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. . , x m ) −→ x i , which assigns to the pointx ∈ Rm its ith coordinate, we haveΔπ i (x; h) = x i + hi − x i = hi ,πithat is, the increment of this function is itself a linear function in h : h −→ hi . Thus,Δπ i (x; h) = dπ i (x)h, and the mapping dπ i (x) = dπ i turns out to be actually independent of x ∈ Rm in the sense that dπ i (x)h = hi at every point x ∈ Rm .
If wewrite x i (x) instead of π i (x), we find that dx i (x)h = dx i h = hi .Taking this fact and formula (8.28) into account, we can now represent the differential of any function as a linear combination of the differentials of the coordinatesof its argument x ∈ Rm . To be specific:df (x) = ∂i f (x) dx i =∂f∂fdx 1 + · · · + m dx m ,∂x∂x 1(8.29)since for any vector h ∈ T Rmx we havedf (x)h = ∂i f (x)hi = ∂i f (x) dx i h.8.2.3 Coordinate Representation of the Differential of a Mapping.The Jacobi MatrixThus we have found formula (8.27) for the differential of a real-valued functionf : E → R. But then, by the equivalence of relations (8.21) and (8.22), for anymapping f : E → Rn of a set E ⊂ Rm that is differentiable at an interior pointx ∈ E, we can write the coordinate representation of the differential df (x) as⎞ ⎛⎞∂i f 1 (x)hidf 1 (x)h....⎟ ⎜⎜⎟df (x)h = ⎝⎠=⎝⎠=..⎛df n (x)h⎛1∂f(x)⎜ ∂x 1.⎜.=⎝ .∂f n(x)∂x 1···...···∂i f n (x)hi⎞⎛ ⎞∂f 1(x)h1m∂x..
⎟⎜. ⎟. ⎟⎠ ⎝ .. ⎠ .∂f nhm∂x m (x)(8.30)8.2 The Differential of a Function of Several Variables437Definition 3 The matrix (∂i f j (x)) (i = 1, . . . , m, j = 1, . . . , n) of partial derivatives of the coordinate functions of a given mapping at the point x ∈ E is called theJacobi matrix3 or the Jacobian4 of the mapping at the point.In the case when n = 1, we are simply brought back to formula (8.27), and whenn = 1 and m = 1, we arrive at the differential of a real-valued function of one realvariable.The equivalence of relations (8.21) and (8.22) and the uniqueness of the differential (8.27) of a real-valued function implies the following result.Proposition 3 If a mapping f : E → Rn of a set E ⊂ Rm is differentiable at aninterior point x ∈ E, then it has a unique differential df (x) at that point, and thencoordinate representation of the mapping df (x) : T Rmx → T Rf (x) is given by relation (8.30).8.2.4 Continuity, Partial Derivatives, and Differentiabilityof a Function at a PointWe complete our discussion of the concept of differentiability of a function at a pointby pointing out some connections among the continuity of a function at a point, theexistence of partial derivatives of the function at that point, and differentiability atthe point.In Sect.
8.1 (relations (8.17) and (8.18)) we established that if L : Rm → Rn isa linear transformation, then Lh → 0 as h → 0. Therefore, one can conclude fromrelation (8.21) that a function that is differentiable at a point is continuous at thatpoint, sincef (x + h) − f (x) = L(x)h + o(h)as h → 0, x + h ∈ E.The converse, of course, is not true because, as we know, it fails even in theone-dimensional case.Thus the relation between continuity and differentiability of a function at a pointin the multidimensional case is the same as in the one-dimensional case.The situation is completely different in regard to the relations between partialderivatives and the differential.
In the one-dimensional case, that is, in the case ofa real-valued function of one real variable, the existence of the differential and theexistence of the derivative for a function at a point are equivalent conditions. Forfunctions of several variables, we have shown (Proposition 2) that differentiability ofa function at an interior point of its domain of definition guarantees the existence ofa partial derivative with respect to each variable at that point. However, the converseis not true.3 C.G.J.4 TheJacobi (1804–1851) – well-known German mathematician.term Jacobian is more often applied to the determinant of this matrix (when it is square).4388 Differential Calculus in Several VariablesExample 5 The functionf x1, x2 =!0, if x 1 x 2 = 0,1, if x 1 x 2 = 0,equals 0 on the coordinate axes and therefore has both partial derivatives at the point(0, 0):f (h1 , 0) − f (0, 0)0−0= lim= 0,111hh →0h →0 h1∂1 f (0, 0) = limf (0, h2 ) − f (0, 0)0−0= lim= 0.222hh →0h →0 h2∂2 f (0, 0) = limAt the same time, this function is not differentiable at (0, 0), since it is obviouslydiscontinuous at that point.The function given in Example 5 fails to have one of its partial derivatives atpoints of the coordinate axes different from (0, 0).
