Kev Ua Haujlwm ntawm Cov Naj Npawb Sib Xyaws
Cov lej sib xyaw yog ib lub tswv yim lej uas muab cov lej tiag tiag thiab cov lej xav ua ke. Lawv yog qhov tseem ceeb rau ntau ceg ntawm kev tshawb fawb, suav nrog physics, engineering, thiab lej nws tus kheej. Hauv tsab xov xwm no, peb yuav tshawb nrhiav ntau yam kev ua haujlwm uas tuaj yeem ua rau cov lej sib xyaw, suav nrog kev ntxiv, kev rho tawm, kev sib npaug, kev faib, thiab ntau ntxiv.
Kev Nkag Siab Txog Cov Naj Npawb Uas Muaj Cov Naj Npawb Sib Txawv
Txhua tus lej nyuaj tuaj yeem sau ua daim ntawv \(a+bi\), qhov twg \(a\) thiab \(b\) yog cov lej tiag tiag thiab \(i\) yog ib chav xav uas txaus siab \(i^2 = -1\). Lo lus \(a\) hu ua qhov tiag tiag, thaum \(b\) yog qhov xav ntawm tus lej nyuaj. Piv txwv li, \(3 + 4i\) yog tus lej nyuaj nrog qhov tiag tiag 3 thiab qhov xav 4.
Ua ib qho kev sib npaug yooj yim, peb muaj:
\[ i^2 = -1 \]
Uas txhais tau tias \(i\) yog lub hauv paus plaub fab ntawm -1.
Kev Ntxiv thiab Kev Rho Tawm
Kev ntxiv thiab rho tawm cov lej nyuaj yog ua los ntawm kev ntxiv thiab rho tawm cov feem tiag thiab xav txog. Xav tias peb muaj ob tus lej nyuaj \( z_1 = a + bi \) thiab \( z_2 = c + di \), ces:
\[z_1 + z_2 = (a + bi) + (c + di) = (a + c) + (b + d)i \]
\[ z_1 – z_2 = (a + bi) – (c + di) = (a – c) + (b – d)i \]
Piv txwv:
Cia \( z_1 = 3 + 4i \) thiab \( z_2 = 1 + 2i \), ces:
\[z_1 + z_2 = (3+1) + (4+2)i = 4 + 6i \]
\[z_1 – z_2 = (3-1) + (4-2)i = 2 + 2i \]
Kev sib npaug
Kev sib npaug ntawm cov lej nyuaj siv cov kev faib tawm zoo li hauv algebra tab sis suav nrog \( i^2 = -1 \). Xav tias \( z_1 = a + bi \) thiab \( z_2 = c + di \), ces:
\[ z_1 \cdot z_2 = (a + bi)(c + di) = ac + adi + bci + bdi^2 \]
\[= ac + adi + bci + bd(-1) \]
\[ = ac + adi + bci – bd \]
\[ = (ac – bd) + (ad + bc)i \]
Piv txwv:
Cia \( z_1 = 3 + 4i \) thiab \( z_2 = 1 + 2i \), ces:
\[ z_1 \cdot z_2 = (3 + 4i)(1 + 2i) \]
\[ = 3 \cdot 1 + 3 \cdot 2i + 4i \cdot 1 + 4i \cdot 2i \]
\[ = 3 + 6i + 4i + 8i^2 \]
\[ = 3 + 10i + 8(-1) \]
\[ = 3 + 10i – 8 \]
\[ = -5 + 10i \]
Kev faib tawm
Kev faib cov lej nyuaj yog ua los ntawm kev muab tus lej suav thiab tus lej denominator sib npaug los ntawm tus lej sib txuas ntawm tus lej denominator. Tus lej sib txuas ntawm tus lej nyuaj \( z = a + bi \) yog \( \overline{z} = a - bi \).
