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  1. @EQEDWIZ@
  2. Ca│ka oznaczona
  3. \EQEDint
  4. {\EQEDplain
  5. {a}
  6. }
  7. {\EQEDplain
  8. {b}
  9. }
  10. {\EQEDplain
  11. {f}
  12. \EQEDbrackets
  13. {\left(}
  14. {\EQEDplain
  15. {x}
  16. }
  17. {\right)}
  18. \EQEDplain
  19. {dx}
  20. }
  21. @
  22. wsp≤│czynniki Fouriera (wzory Eulera)
  23. \EQEDlo
  24. {\EQEDplain
  25. {k}
  26. }
  27. {\EQEDplain
  28. {a}
  29. }
  30. \EQEDplain
  31. {=}
  32. \frac
  33. {\EQEDplain
  34. {2}
  35. }
  36. {\EQEDplain
  37. {T}
  38. }
  39. \EQEDint
  40. {\EQEDplain
  41. {0}
  42. }
  43. {\EQEDplain
  44. {T}
  45. }
  46. {\EQEDplain
  47. {f}
  48. }
  49. \EQEDbrackets
  50. {\left(}
  51. {\EQEDplain
  52. {x}
  53. }
  54. {\right)}
  55. \cos \EQEDplain
  56. {k}
  57. \omega \EQEDplain
  58. {xdx}
  59.  
  60. \EQEDlo
  61. {\EQEDplain
  62. {k}
  63. }
  64. {\EQEDplain
  65. {b}
  66. }
  67. \EQEDplain
  68. {=}
  69. \frac
  70. {\EQEDplain
  71. {2}
  72. }
  73. {\EQEDplain
  74. {T}
  75. }
  76. \EQEDint
  77. {\EQEDplain
  78. {0}
  79. }
  80. {\EQEDplain
  81. {T}
  82. }
  83. {\EQEDplain
  84. {f}
  85. \EQEDbrackets
  86. {\left(}
  87. {\EQEDplain
  88. {x}
  89. }
  90. {\right)}
  91. \sin \EQEDplain
  92. {k}
  93. \omega \EQEDplain
  94. {xdx}
  95. }
  96. @
  97. zastapienie funkcji szeregami Fouriera
  98. \EQEDlo
  99. {\EQEDplain
  100. {n}
  101. }
  102. {\EQEDplain
  103. {s}
  104. }
  105. \EQEDbrackets
  106. {\left(}
  107. {\EQEDplain
  108. {x}
  109. }
  110. {\right)}
  111. \EQEDplain
  112. {=}
  113. \frac
  114. {\EQEDplain
  115. {1}
  116. }
  117. {\EQEDplain
  118. {2}
  119. }
  120. \EQEDlo
  121. {\EQEDplain
  122. {0}
  123. }
  124. {\EQEDplain
  125. {a}
  126. }
  127. \EQEDplain
  128. {+}
  129. \EQEDsum
  130. {\EQEDplain
  131. {k=1}
  132. }
  133. {\EQEDplain
  134. {n}
  135. }
  136. {\EQEDlo
  137. {\EQEDplain
  138. {k}
  139. }
  140. {\EQEDplain
  141. {a}
  142. }
  143. \cos \EQEDplain
  144. {k}
  145. \omega \EQEDplain
  146. {x}
  147. \EQEDplain
  148. {+}
  149. \EQEDsum
  150. {\EQEDplain
  151. {k=1}
  152. }
  153. {\EQEDplain
  154. {n}
  155. }
  156. {\EQEDlo
  157. {\EQEDplain
  158. {k}
  159. }
  160. {\EQEDplain
  161. {b}
  162. }
  163. \sin \EQEDplain
  164. {k}
  165. \omega \EQEDplain
  166. {x}
  167. \EQEDplain
  168. {}
  169. }
  170. }
  171. @
  172. szereg Taylora
  173. \EQEDplain
  174. {f}