However, the function! xyif x 2 + y 2 = 0,22,f (x, y) = x +y0,if x 2 + y 2 = 0(which we encountered in Example 2 of Sect. 7.2) has partial derivatives at all pointsof the plane, but it also is discontinuous at the origin and hence not differentiablethere.Thus the possibility of writing the right-hand side of (8.27) and (8.28) still doesnot guarantee that this expression will represent the differential of the function weare considering, since the function may be nondifferentiable.This circumstance might have been a serious hindrance to the entire differentialcalculus of functions of several variables, if it had not been determined (as willbe proved below) that continuity of the partial derivatives at a point is a sufficientcondition for differentiability of the function at that point.8.3 The Basic Laws of Differentiation8.3.1 Linearity of the Operation of DifferentiationTheorem 1 If the mappings f1 : E → Rn and f2 : E → Rn , defined on a setE ⊂ Rm , are differentiable at a point x ∈ E, then a linear combination of them(λ1 f1 + λ2 f2 ) : E → Rn is also differentiable at that point, and the following equality holds:(λ1 f1 + λ2 f2 ) (x) = λ1 f1 + λ2 f2 (x).(8.31)8.3 The Basic Laws of Differentiation439Equality (8.31) shows that the operation of differentiation, that is, formingthe differential of a mapping at a point, is a linear transformation on the vector space of mappings f : E → Rn that are differentiable at a given point of theset E.
The left-hand side of (8.31) contains by definition the linear transformation (λ1 f1 + λ2 f2 ) (x), while the right-hand side contains the linear combination(λ1 f1 + λ2 f2 )(x) of linear transformations f1 (x) : Rm → Rn , and f2 (x) : Rm →Rn , which, as we know from Sect. 8.1, is also a linear transformation. Theorem 1asserts that these mappings are the same.Proof(λ1 f1 + λ2 f2 )(x + h) − (λ1 f2 + λ2 f2 )(x) = = λ1 f1 (x + h) + λ2 f2 (x + h) − λ1 f1 (x) + λ2 f2 (x) == λ1 f1 (x + h) − f1 (x) + λ2 f2 (x + h) − f2 (x) == λ1 f1 (x)h + o(h) + λ2 f2 (x)h + o(h) == λ1 f1 (x) + λ2 f2 (x) h + o(h).If the functions in question are real-valued, the operations of multiplication anddivision (when the denominator is not zero) can also be performed.
We have thenthe following theorem.Theorem 2 If the functions f : E → R and g : E → R, defined on a set E ⊂ Rm ,are differentiable at the point x ∈ E, thena) their product is differentiable at x and(f · g) (x) = g(x)f (x) + f (x)g (x);(8.32)b) their quotient is differentiable at x if g(x) = 0, and f1 g(x)f (x) − f (x)g (x) .(x) = 2gg (x)(8.33)The proof of this theorem is the same as the proof of the corresponding parts ofTheorem 1 in Sect. 5.2, so that we shall omit the details.Relations (8.31), (8.32), and (8.33) can be rewritten in the other notations for thedifferential.
To be specific:d λ1 f1 (x) + λ2 f2 (x) = (λ1 df1 + λ2 df2 )(x),d(f · g)(x) = g(x) df (x) + f (x) dg(x), 1 f(x) = 2g(x) df (x) − f (x) dg(x) .dgg (x)4408 Differential Calculus in Several VariablesLet us see what these equalities mean in the coordinate representation of themappings. We know that if a mapping ϕ : E → Rn that is differentiable at an interiorpoint x of the set E ⊂ Rm is written in the coordinate form⎛ 1 1⎞ϕ (x , . . . , x m )..⎜⎟ϕ(x) = ⎝⎠,.ϕ n (x 1 , . . .
