Strong/weak limits and strong/weak differentials of multi-variable indeterminate forms. I: Deduction and geometric illustration

1. Introduction In 1696, the French mathematician L. Hospital published a thesis titled Infinitesimal Analysis which dealt with differential and integral calculus created by Newton and Leibniz. The thesis attracted much worldwide attention on its appearance, especially Hospital抯 creation so called 揹ifferential solution method for unitary indeterminate form?is till popular. The author acquired in early 1936 that 22xyzxy=+ is the indeterminate form when P(x,y) O (0,0), and that dtandyykxxa…     

Strong/weak limits and strong/weak differentials of multi-variable indeterminate forms. III: Transformations

1SimplificationinsphericalcoordinatesInthesphericalcoordinatessystem,??y?x=rsinθcosφ,z=rsinθsinφ,=rcosθ,???00≤θ<π,≤θ<2π.Setk=?tanφ,yandK=?zcotθ,then,xxcosφcotθu=r0K(t,t')istransformedintor=r0K(tanφ,cos),φandis,whenφ=0,simplifiedintoatruncatedcurver=r°K(0,cotθ).Thelatteriseasiertoberesolvedandcanreverttotheformerthroughturningaroundfor180°.Example.Thereexistsu=z2=(rcosθ)2x2+y2+z2r2=r°cos2θ?φ°,0≤θ≤π,0≤φ<2π.AsshowninFig.1,XX′isthediameteroftheunitcircle,OP0i…