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. . , f m (x)) is the coordinate representation of the mappingf : X → Rm , then in view of the inequalitiesm= i = i f (x) ≤ =f (x)= ≤f (x)(8.14)i=1one can make the following observation, which will be useful below: f = o(g) over the base B ⇔ f i = o(g) over the base B; i = 1, . . . , m . (8.15)We also make the convention that the statement f = O(g) over the base B in Xwill mean that ,f (x),Rm = O(,g(x),Rn ) over the base B.We then obtain from (8.14) f = O(g) over the base B ⇔ f i = O(g) over the base B; i = 1, . . .
, m .(8.16)Example Consider a linear transformation L : Rm → Rn . Let h = h1 e1 + · · · +hm em be an arbitrary vector in Rm . Let us estimate ,L(h),Rn := m= m m= == ========L(h)= = ==L(ei )=hi ≤=L(ei )= ,h,.hi L(ei )= ≤(8.17)==i=1i=1i=18.1 The Linear Structure on Rm431Thus one can assert thatL(h) = O(h)as h → 0.(8.18)In particular, it follows from this that L(x − x0 ) = L(x) − L(x0 ) → 0 as x → x0 ,that is, a linear transformation L : Rm → Rn is continuous at every point x0 ∈ Rm .From estimate (8.17) it is even clear that a linear transformation is uniformly continuous.8.1.4 The Euclidean Structure on RmThe concept of the inner product in a real vector space is known from algebra asa numerical function &x, y' defined on pairs of vectors of the space and possessingthe properties&x, x' ≥ 0,&x, x' = 0 ⇔ x = 0,&x1 , x2 ' = &x2 , x1 ',&λx1 , x2 ' = λ&x1 , x2 ',where λ ∈ R,&x1 + x2 , x3 ' = &x1 , x3 ' + &x2 , x3 '.It follows in particular from these properties that if a basis {e1 , .
. . , em } is fixedin the space, then the inner product &x, y' of two vectors x and y can be expressedin terms of their coordinates (x 1 , . . . , x m ) and (y 1 , . . . , y m ) as the bilinear form&x, y' = gij x i y j(8.19)(where summation over i and j is understood), in which gij = &ei , ej '.Vectors are said to be orthogonal if their inner product equals 0.A basis {e1 , . . .
, em } is orthonormal if gij = δij , where!0, if i = j,δij =1, if i = j.In an orthonormal basis the inner product (8.19) has the very simple form&x, y' = δij x i y j ,or&x, y' = x 1 · y 1 + · · · + x m · y m .(8.20)Coordinates in which the inner product has this form are called Cartesian coordinates.4328 Differential Calculus in Several VariablesWe recall that the space Rm with an inner product defined in it is called Euclideanspace.Between the inner product (8.20) and the norm of a vector (8.12) there is anobvious connection&x, x' = ,x,2 .The following inequality is known from algebra:&x, y'2 ≤ &x, x'&y, y'.It shows in particular that for any pair of vectors there is an angle ϕ ∈ [0, π] suchthat&x, y' = ,x,,y, cos ϕ.This angle is called the angle between the vectors x and y.
That is the reason weregard vectors whose inner product is zero as orthogonal.We shall also find useful the following simple, but very important fact, knownfrom algebra:any linear function L : Rm → R in Euclidean space has the formL(x) = &ξ, x',where ξ ∈ Rm is a fixed vector determined uniquely by the function L.8.2 The Differential of a Function of Several Variables8.2.1 Differentiability and the Differential of a Function at a PointDefinition 1 A function f : E → Rn defined on a set E ⊂ Rm is differentiable atthe point x ∈ E, which is a limit point of E, iff (x + h) − f (x) = L(x)h + α(x; h),(8.21)where L(x) : Rm → Rn is a function2 that is linear in h and α(x; h) = o(h) ash → 0, x + h ∈ E.The vectorsΔx(h) := (x + h) − x = h,Δf (x; h) := f (x + h) − f (x)2 By analogy with the one-dimensional case, we allow ourselves to write L(x)h instead of L(x)(h).We note also that in the definition we are assuming that Rm and Rn are endowed with the norm ofSect.
8.1.8.2 The Differential of a Function of Several Variables433are called respectively the increment of the argument and the increment of the function (corresponding to this increment of the argument). These vectors are traditionally denoted by the symbols of the functions of h themselves Δx and Δf (x). Thelinear function L(x) : Rm → Rn in (8.21) is called the differential, tangent mapping,or derivative mapping of the function f : E → Rn at the point x ∈ E.The differential of the function f : E → Rn at a point x ∈ E is denoted by thesymbols df (x), Df (x), or f (x).In accordance with the notation just introduced, we can rewrite relation (8.21) asf (x + h) − f (x) = f (x)h + α(x; h)orΔf (x; h) = df (x)h + α(x; h).We remark that the differential is defined on the displacements h from the pointx ∈ Rm .To emphasize this, we attach a copy of the vector space Rm to the point x ∈ Rmmand denote it Tx Rm , T Rm (x), or T Rmx .