  175. \EQEDbrackets
  176. {\left(}
  177. {\EQEDplain
  178. {x}
  179. }
  180. {\right)}
  181. \EQEDplain
  182. {=}
  183. \EQEDplain
  184. {f}
  185. \EQEDbrackets
  186. {\left(}
  187. {\EQEDplain
  188. {a}
  189. }
  190. {\right)}
  191. \EQEDplain
  192. {+}
  193. \frac
  194. {\EQEDplain
  195. {x-a}
  196. }
  197. {\EQEDplain
  198. {1!}
  199. }
  200. \EQEDhi
  201. {\EQEDplain
  202. {,}
  203. }
  204. {\EQEDplain
  205. {f}
  206. }
  207. \EQEDbrackets
  208. {\left(}
  209. {\EQEDplain
  210. {a}
  211. }
  212. {\right)}
  213. \EQEDplain
  214. {+}
  215. \frac
  216. {\EQEDhi
  217. {\EQEDplain
  218. {2}
  219. }
  220. {\EQEDbrackets
  221. {\left(}
  222. {\EQEDplain
  223. {x-a}
  224. }
  225. {\right)}
  226. }
  227. }
  228. {\EQEDplain
  229. {2!}
  230. }
  231. \EQEDhi
  232. {\EQEDplain
  233. {,,}
  234. }
  235. {\EQEDplain
  236. {f}
  237. }
  238. \EQEDbrackets
  239. {\left(}
  240. {\EQEDplain
  241. {a}
  242. }
  243. {\right)}
  244. \EQEDplain
  245. {+}
  246. \EQEDplain
  247. {...}
  248. \EQEDplain
  249. {+}
  250. \frac
  251. {\EQEDhi
  252. {\EQEDplain
  253. {n}
  254. }
  255. {\EQEDbrackets
  256. {\left(}
  257. {\EQEDplain
  258. {x-a}
  259. }
  260. {\right)}
  261. }
  262. }
  263. {\EQEDplain
  264. {n!}
  265. }
  266. \EQEDhi
  267. {\EQEDplain
  268. {n}
  269. }
  270. {\EQEDplain
  271. {f}
  272. }
  273. \EQEDbrackets
  274. {\left(}
  275. {\EQEDplain
  276. {a}
  277. }
  278. {\right)}
  279. @
  280. szereg Maclaurina
  281. \EQEDplain
  282. {f}
  283. \EQEDbrackets
  284. {\left(}
  285. {\EQEDplain
  286. {x}
  287. }
  288. {\right)}
  289. \EQEDplain
  290. {=}
  291. \EQEDplain
  292. {f}
  293. \EQEDbrackets
  294. {\left(}
  295. {\EQEDplain
  296. {0}
  297. }
  298. {\right)}
  299. \EQEDplain
  300. {+}
  301. \frac
  302. {\EQEDplain
  303. {x}
  304. }
  305. {\EQEDplain
  306. {1!}
  307. }
  308. \EQEDhi
  309. {\EQEDplain
  310. {,}
  311. }
  312. {\EQEDplain
  313. {f}
  314. }
  315. \EQEDbrackets
  316. {\left(}
  317. {\EQEDplain
  318. {0}
  319. }
  320. {\right)}
  321. \EQEDplain
  322. {+}
  323. \frac
  324. {\EQEDhi
  325. {\EQEDplain
  326. {2}
  327. }
  328. {\EQEDplain
  329. {x}
  330. }
  331. }
  332. {\EQEDplain
  333. {2!}
  334. }
  335. \EQEDhi
  336. {\EQEDplain
  337. {,,}
  338. }
  339. {\EQEDplain
  340. {f}
  341. }
  342. \EQEDbrackets
  343. {\left(}
  344. {\EQEDplain
  345. {0}
  346. }
