dx du a(x, y) a(w, a(x, as ys==yu. ux. Again, if 0, then u0---e; so, dx if v) =0, then F (u, v)=--c, and so on. a(x, y) From DI we readily pass to the triple I, r aj r b e f c f (x, y, z) dx dy dz, and hence to mut j tiple is in general. The higher Jacobian maintains its role: du dv dtv=I dx dy ds, I= a(u, v, w) a(x, Y. z) =-- us uy us , and so on in general.
Wy Ws Thus, to pass from rectangular to polar co-ordinates, x=p cos 8, y=p sin 1 (x, y; p, +t) =p; under ff ,dx dy-=pdpd19 — this latter is in fact the elementary curvilinear rectangle.—To pass from rectangular to spherical co-ordinates, x=p cos >9 sin 56, y=p sin 0 sin 0, z=- p cos 0, whence J (x, y, s; p, 8, 0) = sin 0, and sin i s dp da dp is in fact the rectangular curvilinear volumetric element under fff Analogy readily extends these forms to n-fold spaces. Thus the Jacobian appears geomet rically as a real derivative, the limit of the ratio of two simultaneous changes.
b The single f f (x)dx—F (b)—F (a), where a, -f=F' expresses the sum-value of f, integrated along the length b—a, through the end-values of some Fat b and a; can the doubleff f(x, y) dx dy integrated over the region R also be expressed through the end-values of some F(x, y) along the contour of R? This query is much harder to answer, but is answered similarly : If f(x, y) is integrable in R, then in generallf(x, y)dx is for every included value of y a uous function of x, F (x, y), and for every included value of x an integrable function of y.
Then the DI f f f (x, y)dx vI3, f F (x „ Y) sin vds, where s is the contour of R and x.-_- slope of the normal (drawn inwards at any point of s) to the +Y-axis. This latter is a curvilinear f , geometrically depicted as a wall built up (resp. down) along the contour s of R.— Similarly afff of f extended throughout a volume (or three-wayed spread) may be expressed through the end-values of a certain F integrated over the entire surface (S) of the volume, whereby a space-integral is turned into a surface-integral and conversely: 111(0 0 8x dy ds ..=
ff ( . an .__ xa) dS, az where 4n along the normal to S corresponds to dx on X, etc. These conversions (of Green and Riemann) are equally important to pure and to applied mathematics.
The most immediate geometric problem of integration is Quadrature, already discussed. Rectification is finding in a straight-length=an arc-length. This latter must be defined as the common limit of the length of inscribed and circumscribed polygons of which each side -= 0.
Since st= (xt)LF s = f V (xt)' (Yt)'dl. If now .1.°0(t), r= (t), be continuous functions of t, with finite limits of value, then this integration is possible, and the curve is rectifiable. Such is not the case in Weierstrass's curve; there the oscillations (maxima and minima) are infinitely many in every neighborhood however small, nor is the variation finite in any such neighborhood the arc is infinite between any two points.
Volume is given by triple Integration, ds, extremes defined by the bounding surface. Often the area of a section dicular (or possibly oblique) to an axis, as X, is a function of x, f(x) ; then In the important case of Revolutes (of an area bounded by X, the curve, and two y-ordinates), V = Quadrature of a curved surface is sometimes called Complanation.
Here again the area must be defined as the common limit of the surface area of polyhedra inscribed and circumscribed, no matter how. The surface element dS or (4.5) anout P may be viewed as projected into the element dx dy in XY and as having a limiting ratio I with the corresponding element JII in the plane tangent at P. The slope of this plane to XY....y= slope of normal to Z;