Calculus

0 Maxima and Minima are points where a curve ceases to ascend and begins to descend; accordingly at such points the D must change sign ; hence must pass through 0, if continuous. This passage is from + to — for a maximum, from — to + for a minimum. Hence the ordinary rule: To maximize or minimize f(x), put f' (x)=0; the x-roots will yield maximal or minimal values of f(x) according as they make f"(x) negative or positive. For f" (x) =0, treat 3d and 4th D's precisely as the 1st and 2d as. The same rules result from expanding f(x), as by Taylor's Theorem. Special cases (as of D discontinuous) call for special treatment, often preferably geometrical or mechanical.

The geometric depiction of a function of two independent variables,..:..—f(x,y) or F(x, y, =3, is of course a surface (S): at any point (x, y)in the plane erect the corresponding value of z; the ends of these z's form S. We may pass on S from P to P' in ce of ways; e.g., parallel to ZX ; then x ands would change, but not y. Hence there would be simultaneous dx dz and 42, but dy= O. Then L--i is written as ax (Jacobi), and is read partial D of z as to x.

Similarly, for a path of P parallel to YZ, there are simultaneous .dy and .1z, but dz L— =— as y =.= partial D of z as to y.

dy a For u=f (x, y, z) institution fails, but we think each P in [x, y, z] as weighted with the proper u instead of erecting this u perpendicular to Ix, y, s]. As P moves, the weight u changes. For motion parallel to X both dy and dis are 0 du au and =-- etc. Of course, the foregoing presumes that s and u actually admit of the derivations in question.

Differentials, Partial and Total.—By Defini au tion, dsu T. • 11.2= partial Differential of u as au au • to x, etc. du x + . . . dx + . . total Differential of u. -F.. . -Fe dx . Difference—u.

Geometrically, on e=f(x, y), the path of x change is parallel to ZX, hence =tan r, as before. Similarly = tan u. The plane ay through the tangents to these paths (at P) is in general the plane tangent at P(x, y, z) to S.

8z Clearly its equation is w a; az 31) ay( u, v, tv being the current co-ordinates for the plane). This equation assumes the sym metric form (u-x) Fs + (v-y) Fy + (w-z) = 0, as since —= — — etc. Hence the equations ax F Fss ' u—x v--y w—s of the normal are — =- .Fx Fs As to existence, the tangent plane is condi tioned like the tangent line. Through P (x, y, z)

and two neighbor points, Q and R (as x + dx, y, z + and x, y z + do), pass a secant plane II. Let R and Q descend any wise upon P. If the tiltings of II approach o, if 11 settles down toward the same fixed position, no matter how dx and dy approach 0, i.e. independently dy d — dx and yx, then the limiting position of II is the tangent plane at P. But at the vertex P of a cone + = II rolls forever around the cone as Q and R circle around P.

az x , as y ay m2s Here, at (0, 0, 0), ax m2z — — = all meaning, as do tangent plane and normal.

In general there is no such notion as Total Derivative of z= f (x, y), x and y independent ; but if both be functions of an arbitrary t, we have the Total D of z as to this t; dz df _Of dx + of dy — — dt ax dt ay • dt an extension of mediate derivation. Of course, the possibility and definiteness of the opera tions are implied. Hence again ar .

dz _-_-. df =--= . dx+ . dy, ax ay or the Total Differential=the sum of the Partial dz Differentials. For t=x, --=--- fx + fy • yz, or ds= fx.dx + a fundamental theorem hold ing when at (x, y) the DQ f(x + dx, y + dy) —f(x, y + dy) dx is an equably continuous function of y and .1x. Higher D's are pure when the same pendent Variable is retained, mixed when it is ats R a2z changed. So are pure, but ay is mixed. In mixed D's the question arises: Is the order of Derivation indifferent! The answer is, Yes, — --= — ax ay ay ax , but only under conditions. For a power-series the case is clear, but the general investigation is subtle, and the result is involved and tedious. The theorem holds: When, for x in fa— lt, a + h] and y in [b — k, b + le], R x, v) is uniquely de fined, and the i(n —1) (n 2) DC's r ----- 0, ....,1 ... m) l< tits- - n— 1 exist and are finite, and all the mixed ones are continuous as to both x and y, then every where in the same rectangle (2h, 2k) all the other mixed DCs below the nth order exist, and the order of derivation is indifferent Also, if, besides, everywhere in the rectangle f (x the (n-1) mixed DC's of nth y) , anf(x,Y) Y) exist finite, and at ax aye-i' [a, 6] are continuous as to both x and y, then all other mixed DCs exist at [a, b], the order of derivation being indifferent. Space is want ing for the proofs.