d2 dx2' nth D's. We variously f'(x), Dx2, 3d, 4th, Dnxm ... Im m-n a Dn(x+6)m xm-n, Dn sin :( − 1 )n; COS see at once, am+n−1 m-1(x+b)m+n' The simultaneous variation o. x and y is vividly depicted by the graph of f(x, y)=0. But we may image it otherwise, thus: Let P and Q depict the variables x and y, one moving on X, the other on Y, and let the elastic rod PQ connect them. As P moves uniformly along X, Q will slide up and down along Y, obeying fo. The assemblage of x-values is strewn evenly along X, the assemblage of y-values is not strewn evenly along Y, but is stretched, compressed, folded, crumpled in countless ways. The study of y becomes the study of the intimate texture of this Y-axis bearing the assemblage of y-values. If now 2-x+iy, w=u+iv be complex variables, and w=f(z), then w is indeed compounded of x and y, not however in just any combination, as x2 y2, x+2iy, but only in the one combination, x+iy. The domain of z is the XY-plane, of w the UV-plane, as which two we may take the floor and a wall or the ceiling. Not having a 4th dimension at command, we cannot envisage w = f(z) as a surface, as we did y= f(x) as a curve. We must again think of two points P and Q, in XY and UV, connected by an elastic rod, PQ: as P moves about in XY, moves about in UV, obeying w=f(2). We may think of the texture of XY as uniform, then the texture of UV will not be uniform, but stretched, compressed, folded, and crumpled in countless ways. The study of ws now becomes the study of this intimate texture of UV. As in y=f(x) the derivative y is the 4y part of the that is entirely independent of 4x, so w, must be independent of 4: as y, is the same whether 4x+o or = -o, so in general w, is the same no matter how 420, along whatever path of value. Geometrically this signifies that if p, p' be two intersecting paths of 2, and q, q' the corresponding intersecting paths of w, then p and p', q and q', intersect under the same angle. Hence the angles of the curvilinear triangle (q, q', q') the angles of the corresponding curvilinear triangle (p, p', p''); hence the two corresponding infinitesimal rectilinear triangles of the chords (or tangents) will be similar, the ratio of similitude being w2, which of course varies from point to point: the one plane is a map of the other and the textures are similar in the smallest parts. Such is the geometric interpretation of the Derivative of Functions of a Complex Variable (q. v.). It would seem that such interpretations might be indefinitely extended. 4x precisely as in the Binomial Theorem, the subscripts denoting the D's. For a quotient we write uvy, u1=v1y+vy1, U2=V2¥ = u ย Applications.-1. Let P be any ordinary point of a curve, S the foot of the ordinate y, T and N the feet (on X) of tangent and normal at P. Since ytan 7,_ SN=y⋅yx, ST =y⋅xy, and we easily express PN, PT, etc. Also, if angle of intersection of y = f(x) and Y=F(x), Yx - Yx then tan If the curves touch, the I + Y x Y x numerator=0; if they are perpendicular, the denominator = 0. = =0 2. Envelopes.-Let F(x, y; p) (1) be a system of curves distinguished by varying values of the parameter p. For any special value of p, F(x, y; p) =o will be one curve and F(x,y; p+4p) = =0 (2) a neighboring curve. Where do they meet? What relation connects x and y of the intersection, I, of any such pair? We must com‐ bine (1) and (2) and eliminate p. If in the result we pass to the limit for 4po, we shall find the locus of the intersection of consecutives (or the Envelope) of the system. For (2) we may put (1)-(2), or still better F(x, y; p+4p) − F(x, y; p) Ар between which and (1) we now eliminate p. This eliminant connects x and y for every intersection of two consecutives of the system, every instantaneous pivot about which the curve starts to turn into a neighboring position. But this is not all. It connects the x and y of all other points where meet two curves (branches) corresponding to the same p, as may thus be seen. Assign any pair of values to x and y, i.e., take any point in the plane, and ask what members