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. . , x m → x 1 + · · · + x mthat the sphere is closed, and from the fact that |x i | ≤ r (i = 1, . . . , m) on the spherethat it is bounded.The function 1 22 2 2x , . . . , x m → x 1 + · · · + x k − x k+1 − · · · − x mis continuous on all of Rm , so that its restriction to the sphere is also continuous,and by the global property c) of continuous functions assumes its minimal and maximal values on the sphere. At the points (1, 0, . .
. , 0) and (0, . . . , 0, 1) this functionassumes the values 1 and −1 respectively. By the connectedness of the sphere (seeProblem 3 at the end of this section), global property d) of continuous functions nowenables us to assert that there is a point on the sphere where this function assumesthe value 0.Example 13 The open set Rm \S(0; r) for r > 0 is not a domain, since it is notconnected.Indeed, if Γ : I → Rm is a path one end of which is at the point x0 = (0, . . . , 0)and the other at some point x1 = (x11 , . . . , x1m ) such that (x11 )2 + · · · + (x1m )2 > r 2 ,then the composition of the continuous functions Γ : I → Rm and f : Rm → R,where 1 f 2 2x , . .
. , x m −→ x 1 + · · · + x m ,is a continuous function on I assuming values less than r 2 at one endpoint andgreater than r 2 at the other. Hence there is a point γ on I at which (f ◦ Γ )(γ ) = r 2 .Then the point xγ = Γ (γ ) in the support of the path turns out to lie on the sphereS(0; r). We have thus shown that it is impossible to get out of the ball B(0; r) ⊂ Rmwithout intersecting its boundary sphere S(0; r).4267 Functions of Several Variables: Their Limits and Continuity7.2.3 Problems and Exercises1.
Let f ∈ C(Rm ; R). Show thata)b)c)d)e)the set E1 = {x ∈ Rm | f (x) < c} is open in Rm ;the set E2 = {x ∈ Rm | f (x) ≤ c} is closed in Rm ;the set E3 = {x ∈ Rm | f (x) = c} is closed in Rm ;if f (x) → +∞ as x → ∞, then E2 and E3 are compact in Rm ;for any f : Rm → R the set E4 = {x ∈ Rm | ω(f ; x) ≥ ε} is closed in Rm .2.
Show that the mapping f : Rm → Rn is continuous if and only if the preimageof every open set in Rn is an open set in Rm .3. Show thata) the image f (E) of a connected set E ⊂ Rm under a continuous mappingf : E → Rn is a connected set;b) the union of connected sets having a point in common is a connected set;c) the hemisphere (x 1 )2 + · · · + (x m )2 = 1, x m ≥ 0, is a connected set;d) the sphere (x 1 )2 + · · · + (x m )2 = 1 is a connected set;e) if E ⊂ R and E is connected, then E is an interval in R (that is, a closedinterval, a half-open interval, an open interval, or the entire real line);f) if x0 is an interior point and x1 an exterior point in relation to the set M ⊂ Rm ,then the support of any path with endpoints x0 , x1 intersects the boundary of theset M.Chapter 8The Differential Calculus of Functions of SeveralVariables8.1 The Linear Structure on Rm8.1.1 Rm as a Vector SpaceThe concept of a vector space is already familiar to you from your study of algebra.If we introduce the operation of addition of elements x1 = (x11 , .
. . , x1m ) and x2 =1(x2 , . . . , x2m ) in Rm by the formulax1 + x2 = x11 + x21 , . . . , x1m + x2m ,(8.1)and multiplication of an element x = (x 1 , . . . , x m ) by a number λ ∈ R via the relationλx = λx 1 , . . . , λx m ,(8.2)then Rm becomes a vector space over the field of real numbers. Its points can nowbe called vectors.The vectorsei = (0, . . . , 0, 1, 0, . . . , 0)(i = 1, . .
. , m)(8.3)(where the 1 stands only in the ith place) form a maximal linearly independent setof vectors in this space, as a result of which it turns out to be an m-dimensionalvector space.Any vector x ∈ Rm can be expanded with respect to the basis (8.3), that is, represented in the formx = x 1 e1 + · · · + x m em .(8.4)When vectors are indexed, we shall write the index as a subscript, while denotingits coordinates, as we have been doing, by superscripts. This is convenient for many© Springer-Verlag Berlin Heidelberg 2015V.A. Zorich, Mathematical Analysis I, Universitext,DOI 10.1007/978-3-662-48792-1_84274288 Differential Calculus in Several Variablesreasons, one of which is that, following Einstein,1 we can make the convention ofwriting expressions like (8.4) briefly in the formx = x i ei ,(8.5)taking the simultaneous presence of subscript and superscript with the same letterto indicate summation with respect to that letter over its range of variation.8.1.2 Linear Transformations L : Rm → RnWe recall that a mapping L : X → Y from a vector space X into a vector space Y iscalled linear ifL(λ1 x1 + λ2 x2 ) = λ1 L(x1 ) + λ2 L(x2 )for any x1 , x2 ∈ X, and λ1 , λ2 ∈ R.
