Higher-Order Derivatives and Taylor’s Formula in Several ...

1 Higher-Order Derivatives and Taylor s Formula in Several VariablesG. B. FollandTraditional notations for partial Derivatives become rather cumbersome for derivativesof order higher than two, and they make it rather difficult to write Taylor s theorem in anintelligible fashion. (In particular, Apostol sDr1,..,rkis pretty ghastly.) However, a betternotation, which is now in common usage in the literature of partial differential equations, ann-tuple of nonnegative integers. Multi-indices are generally denotedby the Greek letters or : = ( 1, 2.)

2 , n), = ( 1, 2,.., n)( j, j {0,1,2,..}).If is a multi-index, we define| |= 1+ 2+ + n, ! = 1! 2! n!,x =x 11x 22 x nn(wherex= (x1,x2,..,xn) Rn), f= 11 22 nnf= | |f x 11 x 22 x nnThe number| |= 1+ + nis called theorderordegreeof . Thus, the order of isthe same as the order ofx as a monomial or the order of as a partial a function of classCk, by Theorem and the discussion following it the orderof differentiation in akth-order partial derivative offis immaterial. Thus, the generickth-order partial derivative offcan be written simply as fwith| |= 3 andx= (x,y,z), we have (0,3,0)f= 3f y3, (1,0,1)f= 2f x z,x(2,1,5)= the notationx indicates, multi-indices are handy for writing not only Derivatives butalso polynomials in Several variables.

3 To illustrate their use, we present a generalization ofthe binomial 1(The Multinomial Theorem).For anyx= (x1,x2,..xn) Rnand any positiveintegerk,(x1+x2+ +xn)k= | |=kk! !x . casen= 2 is just the binomial theorem:(x1+x2)k=k j=0k!j!(k j)!xj1xk j2= 1+ 2=kk! 1! 2!x 11x 22= | |=kk! !x ,1where we have set 1=j, 2=k j, and = ( 1, 2). The general case follows byinduction onn. Suppose the result is true forn < Nandx= (x1,..,xN). By using theresult forn= 2 and then the result forn=N 1, we obtain(x1+ +xN)k=[(x1+ +xN 1) +xN]k= i+j=kk!

4 I!j!(x1+ +xN 1)ixjN= i+j=kk!i!j! | |=ii! ! x xjN,where = ( 1,.., N 1) and x= (x1,..,xN 1). To conclude, we set = ( 1,.., N 1,j),so that !j! = ! and x xjN=x . Observing that runs over all multi-indices of orderkwhen runs over all multi-indices of orderi=k jandjruns from 0 tok, we obtain | |=kk!x / !.A similar argument leads to the product rule for Higher-Order partial Derivatives : (fg) = + = ! ! !( f)( g).The proof is by induction on the numbernof variables, the base casen= 1 being thehigher-order product rule in your Assignment now turn to Taylor s theorem for functions of Several variables.

5 We consider onlyscalar-valued functions for simplicity; the generalization to vector-valued functions is :Rn Ris of classCkon a convex open setS. We can derive a Taylorexpansion forf(x) about a pointa Sby looking at the restriction offto the line joiningaandx. That is, we seth=x aandg(t) =f(a+t(x a)) =f(a+th).By the chain rule,g (t) =h f(a+th),and henceg(j)(t) = (h )jf(a+th),where the expression on the right denotes the result of applying the directional derivativeh =h1 x1+ +hn xn(1)jtimes tof.

6 The Taylor Formula forgwitha= 0 andh= 1,g(1) =k 0g(j)(0)j!1j+ (remainder),2therefore yieldsf(a+h) =k 0(h )jf(a)j!+Ra,k(h),(2)where formulas forRa,k(h) can be obtained from the lagrange or integral formulas forremainders, applied is usually preferable, however, to rewrite (2) and the accompanying formulas for theremainder so that the partial Derivatives offappear more explicitly. To do this, we applythe multinomial theorem to the expression (1) to get(h )j= | |=jj! !h .Substituting this into (2) and the remainder formulas, we obtain the following:Theorem 2(Taylor s Theorem in Several Variables).

7 Supposef:Rn Ris of classCk+1on an open convex setS. Ifa Sanda+h S, thenf(a+h) = | | k f(a) !h +Ra,k(h),(3)where the remainder is given in lagrange s form byRa,k(h) = | |=k+1 f(a+ch)h !for somec (0,1).(4)and in integral form byRa,k(h) = (k+ 1) | |=k+1h ! 10(1 t)k f(a+th)dt.(5)This result bears a pleasing similarity to the single-variable formulas a triumph formulti-index notation! It may be reassuring, however, to see the Formula for the second-orderTaylor polynomial written out in the more familiar notation:Pa,2(h) =f(a) +n j=1 jf(a)hj+12n j,k=1 j kf(a)hjhk(6)=f(a) +n 1 jf(a)hj+12n j=1 2jf(a)h2j+ 1 j<k n j kf(a)hjhk.

8 (7)The first of these formulas is (2) withk= 2; the second one is (3). (Every multi-index of order 2 is either of the form (..,2,..) or (..,1,..,1,..), where the dots denote zeroentries, so the sum over| |= 2 in (3) breaks up into the last two sums in (7).) Notice that3the mixed Derivatives j k(j6=k) occur twice in (6) (since j k= k j) but only once in(7) (sincej < kthere); this accounts for the disappearance of the factor of12in the last sumin (7).As in the one-variable case, the following estimate for the remainder term follows fromthe lagrange or integral formulas for it:Corollary of classCk+1onSand| f(x)| Mforx Sand| |=k+ 1, then|Ra,k(h)| M(k+ 1)!

9 H k+1,where h =|h1|+|h2|+ +|hn|. follows easily from either (5) or (4) that|Ra,k(h)| M | |=k+1|h | !,and this last expression equalsM h k+1/(k+ 1)! by the multinomial in the one-variable case, the Taylor polynomial | | k( f(a)/ !)(x a) is theonlypolynomial of degree kthat agrees withf(x) to orderkatx a, so the samealgebraic devices are available to derive Taylor expansions of complicated functions fromTaylor expansions of simpler the 3rd-order Taylor polynomial off(x,y) =ex2+yabout (x,y) = (0,0).

10 Direct method is to calculate all the partial Derivatives offof order 3and plug the results into (3), but only a masochist would do this. Instead, use the familiarexpansion for the exponential function, neglecting all terms of order higher than 3:ex2+y= 1 + (x2+y) +12(x2+y)2+16(x2+y)3+ (order>3)= 1 +x2+y+12(x4+ 2x2y+y2) +16(x6+ 3x4y+ 3x2y2+y3)+ (order>3)= 1 +y+x2+12y2+x2y+16y3+ (order>3).In the last line we have thrown the termsx4,x6,x4y, andx2y2into the garbage pail, sincethey are themselves of order>3.