of Fo pass through it, i.e. what are the corresponding values of p? There are n such, if F be of nth degree in p. When will two of these p-roots be equal? ishes; i.e., when the eliminant of p between Only when the p-discriminant of F = F=0 and Fp= =o vanishes, as we know from Algebra. Hence this eliminant connects and y for all points where meet two curves corresponding to the same or equal p's. This will include all cusps and nodes as well as instantaneous pivots; hence the p-eliminant =0 will be the equation of all cusp-loci and node. loci as well as of envelope proper. CALCULUS, THE INFINITESIMAL The Intersections I' and I" of these two with AB are definite. As A' and A" close down upon and coalesce in A, I' and I", always definite, close down upon and coalesce in their common limit I, the instantaneous pivot about which AB starts to turn. Differentiating and eliminating a we find the 4-cusped Hypocycloid, x+y=c, as the envelope, or path of the instantaneous pivot I. Hence Plücker's double conception of a curve as path of a point gliding along a straight line that turns about the point, and envelope of a straight line that turns about a point that glides on the straight line. The relation connecting the corresponding magnitudes, arc-length s (of the path) and angle a (through which the straight line turns), is the intrinsic Δα equation of the curve. The DQ, As da ds is named average curvature of 4s; its limit is named instantaneous curvature (k) at P. Plainly 4a=47; hence I I cor The coordinates (u and v) of K are given by u-x-(1+y), v=y+(1+). Elimiy22), +−(1+y'2). nating x and y between these equations and the equation of the curve, we get the equation connecting u and v for every K, i.e. the equation of the Evolute, or locus of the centers of curvature of the original curve (the Involute). Since Vu yx+1=0, the tangents at responding P and K are perpendicular, the normal to the involute is tangent to the evolute. Also, it is easy to prove that the arc-length in Evolute can differ only by a constant from the radius of curvature (p) in Involute. Hence a point of a cord held tight while being unwound from the evolute must trace an involute; hence the former name. To any involute there is only one evolute, but to any evolute there are infinitely many (so-called parallel) invo lutes, an excellent illustration of a onevalued determination with many-valued inverse, and also of the definiteness of dif ferentiation as compared with the indefiniteness of its inverse, Integration (see below). Some curves reproduce themselves in their evolutes, notably Cycloid and Logarithmic Spiral, which latter inspired the engraving and epitaph on the tomb of Jacob Bernoulli (1654-1705): Eadem mutata resurgo. The general theory of the Contact of curves, Asymptotes, etc., beautifully exemplifies this Calculus, but cannot be treated here (see Curves, Higher Plane). Indeterminates.-If for x=a both terms of a Vr :) $(r) In the circle is the constant, hence the curvature of the circle is the reciprocal of the radius. For any point of any curve the reciprocal of this curvature is called the radius of curvature, p; hence this p at any point is the radius of a circle of equal curvature, hence called circle of curvature. To illustrate.-Draw PT and PV, tangent and normal to the curve; about K' and K" on PN, with radii > and "<p, through P draw two circles, one less, the other greater than the circle of curvature. Leto' and ' approach and coalesce in p; then K' and K" approach and coalesce in K, the center of curvature, and the O about K is the osculatory circle. p(a) O 4(a) = However, we may still seek Lim. for xao,though it would be arbitrary to assume this limit as the value of x-a. Thus ༡3 — a3 x-a boux). If x-a be removable from both hence x= a x-α for x=a, unless arbitrarily. Now Lim. (x) Otherwise, through P, and Q' and Q" on opposite sides of P, draw a circle. Let midnormals to PQ', PQ" meet PN at S. S", and each other at S""; as Q' and Q" approach and coalesce in P, S', S", S"" all approach and coalesce in