We shall be interested in linear mappings L :Rm → Rn .If {e1 , . . . , em } and {ẽ1 , . . . , ẽn } are fixed bases of Rm and Rn respectively, then,knowing the expansionjL(ei ) = ai1 ẽ1 + · · · + ain ẽn = ai ẽj(i = 1, . . . , m)(8.6)of the images of the basis vectors under the linear mapping L : Rm → Rn , we canuse the linearity of L to find the expansion of the image L(h) of any vector h =h1 e1 + · · · + hm em = hi ei in the basis {ẽ1 , . . .
, ẽn }. To be specific,jj(8.7)L(h) = L hi ei = hi L(ei ) = hi ai ẽj = ai hi ẽj .Hence, in coordinate notation:L(h) = ai1 hi , . . . , ain hi .(8.8)For a fixed basis in Rn the mapping L : Rm → Rn can thus be regarded as a setL = L1 , . . . , L n(8.9)of n (coordinate) mappings Lj : Rm → R.Taking account of (8.8), we easily conclude that a mapping L : Rm → Rn islinear if and only if each mapping Lj in the set (8.9) is linear.If we write (8.9) as a column, taking account of relation (8.8), we have⎞⎛ 1 ⎞⎛ 1 ⎞ ⎛ 11hL (h)a1 · · · am..
⎟ ⎜ .. ⎟⎜ .. ⎟ ⎜ .. . .(8.10)L(h) = ⎝ . ⎠ = ⎝ .. . ⎠⎝ . ⎠.Ln (h)a1n···namhm1 A. Einstein (1879–1955) – greatest physicist of the twentieth century. His work in quantum theoryand especially in the theory of relativity exerted a revolutionary influence on all of modern physics.8.1 The Linear Structure on Rm429Thus, fixing bases in Rm and Rn enables us to establish a one-to-one correjspondence between linear transformations L : Rm → Rn and m × n-matrices (ai )j(i = 1, . .
. , m, j = 1, . . . , n). When this is done, the ith column of the matrix (ai )corresponding to the transformation L consists of the coordinates of L(ei ), the image of the vector ei ∈ {e1 , . . . , em }. The coordinates of the image of an arbitraryvector h = hi ei ∈ Rm can be obtained from (8.10) by multiplying the matrix of thelinear transformation by the column of coordinates of h.Since Rn has the structure of a vector space, one can speak of linear combinationsλ1 f1 + λ2 f2 of mappings f1 : X → Rn and f2 : X → Rn , setting(λ1 f1 + λ2 f2 )(x) := λ1 f1 (x) + λ2 f2 (x).(8.11)In particular, a linear combination of linear transformations L1 : Rm → Rn andL2 : Rm → Rn is, according to the definition (8.11), a mappingh → λ1 L1 (h) + λ2 L2 (h) = L(h),which is obviously linear.
The matrix of this transformation is the correspondinglinear combination of the matrices of the transformations L1 and L2 .The composition C = B ◦ A of linear transformations A : Rm → Rn and B :nR → Rk is obviously also a linear transformation, whose matrix, as follows from(8.10), is the product of the matrix of A and the matrix of B (which is multiplied onthe left). Actually, the law of multiplication for matrices was defined in the way youare familiar with precisely so that the product of matrices would correspond to thecomposition of the transformations.8.1.3 The Norm in RmThe quantity,x, = 2x 1 + · · · + (xm )2(8.12)is called the norm of the vector x = (x 1 , .
. . , x m ) ∈ Rm .It follows from this definition, taking account of Minkowski’s inequality, that10203040,x, ≥ 0,(,x, = 0) ⇔ (x = 0),,λx, = |λ| · ,x,, where λ ∈ R,,x1 + x2 , ≤ ,x1 , + ,x2 ,.In general, any function , , : X → R on a vector space X satisfying conditions 10 –40 is called a norm on the vector space. Sometimes, to be precise as towhich norm is being discussed, the norm sign has a symbol attached to it to denotethe space in which it is being considered. For example, we may write ,x,Rm or4308 Differential Calculus in Several Variables,y,Rn .
As a rule, however, we shall not do that, since it will always be clear fromthe context which space and which norm are meant.We remark that by (8.12),x2 − x1 , = d(x1 , x2 ),(8.13)where d(x1 , x2 ) is the distance in Rm between the vectors x1 and x2 , regarded aspoints of Rm .It is clear from (8.13) that the following conditions are equivalent:x → x0 ,d(x, x0 ) → 0,,x − x0 , → 0.In view of (8.13), we have, in particular,,x, = d(0, x).Property 40 of a norm is called the triangle inequality, and it is now clear why.The triangle inequality extends by induction to the sum of any finite number ofterms. To be specific, the following inequality holds:,x1 + · · · + xk , ≤ ,x1 , + · · · + ,xk ,.The presence of the norm of a vector enables us to compare the size of values offunctions f : X → Rm and g : X → Rn .Let us agree to write f (x) = o(g(x)) or f = o(g) over a base B in X if,f (x),Rm = o(,g(x),Rn ) over the base B.If f (x) = (f 1 (x), .
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