  347. {\right)}
  348. \EQEDplain
  349. {+}
  350. \EQEDplain
  351. {...}
  352. \EQEDplain
  353. {+}
  354. \frac
  355. {\EQEDhi
  356. {\EQEDplain
  357. {n}
  358. }
  359. {\EQEDplain
  360. {x}
  361. }
  362. }
  363. {\EQEDplain
  364. {n!}
  365. }
  366. \EQEDhi
  367. {\EQEDplain
  368. {n}
  369. }
  370. {\EQEDplain
  371. {f}
  372. }
  373. \EQEDbrackets
  374. {\left(}
  375. {\EQEDplain
  376. {0}
  377. }
  378. {\right)}
  379. \EQEDplain
  380. {+}
  381. \EQEDplain
  382. {...}
  383. @
  384. wz≤r Newtona
  385. \EQEDhi
  386. {\EQEDplain
  387. {n}
  388. }
  389. {\EQEDbrackets
  390. {\left(}
  391. {\EQEDplain
  392. {a+b}
  393. }
  394. {\right)}
  395. }
  396. \EQEDplain
  397. {=}
  398. \EQEDhi
  399. {\EQEDplain
  400. {n}
  401. }
  402. {\EQEDplain
  403. {a}
  404. }
  405. \EQEDplain
  406. {+}
  407. \EQEDhi
  408. {\EQEDplain
  409. {n-1}
  410. }
  411. {\EQEDplain
  412. {na}
  413. }
  414. \EQEDplain
  415. {b}
  416. \EQEDplain
  417. {+}
  418. \frac
  419. {\EQEDplain
  420. {n}
  421. \EQEDbrackets
  422. {\left(}
  423. {\EQEDplain
  424. {n-1}
  425. }
  426. {\right)}
  427. }
  428. {\EQEDplain
  429. {2!}
  430. }
  431. \EQEDhi
  432. {\EQEDplain
  433. {n-2}
  434. }
  435. {\EQEDplain
  436. {a}
  437. }
  438. \EQEDhi
  439. {\EQEDplain
  440. {2}
  441. }
  442. {\EQEDplain
  443. {b}
  444. }
  445. \EQEDplain
  446. {+}
  447. \frac
  448. {\EQEDplain
  449. {n}
  450. \EQEDbrackets
  451. {\left(}
  452. {\EQEDplain
  453. {n-1}
  454. }
  455. {\right)}
  456. \EQEDbrackets
  457. {\left(}
  458. {\EQEDplain
  459. {n-2}
  460. }
  461. {\right)}
  462. }
  463. {\EQEDplain
  464. {3!}
  465. }
  466. \EQEDhi
  467. {\EQEDplain
  468. {n-3}
  469. }
  470. {\EQEDplain
  471. {a}
  472. }
  473. \EQEDhi
  474. {\EQEDplain
  475. {3}
  476. }
  477. {\EQEDplain
  478. {b}
  479. }
  480. \EQEDplain
  481. {+...+}
  482. \frac
  483. {\EQEDplain
  484. {n}
  485. \EQEDbrackets
  486. {\left(}
  487. {\EQEDplain
  488. {n-1}
  489. }
  490. {\right)}
  491. \EQEDplain
  492. {...}
  493. \EQEDbrackets
  494. {\left(}
  495. {\EQEDplain
  496. {n-m+1}
  497. }
  498. {\right)}
  499. }
  500. {\EQEDplain
  501. {3!}
  502. }
  503. \EQEDhi
  504. {\EQEDplain
  505. {n-m}
  506. }
  507. {\EQEDplain
  508. {a}
  509. }
  510. \EQEDhi
  511. {\EQEDplain
  512. {m}
  513. }
  514. {\EQEDplain
  515. {b}
  516. }
  517. \EQEDplain
  518. {+...+}
  519. \EQEDhi