their common limit, S (or K). Hence the osculatory circle = circle through exponentials like 1°, o°, etc., are first reduced three consecutive points of a curve, and center of curvature=intersection of two consecutive normals. ∞ I to by passing to Logarithms. Maxima, Minima, are already defined, and CALCULUS, THE INFINITESIMAL Αν of 4x accordingly at such points the D must change how 4x and 4y approach o, i.e. independently sign; hence must pass through o, if continuous. This passage is from + to for a maximum, and yz, then the limiting position of II from to for a minimum. Hence the is the tangent plane at P. But at the vertex ordinary rule: To maximize or minimize P of a cone (x2 + y2=m2x2), II rolls forever f(x), put f'(x) =0; the x-roots will yield round the cone as Q and R circle round P. maximal or minimal values of f(x) according as მა y they make f'(x) negative or positive. For Here, at (0, 0, 0), and f'(x)=0, treat 3d and 4th D's precisely as the all meaning, as do tangent plane and normal. ду m2z 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: 1st and 2d D's. The same rules result from expanding f(x), as by Taylor's Theorem. Special cases (as of D discontinuous) call for special treatment, often geometrical or mechanical preferably. The geometric depiction of a function of two independent variables, z= f(x, y) or F(x, y, z) =0, 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 ∞ of ways; e.g., parallel to ZX; then x and z would change, but not y. Hence there would be simultaneous 4x and 4z, but 4y= =0. Then L- is written Ax Əz дх Az (Jacobi), and is read partial D of z as to x. Similarly, for a path of P parallel to YZ, there are simultaneous 4y and 4z, but 4x=0; Az az LДу ду = = partial D of z as to y. For u =f(x, y, z) intuition fails, but we think each Pin (x, y, z) as weighted with the proper u instead of erecting this u perpendicular to [x, y, z). As P moves, the weight u changes. For motion parallel to X both Дy and Az are o Au du and L etc. Of course, the foregoing 4x ax' presumes that z and u actually admit of the Derivations in question. Differentials, Partial and Total.-By Definidxu • 4x=partial Differential of u as дх tion, to x, etc. ди total Differential of u. Au дх Clearly its equation is w― z = (u — x) 5 x + მა (vy) (u, v, w, the current coördinates for ay the plane). This equation assumes the symmetric form (u−x) Fx+(v− y) F ̧+(w−2) F2 =0, მო since etc. Hence the equations of მა Fz' u-x V-V W-2 the normal are Fx Fz As to existence, the tangent plane is conditioned like the tangent line. Through P (x, y, z) and two neighbor points, Q and R (as x+4x, y, z+4, and x, y+4y, z+4), pass a secant plane II. Let R and O descend any wise upon P. If the tiltings of II approach o, if II settlez down toward the same fixed position, no matter Fy f(x+4x, y+4y) − f(x, y+4y) 4x is an equably continuous function of y and 4x. Higher D's are pure when the same Independent Variable is retained, mixed when it is changed. So მx2' მy3 are pure, but is mixed. дх ду In mixed D's the question arises: Is the order of Derivation indifferent? The a2 answer is, Yes, but only under дх ду дудх' 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 [a−h, a+h] and y in [b-k, b+k], f(x, y) is uniquely defined, and the 1(n−1)(n+2) DC's (r =o, I, amf əxm-rəyr 1≤m≤n- exist and are finite, and all the mixed ones CALCULUS, THE INFINITESIMAL f(x+h, y+k) = Ze" ox ·+h a oyf(x, y). I '+/2, A sin '-2B sin cos +C cos I hence + =A+B, a constant for all pairs of perpendicular normal sections (Euler), important in Physics and formerly taken as measure of the curvature at P(o, o, o). Consider the surface 23=Ax+2Bxy+Cy2. It is a Paraboloid (Pd); it fits on S only at P(o, o, o), elsewhere departs from S. The sections of S and Pd are not the same for z=c, but close down on each other for co. The Pd-section is an ellipse, an hyperbola, or a parabola, according as B2-AC<o, >o, or =0. Suppose it enlarged under the microscope to a constant size as co; then the S-section steadily closes down on it as limit. Hence