Xav tias \( z_1 = a + bi \) thiab \( z_2 = c + di \), ces:
\[ \frac{z_1}{z_2} = \frac{a + bi}{c + di} \]
Muab ntau los ntawm conjugate ntawm tus denominator:
\[ = \frac{(a + bi)(c - di)}{(c + di)(c - di)} \]
\[ = \frac{(ac + bd) + (bc – ad)i}{c^2 + d^2} \]
Piv txwv:
Cia \( z_1 = 3 + 4i \) thiab \( z_2 = 1 + 2i \), ces:
Txuas \( z_2 = 1 – 2i \).
\[ \frac{z_1}{z_2} = \frac{3 + 4i}{1 + 2i} \cdot \frac{1 – 2i}{1 – 2i} \]
\[ = \frac{(3 + 4i)(1 - 2i)}{(1 + 2i)(1 - 2i)} \]
\[ = \frac{3 – 6i + 4i – 8i^2}{1 – 4i^2} \]
Nws paub tias \( i^2 = -1 \):
\[ = \frac{3 – 6i + 4i + 8}{1 + 4} \]
\[ = \frac{11 – 2i}{5} \]
\[ = \frac{11}{5} – \frac{2i}{5} \]
\[ = 2.2 – 0.4i \]
Modulus thiab Arguments
Tus modulus ntawm tus lej complex yog qhov deb ntawm qhov pib hauv lub dav hlau complex mus rau qhov chaw uas tus lej complex sawv cev. Tus modulus ntawm tus lej complex \(z = a + bi \) yog qhia ua \(|z| \) thiab yog xam los ntawm:
\[ |z| = \sqrt{a^2 + b^2} \]
Piv txwv:
Yog tias \( z = 3 + 4i \), ces:
\[ |z| = \sqrt{3^2 + 4^2} = \sqrt{9 + 16} = \sqrt{25} = 5 \]
Qhov kev sib cav ntawm tus lej nyuaj yog lub kaum sab xis uas tus lej nyuaj tsim nrog lub axis tiag tiag hauv lub dav hlau nyuaj thiab feem ntau yog qhia ua radians lossis degrees.
Daim Ntawv Polar
Cov lej sib xyaw kuj tseem tuaj yeem qhia ua daim ntawv polar. Daim ntawv no feem ntau ua kom yooj yim rau kev suav nrog lub zog thiab cov hauv paus ntawm cov lej sib xyaw. Ib tus lej sib xyaw tuaj yeem qhia ua:
\[ z = r(\cos \theta + i\sin \theta) \]
qhov twg \( r \) yog modulus thiab \( \theta \) yog qhov sib cav ntawm tus lej complex.
Lwm Cov Neeg Ua Haujlwm Sib Xyaws: Exponential thiab Logarithmic
Kev hloov cov lej nyuaj mus rau daim ntawv exponential tuaj yeem ua tiav los ntawm kev siv Euler tus qauv:
\[ z = re^{i\theta} \]
qhov twg \(e \) yog lub hauv paus ntawm lub logarithm ntuj, thiab \(\theta \) yog qhov sib cav ntawm \(z \).
Kev nthuav tawm ntawm cov lej nyuaj yog qhov muaj txiaj ntsig zoo heev rau ntau yam kev ua haujlwm, tshwj xeeb tshaj yog hauv Fourier kev tshuaj xyuas thiab Laplace transforms.
Xaus
Cov lej sib xyaw yog ib qho cuab yeej tseem ceeb uas muaj txiaj ntsig zoo kawg nkaus hauv kev daws ntau yam teeb meem hauv kev suav lej thiab kev tshawb fawb. Kev paub txog cov haujlwm yooj yim xws li kev ntxiv, kev rho tawm, kev sib npaug, thiab kev faib yog thawj kauj ruam tseem ceeb. Ntxiv mus, kev nkag siab txog cov ntsiab lus ntawm modulus, kev sib cav, thiab kev hloov pauv mus rau cov ntawv polar thiab exponential txhim kho peb lub peev xwm los tshawb nrhiav cov kev siv ntawm cov lej sib xyaw hauv ntau qhov chaw.
Los ntawm kev nkag siab thiab siv cov lej nyuaj, peb tuaj yeem daws cov teeb meem uas yuav nyuaj lossis tsis yooj yim sua kom daws tau siv cov lej tiag tiag xwb. Ua ib lub cuab yeej muaj zog hauv kev tshuaj xyuas, cov lej nyuaj tseem yog ib feem tseem ceeb ntawm kev suav lej thiab kev siv tshuab txog niaj hnub no.
