By Luther Pfahler Eisenhart
Created particularly for graduate scholars, this introductory treatise on differential geometry has been a hugely winning textbook for a few years. Its strangely specific and urban procedure incorporates a thorough clarification of the geometry of curves and surfaces, targeting difficulties that might be such a lot precious to scholars. 1909 version.
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Extra resources for A Treatise on the Differential Geometry of Curves and Surfaces
J- (e o, Pl sin co + a cos da> A = 0. G)J Since these equations must be true for every fixed line, the coefficients of a, 6, c in each of these equations must be zero. , (100) I sin cos 6) Since sin o> ^_ *"*~ 1 y I A V/ a constant, equation (97) is a linear relation between and second curvatures of the curve C. , are constants different from zero COt 6) = A , ; if we take TANG-ENT SURFACE' OF A CURVE 41 and for p l and TX the values given by (100), equations isfied identically, and the point (0, fc, 0) on the (99) are sat- principal normal We generates the curve Cv conjugate to C.
In the latter general case, M if P 1 on C, the coordinates denotes the point on T corresponding to relative to the axes at 11 and J/' may be written of P r f Thus A 2 moving indicates the variation of a function trihedral, and A X relative to the the variation due to the latter and to the motion of the trihedral. M To f are within terms of higher order the coordinates of of the aid and with axes at to the with M, (81) respect (As, 0, 0) the equations of the transformation of coordinates with respect to the two axes are expressible thus: A o CURVES IN SPACE 32 These reduce to As ^ 2 As "" In the limit as JLf T p ' ~--- approaches M these equations become > *.
From (iii) it 1/pj. ILLUSTRATIVE EXAMPLES Erorn the second of these equations the former we get, by integration, log (a where c is take = c 3 follows that it + P 2 C miust 35 be plane, and from ):::-* -he, a a constant of integration If the point 2 log a this equation reduces to be chosen so that s we may , If 6 denotes the angle which the line GI makes with the -axis, p 5 we have, from (i), /Ts \e~-l Differentiating this equation with respect to s, we can put the result in the form ds consequently ( C), -e (89) When these values are substituted in equations (75), in the form or, we obtain the coordinates of C m terms ot 6, x (90) = a log tan - -f cos0 = a sin0.
A Treatise on the Differential Geometry of Curves and Surfaces by Luther Pfahler Eisenhart