  520. {\EQEDplain
  521. {n}
  522. }
  523. {\EQEDplain
  524. {b}
  525. }
  526. @
  527. nier≤wno£µ Cauchy'ego
  528. \frac
  529. {\EQEDlo
  530. {\EQEDplain
  531. {1}
  532. }
  533. {\EQEDplain
  534. {a}
  535. }
  536. \EQEDplain
  537. {+}
  538. \EQEDlo
  539. {\EQEDplain
  540. {2}
  541. }
  542. {\EQEDplain
  543. {a}
  544. }
  545. \EQEDplain
  546. {+}
  547. \EQEDplain
  548. {...}
  549. \EQEDplain
  550. {+}
  551. \EQEDlo
  552. {\EQEDplain
  553. {n}
  554. }
  555. {\EQEDplain
  556. {a}
  557. }
  558. }
  559. {\EQEDplain
  560. {n}
  561. }
  562. \geq \EQEDroot
  563. {\EQEDplain
  564. {n}
  565. }
  566. {\EQEDlo
  567. {\EQEDplain
  568. {1}
  569. }
  570. {\EQEDplain
  571. {a}
  572. }
  573. \EQEDlo
  574. {\EQEDplain
  575. {2}
  576. }
  577. {\EQEDplain
  578. {a}
  579. }
  580. \EQEDplain
  581. {...}
  582. \EQEDlo
  583. {\EQEDplain
  584. {n}
  585. }
  586. {\EQEDplain
  587. {a}
  588. }
  589. }
  590. @
  591. nier≤wno£µ Buniakowskiego-Cauchy'ego
  592. \EQEDlo
  593. {\EQEDplain
  594. {1}
  595. }
  596. {\EQEDplain
  597. {a}
  598. }
  599. \EQEDlo
  600. {\EQEDplain
  601. {1}
  602. }
  603. {\EQEDplain
  604. {b}
  605. }
  606. \EQEDplain
  607. {+}
  608. \EQEDlo
  609. {\EQEDplain
  610. {2}
  611. }
  612. {\EQEDplain
  613. {a}
  614. }
  615. \EQEDlo
  616. {\EQEDplain
  617. {2}
  618. }
  619. {\EQEDplain
  620. {b}
  621. }
  622. \EQEDplain
  623. {+}
  624. \EQEDlo
  625. {\EQEDplain
  626. {3}
  627. }
  628. {\EQEDplain
  629. {a}
  630. }
  631. \EQEDlo
  632. {\EQEDplain
  633. {3}
  634. }
  635. {\EQEDplain
  636. {b}
  637. }
  638. \EQEDplain
  639. {+}
  640. \EQEDplain
  641. {...}
  642. \EQEDplain
  643. {+}
  644. \EQEDlo
  645. {\EQEDplain
  646. {n}
  647. }
  648. {\EQEDplain
  649. {a}
  650. }
  651. \EQEDlo
  652. {\EQEDplain
  653. {n}
  654. }
  655. {\EQEDplain
  656. {b}
  657. }
  658. \leq \EQEDroot
  659. {\EQEDplain
  660. {}
  661. }
  662. {\EQEDboth
  663. {\EQEDplain
  664. {1}
  665. }
  666. {\EQEDplain
  667. {2}
  668. }
  669. {\EQEDplain
  670. {a}
  671. }
  672. \EQEDplain
  673. {+}
  674. \EQEDboth
  675. {\EQEDplain
  676. {2}
  677. }
  678. {\EQEDplain
  679. {2}
  680. }
  681. {\EQEDplain
  682. {a}
  683. }
  684. \EQEDplain
  685. {+...+}
  686. \EQEDboth
  687. {\EQEDplain
  688. {n}
  689. }
  690. {\EQEDplain
  691. {2}
  692. }
  693. {\EQEDplain
  694. {a}
  695. }
  696. }
  697. \EQEDroot
  698. {\EQEDplain