Pd and S agree elementally at P(0, 0, 0); also they agree in curvature (of their own normal sections), hence Pd is called the osculating paraboloid of S at (0, 0, 0). All these parallel sections, for changing c, are similar, hence Ax2+2Bxy+Cy2 = 1 is taken as type and called Indicatrix (Dupin). This indicatrix is an ellipse for B2-AC<o, the S is cup-shaped or synclastic; it is an hyperbola for B2-AC>o, the S is saddle-shaped (anticlastic), like a mountain-pass. The indicatrix has two Axes, tangents to sections of greatest and least curvature both of Pd and S at P (any point of S) which are mutually perpendicular and named principal sections. Now let P start to move on S facing along either (say the least) axis or principal section. This axis starts to turn about P. Let P continue to move on S facing always along the turning axis. The tangent to its path will give the direction of this axis at every point of its path, which path ; is called a Line of Curvature (LC). Plainly through every point of S there pass in general two and only two LC's (Monge), each the envelope on S of a system of principal tangents to S. These LC's cut up S into elementary curvilinear rectangles and yield an excellent system of coordinates (u, v). If the indicatrix be a circle, then all its axes are principal, through the point P there pass an of LC's, every normal section is principal, the point P is an umbilic or cyclic point. If the indicatrix be a parabola, then S is edged or ridged (cylindric) at P. The notion of surface-curvature is generated and defined quite like that of line-curvature. Draw the normal N to S at every point of the border B of 4S, forming a ruled surface, R. Draw parallel to each V a radius of a unitsphere, forming a cone C cutting out 45, on the sphere-surface, which subtends a (so-called) solid angle da at the center. This we also define as the solid angle of the V's, and further define the average curvature of 4S as the ratio Δα S (Think of a cord passed round the gorge of R and then tightened, compressing R into If the C without changing the solid angie.) unit solid angle or stereradian (Halsted) be subtended by r2, the whole solid angle about center=47; then the metric numbers of 4a and 4S' are equal, hence the curvature of AS' 4S' 4S= and Lim. AS' curvature at P (in 4S) =K. ASS=α=instantaneous = I RR If R, R' be the principal radii of curvature at any P on S, then K moreover, this K is not affected by bending S in any way without stretching or tearing a beautiful Gaussian theorem of profound philosophic import. In this sense, an S that may be flattened out into a plane (a Developable) has o-curvature; for such, RR' must become; hence either But RR'=(1+p2+q2) / (rt-s2), where p2r, q=y, r=Zxx, S=Zry, t=zyy (Euler). Hence rt-s2-o is the equation of Developables. For Applicables and further illustrations, see SURFACES, THEORY OF. R = ∞ or R'∞. The difficulty in dealing with Implicit Functions (defined by unsolved equations) le; in the Existence-theorems, which can only be stated. I. Let F(x, y): =o at (xo, yo), and have 1st partial D's finite and continuous about (x, y), and Fyo at (xo, Yo): then there is a (r) that becomes y, for x=x, and satisfies identically F(x, y): =o in the vicinity, and is unique, and has a D. Yx=9′(x) = -Fx/Fy. II. Quite similarly for F(x, y, z) =0, z = p(x, y), the last statement being: has two partial D's. Fy. and so on for n variables. Most generally (III) let F1, . . ., Fn be n functions of m variables x, y, and n variables u, v, ..., all the F's vanishing at (x, Yo,... u, v,...), all admitting partial D's in that vicinity, and J(F1,..., Fn; u, v, ...) #0 (p. 13), at (xo, Yo 2x= CALCULUS, THE INFINITESIMAL regarding y as a function of x, as in mediate Definite Integral (DI) of f(x) (the Integrand) derivation, and solve the result as to yr. as to x, between the extremes (end-values) a and b. The total sign of Integration is To find now maximum or minimum y in y=0, .'