, x m )then the Jacobi matrix⎛∂1 ϕ 1⎜ ..ϕ (x) = ⎝ .∂1 ϕ n···...···⎞∂m ϕ 1 j.. ⎟. ⎠ (x) = ∂1 ϕ (x)∂m ϕ nwill correspond to its differential dϕ(x) : Rm → Rn at this point.For fixed bases in Rm and Rn the correspondence between linear transformationsL : Rm → Rn and m × n matrices is one-to-one, and hence the linear transformationL can be identified with the matrix that defines it.Even so, we shall as a rule use the symbol f (x) rather than df (x) to denote theJacobi matrix, since it corresponds better to the traditional distinction between theconcepts of derivative and differential that holds in the one-dimensional case.Thus, by the uniqueness of the differential, at an interior point x of E we obtainthe following coordinate notation for (8.31), (8.32), and (8.33), denoting the equalityof the corresponding Jacobi matrices: jj jj∂i λ1 f1 + λ2 f2 (x) = λ1 ∂i f1 + λ2 ∂i f2 (x)(i = 1, .
. . , m, j = 1, . . . , n),(8.31 )∂i (f · g) (x) = g(x)∂i f (x) + f (x)∂i g(x) (i = 1, . . . , m),(8.32 ) f1 ∂i(x) = 2g(x)∂i f (x) − f (x)∂i g(x) (i = 1, . . . , m). (8.33 )gg (x)It follows from the elementwise equality of these matrices, for example, thatthe partial derivative with respect to the variable x i of the product of real-valuedfunctions f (x 1 , . . . , x m ) and g(x 1 , . . . , x m ) should be taken as follows: ∂f 1∂(f · g) 1x , . .
. , xm = g x1, . . . , xmx , . . . , xm +ii∂x∂x 1 ∂g 1+ f x , . . . , xmx , . . . , xm .i∂xWe note that both this equality and the matrix equalities (8.31 ), (8.32 ), and(8.33 ) are obvious consequences of the definition of a partial derivative and theusual rules for differentiating real-valued functions of one real variable. However,8.3 The Basic Laws of Differentiation441we know that the existence of partial derivatives may still turn out to be insufficientfor a function of several variables to be differentiable.
For that reason, along withthe important and completely obvious equalities (8.31 ), (8.32 ), and (8.33 ), theassertions about the existence of a differential for the corresponding mapping inTheorems 1 and 2 acquire a particular importance.We remark finally that by induction using (8.32) one can obtain the relationd(f1 , . . . , fk )(x) = (f2 · · · fk )(x) df1 (x) + · · · + (f1 · · · fk−1 ) dfk (x)for the differential of a product (f1 · · · fk ) of differentiable real-valued functions.8.3.2 Differentiation of a Composition of Mappings (Chain Rule)a. The Main TheoremTheorem 3 If the mapping f : X → Y of a set X ⊂ Rm into a set Y ⊂ Rn is differentiable at a point x ∈ X, and the mapping f : Y → Rk is differentiable at the pointy = f (x) ∈ Y , then their composition g ◦ f : X → Rk is differentiable at x and thekdifferential d(g ◦ f ) : T Rmx → T Rg(f (x)) of the composition equals the compositiondg(y) ◦ df (x) of the differentialsndf (x) : T Rmx → T Rf (x)=y ,dg(y) : T Rny → T Rkg(y) .The proof of this theorem repeats almost completely the proof of Theorem 2 ofSect.
5.2. In order to call attention to a new detail that arises in this case, we shallnevertheless carry out the proof again, without going into technical details that havealready been discussed, however.Proof Using the differentiability of the mappings f and g at the points x and y =f (x), and also the linearity of the differential g (x), we can write(g ◦ f )(x + h) − (g ◦ f )(x) == g f (x + h) − g f (x) == g f (x) f (x + h) − f (x) + o f (x + h) − f (x) == g (y) f (x)h + o(h) + o f (x + h) − f (x) == g (y) f (x)h + g (y) o(h) + o f (x + h) − f (x) == g (y) ◦ f (x) h + α(x; h),where g (y) ◦ f (x) is a linear mapping (being a composition of linear mappings),andα(x; h) = g (y) o(h) + o f (x + h) − f (x) .4428 Differential Calculus in Several VariablesBut, as relations (8.17) and (8.18) of Sect.
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