  699. {}
  700. }
  701. {\EQEDboth
  702. {\EQEDplain
  703. {1}
  704. }
  705. {\EQEDplain
  706. {2}
  707. }
  708. {\EQEDplain
  709. {b}
  710. }
  711. \EQEDplain
  712. {+}
  713. \EQEDboth
  714. {\EQEDplain
  715. {2}
  716. }
  717. {\EQEDplain
  718. {2}
  719. }
  720. {\EQEDplain
  721. {b}
  722. }
  723. \EQEDplain
  724. {+...+}
  725. \EQEDboth
  726. {\EQEDplain
  727. {n}
  728. }
  729. {\EQEDplain
  730. {2}
  731. }
  732. {\EQEDplain
  733. {b}
  734. }
  735. }
  736. @
  737. wyraz og≤lny ci╣gu arytmetycznego
  738. \EQEDlo
  739. {\EQEDplain
  740. {n}
  741. }
  742. {\EQEDplain
  743. {a}
  744. }
  745. \EQEDplain
  746. {=}
  747. \EQEDlo
  748. {\EQEDplain
  749. {1}
  750. }
  751. {\EQEDplain
  752. {a}
  753. }
  754. \EQEDplain
  755. {+}
  756. \EQEDbrackets
  757. {\left(}
  758. {\EQEDplain
  759. {n-1}
  760. }
  761. {\right)}
  762. \EQEDplain
  763. {r}
  764. @
  765. suma n wyraz≤w ci╣gu arytmetycznego
  766. \EQEDlo
  767. {\EQEDplain
  768. {n}
  769. }
  770. {\EQEDplain
  771. {S}
  772. }
  773. \EQEDplain
  774. {=}
  775. \frac
  776. {\EQEDplain
  777. {n}
  778. \EQEDbrackets
  779. {\left(}
  780. {\EQEDlo
  781. {\EQEDplain
  782. {1}
  783. }
  784. {\EQEDplain
  785. {a}
  786. }
  787. \EQEDplain
  788. {+}
  789. \EQEDlo
  790. {\EQEDplain
  791. {n}
  792. }
  793. {\EQEDplain
  794. {a}
  795. }
  796. }
  797. {\right)}
  798. }
  799. {\EQEDplain
  800. {2}
  801. }
  802. @
  803. wyraz og≤lny ci╣gu geometrycznego
  804. \EQEDlo
  805. {\EQEDplain
  806. {n}
  807. }
  808. {\EQEDplain
  809. {a}
  810. }
  811. \EQEDplain
  812. {=}
  813. \EQEDlo
  814. {\EQEDplain
  815. {1}
  816. }
  817. {\EQEDplain
  818. {a}
  819. }
  820. \EQEDhi
  821. {\EQEDplain
  822. {n-1}
  823. }
  824. {\EQEDplain
  825. {q}
  826. }
  827. @
  828. suma szeregu geometrycznego
  829. \EQEDlo
  830. {\EQEDplain
  831. {n}
  832. }
  833. {\EQEDplain
  834. {S}
  835. }
  836. \EQEDplain
  837. {=}
  838. \frac
  839. {\EQEDlo
  840. {\EQEDplain
  841. {1}
  842. }
  843. {\EQEDplain
  844. {a}
  845. }
  846. \EQEDbrackets
  847. {\left(}
  848. {\EQEDhi
  849. {\EQEDplain
  850. {n}
  851. }
  852. {\EQEDplain
  853. {q}
  854. }
  855. \EQEDplain
  856. {-}
  857. \EQEDplain
  858. {1}
  859. }
  860. {\right)}
  861. }
  862. {\EQEDplain
  863. {q-1}
  864. }
  865. \EQEDplain
  866. {,}
  867. \EQEDplain
  868. {q}
  869. \neq \EQEDplain
  870. {1}
  871.  