. Fx F(x, y) =o, we have y2 =offo, there is ... dx or f. f being an extended S, meaning therefore, Fxx+Fy• Yxx=0. If Fyo, maximum for Fxx and Fy like-signed, minimum for Fax and Fy unlike-signed, and no determination for Fxx=0.* Often we seek (so-called) relative maximum and minimum of z=f(x, y) when F(x, y) =o. The former equation is a surface S, the latter an intersecting surface determining a path over S,—we seek the peak and valley points in this path. Differentiating we find as the prime condition, fx F¬†y•Fx =0 = =J(f, F; x, y), from which and F =o we find the x and y that maximize or minimize f. More generally we seek maximum or minimum of a function of (m+n) variables, f(x, y, . u, v, . . .), under n conditions F,(x, y, 1, 2,..) ... =0,... Fn(x, y,... u, v, ...) =o. Theoreti cally we might eliminate n variables u, v, leaving the other m independent; better to let them remain considered as functions of the m independents, x, y, ... Hence, on putting each partial D=o, we get m equations which, with the n F, =0,... Fn =0, form (m+n) equations for finding (m+n) unknowns x, y, . . . u, To discriminate between maximum and minimum by the sign of d2f will now be tedious, but often geometrically or mechanically unnecessary. v, w, ... Swifter and simpler is Lagrange's 'Method of Multipliers. We form a new function, f(x, y, . u, v, ...) = f(x, y, ... u, v, ...) - ΣF(x,y, u, v,...). Only so long as each Fo will of identically for all values (under consideration) of the variables. We now determine these 's so as to make vanish simultaneously all the partial D's of as to x, y, ... u, The n conditions are rolled off from the u, v, ... upon the n 's. We may proceed similarly in dealing with Envelopes, where (n+1) parameters are connected by n conditions. V. ... Transformation of Variables is often necessary, like transformation of Coördinates. The formulæ, simple at first, soon become highly complicated and we are led into the Theory of Substitutions, Invariants, Reciprocants and the like, which cannot be treated here. SO Integration. As the Differential Calculus is the doctrine of Limits of Quotients of Simultaneous Infinitesimal Differences, the INTEGRAL CALCULUS is the doctrine of Limits of the Sums of Infinitesimal Products that increase in number while decreasing in size, both indefinitely. The type is the quadrature of an area (A) bounded by X, a curve y=f(x), and two end-ordinates, x=a and x =b. Cut it into n strips, 4A, standing each on a 4x; their sum is A; plainly Yx> 4A>y4x, Y being the greatest, y the least, ordinate standing on its own particular base 4x. Since x=b-a, if y = f(x) be continuous, finite, onevalued throughout [a, b], each Y-y is an e, hence (Y-y) 4x <ex, or <(b-a); hence Lim. (Y-y) 4x=o. Hence A is the common limit of every Ey4x, written ydx, named From foregoing sections it is seen that, to max- resp. minimize z a function of independent x and v, we must have, as first condition, =0=2; and similarly for any number of independents. The secondary conditions are too complicate for discussion here. f... Limit of Sum. Plainly, exchanging extremes (a and b) merely reverses the integral by reversing Also 4x, C a So = ↓ + √ √(x) being a a seen from 2-x2=r2. We perceive that the f(x) is always the D of the so-called Indefinite Integral, the expression to be evaluated at b This is easily proved variously to be and a. Thus, if f(x) = '(x) for x always the case. in [a, b], then {1-(b)} b = f(b) — p′(b) =0, for every value of b in the range of Integrability. Hence I-(b) =C. For b=a, Ĉ is -p(a); hence found to be Hence, to calculate the integral of any integrand from a to b, find the function of which the integrand is the D, and take the difference of its values at b and a. The D of the integral is the integrand, so far as form goes, but the value depends on the extremes. Since b may be any x in the range of integrability, it is common to write it x, using x in double sense, not necessarily confusing. So long as a is unassigned, C is undetermined; hence it is common to omit a and write ff(x)dx=6(x)+C, where under f put z or any other symbol for x. we mav The integral |