Daim ntawv dav dav ntawm Pfaffian equations hauv ob qhov sib txawv x thiab y yog P dx + Qdy=0, qhov twg P=P(x, y) thiab Q=Q(x, y) yog cov haujlwm ntawm x thiab y. … Yog hais tias peb tuaj yeem nrhiav cov haujlwm f=f (x, y) thiab g=g(x, y) xws li ω=gdf, ces ω=0 tuaj yeem raug txo kom df=0 nrog kev daws f(x, y)=c (c yog qhov tsis tu ncua).
Pfaffian daim ntawv yog dab tsi?
A Pfaffian saw ntawm kev txiav txim r ≥ 0 thiab degree α ≥ 1 hauv U yog ib ntu ntawm real analytic functions f1, …, fr hauv U txaus siab qhov sib npaug sib txawv. rau kuv=1, …, r qhov twg Pi, j ∈ R[x 1, …, x , y1, …, yi] yog polynomials degree ≤ α. A muaj nuj nqi f ntawm U yog hu ua Pfaffian muaj nuj nqi ntawm kev txiav txim r thiab degree (α, β) yog.
Dab tsi yog qhov tsim nyog thiab txaus txaus rau Pfaffian qhov sib txawv sib npaug?
Theorem Ib qho tsim nyog thiab txaus tias Pfaffian qhov sib txawv sib npaug X · r=0 yuav tsum sib koom ua ke yog tias X · rot X=0.
Dab tsi yog qhov sib npaug sib npaug sib npaug?
SIMULTANEOUS DIFFERENTIAL EQUATIONS
Yog tias ob lossis ntau qhov sib txawv yog cov haujlwm ntawm ib qho kev hloov pauv ywj pheej, tus. equations uas muaj lawv cov derivatives hu ua simultaneous equations, e.g., ty. ua dt. dx.
Yuav ua li cas homogeneous hauv kev sib npaug sib txawv?
Ib qhov sib txawv ntawm daim ntawv f(x, y)dy=g(x, y)dx yog hais tias yog homogeneous sib txawv yog tias qhov degree ntawm f(x, y) thiab g(x, y) yog tib yam. Ib txoj haujlwm ntawm daim ntawv F(x, y) uas tuaj yeem sau rau hauv daim ntawv k F(x, y) tau hais tias yog homogeneous muaj nuj nqi ntawm degree n, rau k≠0.