  872. \EQEDlo
  873. {\EQEDplain
  874. {n}
  875. }
  876. {\EQEDplain
  877. {S}
  878. }
  879. \EQEDplain
  880. {=}
  881. \EQEDplain
  882. {n}
  883. \EQEDlo
  884. {\EQEDplain
  885. {1}
  886. }
  887. {\EQEDplain
  888. {a}
  889. }
  890. \EQEDplain
  891. {,}
  892. \EQEDplain
  893. {q}
  894. \EQEDplain
  895. {=}
  896. \EQEDplain
  897. {1}
  898. @
  899. funkcja gamma
  900. \Gamma \EQEDbrackets
  901. {\left(}
  902. {\EQEDplain
  903. {x}
  904. }
  905. {\right)}
  906. \EQEDplain
  907. {=}
  908. \EQEDlim
  909. {\lim}
  910. {\EQEDplain
  911. {n}
  912. \rightarrow \infty }
  913. {\frac
  914. {\EQEDplain
  915. {n!}
  916. \EQEDhi
  917. {\EQEDplain
  918. {x-1}
  919. }
  920. {\EQEDplain
  921. {n}
  922. }
  923. }
  924. {\EQEDplain
  925. {x}
  926. \EQEDbrackets
  927. {\left(}
  928. {\EQEDplain
  929. {x+1}
  930. }
  931. {\right)}
  932. \EQEDbrackets
  933. {\left(}
  934. {\EQEDplain
  935. {x+2}
  936. }
  937. {\right)}
  938. \EQEDplain
  939. {...}
  940. \EQEDbrackets
  941. {\left(}
  942. {\EQEDplain
  943. {x+n-1}
  944. }
  945. {\right)}
  946. }
  947. }
  948. @
  949. wz≤r de Moivre'a
  950. \EQEDhi
  951. {\EQEDplain
  952. {n}
  953. }
  954. {\EQEDbrackets
  955. {\left[}
  956. {\EQEDplain
  957. {q}
  958. \EQEDbrackets
  959. {\left(}
  960. {\cos \varphi \EQEDplain
  961. {+}
  962. \EQEDplain
  963. {i}
  964. \sin \varphi }
  965. {\right)}
  966. }
  967. {\right]}
  968. }
  969. \EQEDplain
  970. {=}
  971. \EQEDhi
  972. {\EQEDplain
  973. {n}
  974. }
  975. {\varphi }
  976. \EQEDbrackets
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  981. {+i}
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  983. {n}
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  987. hiperboloida jednopow│okowa
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  990. {\EQEDplain
  991. {2}
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  993. {\EQEDplain
  994. {x}
  995. }
  996. }
  997. {\EQEDhi
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  1001. {\EQEDplain
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  1004. }
  1005. \EQEDplain
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  1015. }
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  1023. }
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  1042. }
  1043. \EQEDplain
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  1046. {1}
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  1054. {\EQEDplain
  1055. {x}
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  1057. }
  1058. {\EQEDhi
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  1065. }
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  1076. }
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  1084. }
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  1095. }
  1096. {\EQEDhi
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  1098. {2}
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  1100. {\EQEDplain
  1101. {c}
  1102. }
  1103. }
  1104. \EQEDplain
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  1107. {-1}
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  1109. sto┐ek
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  1113. {2}
  1114. }
  1115. {\EQEDplain
