Qhov sib txawv ntawm kab sib npaug thiab tsis sib npaug

2024 Tus sau: Alex Aldridge | [email protected]. Kawg hloov kho: 2023-12-17 13:40

Linear Equation vs Nonlinear Equation

Nyob rau hauv lej, algebraic equations yog equations, uas yog tsim siv polynomials. Thaum sau meej meej cov kab zauv yuav yog ntawm daim ntawv P(x)=0, qhov twg x yog ib tug vector ntawm n tsis paub txawv txawv thiab P yog ib tug polynomial. Piv txwv li, P(x, y)=4x⁵ + xy³ + y + 10=0 yog ib qho kev sib npaug ntawm cov lej hauv ob qhov sib txawv uas sau meej meej.. Tsis tas li ntawd, (x+y)³ =3x²y – 3zy⁴ yog ib qho kev sib npaug ntawm algebraic, tab sis hauv daim ntawv implicit thiab nws yuav siv daim ntawv Q(x, y, z)=x³ + y³ + 3xy ² +3zy^{4=0, ib zaug sau meej.}

Tus yam ntxwv tseem ceeb ntawm kev sib npaug ntawm algebraic yog nws qib. Nws txhais tau tias yog lub zog siab tshaj plaws ntawm cov ntsiab lus tshwm sim hauv qhov sib npaug. Yog hais tias ib lub sij hawm muaj ob los yog ntau qhov sib txawv, qhov sib npaug ntawm cov exponents ntawm txhua qhov sib txawv yuav raug coj los ua lub zog ntawm lub sij hawm. Ua raws li qhov txhais no P(x, y)=0 yog degree 5, hos Q(x, y, z)=0 yog degree 5.

Linear equations thiab nonlinear equations yog ob qho kev faib tawm uas tau hais tseg rau ntawm cov kab zauv sib npaug. Qhov degree ntawm qhov sib npaug yog qhov sib txawv ntawm ib leeg.

Dab tsi yog linear equation?

A linear equation is an algebraic equation of degree 1. Piv txwv li, 4x + 5=0 is a linear equation of one variable. x + y + 5z=0 thiab 4x=3w + 5y + 7z yog linear equations ntawm 3 thiab 4 variables feem. Feem ntau, ib qho kab zauv sib npaug ntawm n qhov sib txawv yuav siv daim ntawv m₁x₁ + m₂x ₂ +…+ m_n-1x_n-1 + m_{n x}n_{=b. Ntawm no, x}i_{'s yog cov tsis paub hloov pauv, m}i_{'s thiab b yog tus lej tiag uas txhua tus ntawm m}i _{tsis yog xoom.}

Qhov kev sib npaug no sawv cev rau lub dav hlau siab hauv n-dimensional Euclidean qhov chaw. Tshwj xeeb, ob qhov sib npaug sib npaug sib npaug sawv cev rau txoj kab ncaj nraim hauv Cartesian dav hlau thiab peb qhov sib npaug sib npaug sib npaug sawv cev rau lub dav hlau ntawm Euclidean 3-space.

Dab tsi yog qhov sib npaug tsis sib npaug?

Ib qho kev sib npaug plaub yog ib qho kev sib npaug ntawm cov lej, uas tsis yog kab. Hauv lwm lo lus, ib qho kev sib npaug nonlinear yog ib qho kev sib npaug ntawm qib 2 lossis siab dua. x² + 3x + 2=0 yog ib qho sib txawv tsis sib xws. x² + y³+ 3xy=4 and 8yzx² + y² + 2z^{2 + x + y + z=4 yog piv txwv ntawm kev sib npaug tsis sib luag ntawm 3 thiab 4 qhov sib txawv raws li.}

Ib qib thib ob uas tsis yog kab sib npaug yog hu ua quadratic equation. Yog tias qhov degree yog 3, ces nws yog hu ua cubic equation. Lub degree 4 thiab degree 5 sib npaug yog hu ua quartic thiab quintic equations feem. Nws tau raug pov thawj tias tsis muaj ib txoj hauv kev tshuaj ntsuam los daws cov kev sib npaug tsis sib npaug ntawm qib 5, thiab qhov no muaj tseeb rau qib siab dua ib yam nkaus. Solvable nonlinear equations sawv cev rau qhov chaw siab uas tsis yog cov dav hlau hyper.

Dab tsi yog qhov sib txawv ntawm kab sib npaug thiab qhov tsis sib npaug?

• Ib kab sib npaug yog qhov sib npaug ntawm qib 1, tab sis qhov sib npaug tsis yog kab zauv yog qhov sib npaug ntawm qib 2 lossis siab dua.

• Txawm hais tias ib qho kev sib npaug kab sib luag yog analytically solvable, nws tsis yog nyob rau hauv nonlinear equations.

• Hauv n-dimensional Euclidean qhov chaw, qhov chaw daws teeb meem ntawm n-variable linear equation yog lub dav hlau hyper, thaum lub n-variable nonlinear equation yog ib qho chaw hyper, uas tsis yog lub dav hlau hyper. (Quadrics, cubic nto thiab lwm yam.)