The space T Rx can be interpreted as a setmof vectors attached at the point x ∈ R . The vector space T Rmx is called the tangentspace to Rm at x ∈ Rm . The origin of this terminology will be explained below.nThe value of the differential on a vector h ∈ T Rmx is the vector f (x)h ∈ T Rf (x)attached to the point f (x) and approximating the increment f (x + h) − f (x) of thefunction caused by the increment h of the argument x.nThus df (x) or f (x) is a linear transformation f (x) : T Rmx → T Rf (x) .We see that, in complete agreement with the one-dimensional case that we studied, a vector-valued function of several variables is differentiable at a point if itsincrement Δf (x; h) at that point is linear as a function of h up to the correctionterm α(x; h), which is infinitesimal as h → 0 compared to the increment of theargument.8.2.2 The Differential and Partial Derivatives of a Real-ValuedFunctionIf the vectors f (x + h), f (x), L(x)h, α(x; h) in Rn are written in coordinates,Eq.
(8.21) becomes equivalent to the n equalitiesf i (x + h) − f i (x) = Li (x)h + α i (x; h)(i = 1, . . . , n)(8.22)between real-valued functions, in which, as follows from relations (8.9) and (8.15)of Sect. 8.1, Li (x) : Rm → R are linear functions and α i (x; h) = o(h) as h → 0,x + h ∈ E, for every i = 1, . . . , n.Thus we have the following proposition.4348 Differential Calculus in Several VariablesProposition 1 A mapping f : E → Rn of a set E ⊂ Rm is differentiable at a pointx ∈ E that is a limit point of E if and only if the functions f i : E → R (i = 1, .
. . , n)that define the coordinate representation of the mapping are differentiable at thatpoint.Since relations (8.21) and (8.22) are equivalent, to find the differential L(x) ofa mapping f : E → Rn it suffices to learn how to find the differentials Li (x) of itscoordinate functions f i : E → R.Thus, let us consider a real-valued function f : E → R, defined on a set E ⊂ Rmand differentiable at an interior point x ∈ E of that set. We remark that in the futurewe shall mostly be dealing with the case when E is a domain in Rm .
If x is aninterior point of E, then for any sufficiently small displacement h from x the pointx +h will also belong to E, and consequently will also be in the domain of definitionof the function f : E → R.If we pass to the coordinate notation for the point x = (x 1 , . . . , x m ), the vectorh = (h1 , . . . , hm ), and the linear function L(x)h = a1 (x)h1 + · · · + am (x)hm , thenthe conditionf (x + h) − f (x) = L(x)h + o(h)as h → 0(8.23)can be rewritten asf x 1 + h1 , . . . , x m + hm − f x 1 , . . .
, x m == a1 (x)h1 + · · · + am (x)hm + o(h)as h → 0,(8.24)where a1 (x), . . . , am (x) are real numbers connected with the point x.We wish to find these numbers. To do this, instead of an arbitrary displacement hwe consider the special displacementhi = hi ei = 0 · e1 + · · · + 0 · ei−1 + hi ei + 0 · ei+1 + · · · + 0 · emby a vector hi collinear with the vector ei of the basis {e1 , . . . , em } in Rm .When h = hi , it is obvious that ,h, = |hi |, and so by (8.24), for h = hi we obtainf x 1 , . . . , x i−1 , x i + hi , x i+1 , .
. . , x m − f x 1 , . . . , x i , . . . , x m = as hi → 0.(8.25)= ai (x)hi + o hiThis means that if we fix all the variables in the function f (x 1 , . . . , x m ) exceptthe ith one, the resulting function of the ith variable alone is differentiable at thepoint x i .In that way, from (8.25) we find thatf (x 1 , . . .
, x i−1 , x i + hi , x i+1 , . . . , x m ) − f (x 1 , . . . , x i , . . . , x m ).hihi →0(8.26)ai (x) = lim8.2 The Differential of a Function of Several Variables435Definition 2 The limit (8.26) is called the partial derivative of the function f (x) atthe point x = (x 1 , .
. . , x m ) with respect to the variable x i . We denote it by one ofthe following symbols:∂f(x),∂x i∂i f (x),Di f (x),fx i (x).Example 1 If f (u, v) = u3 + v 2 sin u, then∂f(u, v) = 3u2 + v 2 cos u,∂u∂f∂2 f (u, v) =(u, v) = 2v sin u.∂v∂1 f (u, v) =Example 2 If f (x, y, z) = arctan(xy 2 ) + ez , then∂1 f (x, y, z) =y2∂f(x, y, z) =,∂x1 + x2y4∂2 f (x, y, z) =2xy∂f(x, y, z) =,∂y1 + x2y4∂3 f (x, y, z) =∂f(x, y, z) = ez .∂zThus we have proved the following result.Proposition 2 If a function f : E → R defined on a set E ⊂ Rm is differentiable atan interior point x ∈ E of that set, then the function has a partial derivative at thatpoint with respect to each variable, and the differential of the function is uniquelydetermined by these partial derivatives in the formdf (x)h =∂f∂f(x)h1 + · · · + m (x)hm .1∂x∂x(8.27)Using the convention of summation on an index that appears as both a subscriptand a superscript, we can write formula (8.27) succinctly:df (x)h = ∂i f (x)hi .(8.28)Example 3 If we had known (as we soon will know) that the function f (x, y, z)considered in Example 2 is differentiable at the point (0, 1, 0), we could have writtenimmediatelydf (0, 1, 0)h = 1 · h1 + 0 · h2 + 1 · h3 = h1 + h3and accordinglyf h1 , 1 + h2 , h3 − f (0, 1, 0) = df (0, 1, 0)h + o(h)4368 Differential Calculus in Several Variablesor 2 3arctan h1 1 + h2 + eh = 1 + h1 + h3 + o(h)as h → 0.πiExample 4 For the function x = (x 1 , .
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