  1116. {x}
  1117. }
  1118. }
  1119. {\EQEDhi
  1120. {\EQEDplain
  1121. {2}
  1122. }
  1123. {\EQEDplain
  1124. {a}
  1125. }
  1126. }
  1127. \EQEDplain
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  1132. {2}
  1133. }
  1134. {\EQEDplain
  1135. {y}
  1136. }
  1137. }
  1138. {\EQEDhi
  1139. {\EQEDplain
  1140. {2}
  1141. }
  1142. {\EQEDplain
  1143. {b}
  1144. }
  1145. }
  1146. \EQEDplain
  1147. {-}
  1148. \frac
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  1150. {\EQEDplain
  1151. {2}
  1152. }
  1153. {\EQEDplain
  1154. {z}
  1155. }
  1156. }
  1157. {\EQEDhi
  1158. {\EQEDplain
  1159. {2}
  1160. }
  1161. {\EQEDplain
  1162. {c}
  1163. }
  1164. }
  1165. \EQEDplain
  1166. {=}
  1167. \EQEDplain
  1168. {0}
  1169. @
  1170. paraboloida eliptyczna
  1171. \frac
  1172. {\EQEDhi
  1173. {\EQEDplain
  1174. {2}
  1175. }
  1176. {\EQEDplain
  1177. {x}
  1178. }
  1179. }
  1180. {\EQEDhi
  1181. {\EQEDplain
  1182. {2}
  1183. }
  1184. {\EQEDplain
  1185. {a}
  1186. }
  1187. }
  1188. \EQEDplain
  1189. {+}
  1190. \frac
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  1192. {\EQEDplain
  1193. {2}
  1194. }
  1195. {\EQEDplain
  1196. {y}
  1197. }
  1198. }
  1199. {\EQEDhi
  1200. {\EQEDplain
  1201. {2}
  1202. }
  1203. {\EQEDplain
  1204. {b}
  1205. }
  1206. }
  1207. \EQEDplain
  1208. {=}
  1209. \EQEDplain
  1210. {z}
  1211. @
  1212. paraboloida hiperboliczna
  1213. \frac
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  1215. {\EQEDplain
  1216. {2}
  1217. }
  1218. {\EQEDplain
  1219. {x}
  1220. }
  1221. }
  1222. {\EQEDhi
  1223. {\EQEDplain
  1224. {2}
  1225. }
  1226. {\EQEDplain
  1227. {a}
  1228. }
  1229. }
  1230. \EQEDplain
  1231. {-}
  1232. \frac
  1233. {\EQEDhi
  1234. {\EQEDplain
  1235. {2}
  1236. }
  1237. {\EQEDplain
  1238. {y}
  1239. }
  1240. }
  1241. {\EQEDhi
  1242. {\EQEDplain
  1243. {2}
  1244. }
  1245. {\EQEDplain
  1246. {b}
  1247. }
  1248. }
  1249. \EQEDplain
  1250. {=}
  1251. \EQEDplain
  1252. {z}
  1253. @
  1254. walec hiperboliczny
  1255. \frac
  1256. {\EQEDhi
  1257. {\EQEDplain
  1258. {2}
  1259. }
  1260. {\EQEDplain
  1261. {x}
  1262. }
  1263. }
  1264. {\EQEDhi
  1265. {\EQEDplain
  1266. {2}
  1267. }
  1268. {\EQEDplain
  1269. {a}
  1270. }
  1271. }
  1272. \EQEDplain
  1273. {-}
  1274. \frac
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  1276. {\EQEDplain
  1277. {2}
  1278. }
  1279. {\EQEDplain
  1280. {y}
  1281. }
  1282. }
  1283. {\EQEDhi
  1284. {\EQEDplain
  1285. {2}
  1286. }
  1287. {\EQEDplain
  1288. {b}
  1289. }
  1290. }
  1291. \EQEDplain
  1292. {=}
  1293. \EQEDplain
  1294. {1}
  1295. @
  1296. walec eliptyczny
  1297. \frac
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  1299. {\EQEDplain
  1300. {2}
  1301. }
  1302. {\EQEDplain
  1303. {x}
  1304. }
  1305. }
  1306. {\EQEDhi
  1307. {\EQEDplain
  1308. {2}
  1309. }
  1310. {\EQEDplain
  1311. {a}
  1312. }
  1313. }
  1314. \EQEDplain
  1315. {+}
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  1318. {\EQEDplain
  1319. {2}
  1320. }
  1321. {\EQEDplain
  1322. {y}
  1323. }
  1324. }
  1325. {\EQEDhi
  1326. {\EQEDplain
  1327. {2}
  1328. }
  1329. {\EQEDplain
  1330. {b}
  1331. }
