the domain of f andwhere()|(),()R fw wf x xD fcalled the dependent variable.()D fA is called the range of f.RnA 12(,),nwf x xx R
18、.nA 12(x)(,).nff x xx The 12x(,)nx xx can be regarded as a function of the point x is a point or a vector in the set Since function or the vector x,denoted by12(,)nf x xx16The Concept of a Multivariable FunctionExample Find the domain of the following functions:(1)Solution so22ln(12)zxy22120,xy Sinc
19、e the logarithm function needs that the domain of this function is222(,)R|21.Dx yxyOxyThis is a bounded region in the xOyplane whose boundary is the ellipse 2221.xy17The Concept of a Multivariable Function(2)Solution 2211zxyIt is easy to see that the domain of the function is 2(,)R|1|1.Dx yxy and Ox
20、yThese are two unbounded closed regionin the xOy plane.Example Find the domain of the following functions:18The Concept of a Multivariable Function(3)Solution 221wzxy It is easy to see that the domain of the function is 322(,)R,.Dx y zzxyThis is an unbounded region in three-dimensional space R3,whos
21、e boundary isOxyzthe paraboloid22.zxyExample Find the domain of the following functions:19Visualization of Multi-variable Functions12(,)R):nnx xxAIn general,the graph of a function of n variables12(,),nzf x xx is a point set in space Rn+1.121212Gr(,)|(,),(,)nnnfx xx zx xxA zf x xxWhile the function1
22、2a,x,(x(,)R)nnx xx121122(,)nnnf x xxa xa xa xis called a linear function of n variables and describes a hyperplane inis a constant vector.12a(,)Rnna aathe space Rn+1,where,Graphing a Function of Two Variables20(,)f x yC Definition(Level Curve（等高线）（等高线）)has a constantThe set of points in the plane wh
23、ere a function is called a level curve of f or contour curve of f.(,)f x yvalue It is easy to see that a level curve is actually the projection of the(,).f x yC on the xOy plane,and is given implicitly by the equation and a horizontal plane z=C,intersection of the surface(,)zf x y 21Graphing a Function of Two Variablesand plot the level curvesGraph 22(,)100f x yxy(,)75f x y in the domain(,)51f x y andSolution(,)0,f x y The level curves2225xyor22(,)10075f x yxy22100 xyor22(,)1000f x yxy2249xyor22(,)10051f x yxyof f in the plane.Example