  1332. }
  1333. \EQEDplain
  1334. {=}
  1335. \EQEDplain
  1336. {1}
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  1338. r≤wnanie og≤lne powierzchni stopnia II
  1339. \EQEDlo
  1340. {\EQEDplain
  1341. {11}
  1342. }
  1343. {\EQEDplain
  1344. {a}
  1345. }
  1346. \EQEDhi
  1347. {\EQEDplain
  1348. {2}
  1349. }
  1350. {\EQEDplain
  1351. {x}
  1352. }
  1353. \EQEDplain
  1354. {+}
  1355. \EQEDlo
  1356. {\EQEDplain
  1357. {22}
  1358. }
  1359. {\EQEDplain
  1360. {a}
  1361. }
  1362. \EQEDhi
  1363. {\EQEDplain
  1364. {2}
  1365. }
  1366. {\EQEDplain
  1367. {y}
  1368. }
  1369. \EQEDplain
  1370. {+}
  1371. \EQEDlo
  1372. {\EQEDplain
  1373. {33}
  1374. }
  1375. {\EQEDplain
  1376. {a}
  1377. }
  1378. \EQEDhi
  1379. {\EQEDplain
  1380. {2}
  1381. }
  1382. {\EQEDplain
  1383. {z}
  1384. }
  1385. \EQEDplain
  1386. {+2}
  1387. \EQEDlo
  1388. {\EQEDplain
  1389. {12}
  1390. }
  1391. {\EQEDplain
  1392. {a}
  1393. }
  1394. \EQEDplain
  1395. {xy}
  1396. \EQEDplain
  1397. {+}
  1398. \EQEDlo
  1399. {\EQEDplain
  1400. {23}
  1401. }
  1402. {\EQEDplain
  1403. {2a}
  1404. }
  1405. \EQEDplain
  1406. {yz}
  1407. \EQEDplain
  1408. {+2}
  1409. \EQEDlo
  1410. {\EQEDplain
  1411. {31}
  1412. }
  1413. {\EQEDplain
  1414. {a}
  1415. }
  1416. \EQEDplain
  1417. {zx}
  1418. \EQEDplain
  1419. {+}
  1420. \EQEDplain
  1421. {2}
  1422. \EQEDlo
  1423. {\EQEDplain
  1424. {14}
  1425. }
  1426. {\EQEDplain
  1427. {a}
  1428. }
  1429. \EQEDplain
  1430. {x+2}
  1431. \EQEDlo
  1432. {\EQEDplain
  1433. {24}
  1434. }
  1435. {\EQEDplain
  1436. {a}
  1437. }
  1438. \EQEDplain
  1439. {y+2}
  1440. \EQEDlo
  1441. {\EQEDplain
  1442. {34}
  1443. }
  1444. {\EQEDplain
  1445. {a}
  1446. }
  1447. \EQEDplain
  1448. {z+}
  1449. \EQEDlo
  1450. {\EQEDplain
  1451. {44}
  1452. }
  1453. {\EQEDplain
  1454. {a}
  1455. }
  1456. \EQEDplain
  1457. {=0}
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  1459. r≤wnanie stycznej
  1460. \EQEDplain
  1461. {Y-y=}
  1462. \frac
  1463. {\EQEDplain
  1464. {dy}
  1465. }
  1466. {\EQEDplain
  1467. {dx}
  1468. }
  1469. \EQEDbrackets
  1470. {\left(}
  1471. {\EQEDplain
  1472. {X-x}
  1473. }
  1474. {\right)}
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  1476. r≤wnanie normalnej
  1477. \EQEDplain
  1478. {Y-y=-}
  1479. \frac
  1480. {\EQEDplain
  1481. {1}
  1482. }
  1483. {\frac
  1484. {\EQEDplain
  1485. {y}
  1486. }
  1487. {\EQEDplain
  1488. {dx}
  1489. }
  1490. }
  1491. \EQEDbrackets
  1492. {\left(}
  1493. {\EQEDplain
  1494. {X-x}
  1495. }
  1496. {\right)}
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  1498. pochodna
  1499. \EQEDhi
  1500. {\EQEDplain
  1501. {,}
  1502. }
  1503. {\EQEDplain
  1504. {f}
  1505. }
  1506. \EQEDbrackets
  1507. {\left(}
  1508. {\EQEDplain
  1509. {x}
  1510. }
  1511. {\right)}
  1512. \EQEDplain
  1513. {=}
  1514. \EQEDlim
  1515. {\lim}
  1516. {\Delta \EQEDplain
  1517. {x}
  1518. \rightarrow \EQEDplain
  1519. {0}
  1520. }
  1521. {\frac
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  1523. {f}
  1524. \EQEDbrackets
  1525. {\left(}
  1526. {\EQEDplain
  1527. {x+}
  1528. \Delta \EQEDplain
  1529. {x}
  1530. }
  1531. {\right)}
  1532. \EQEDplain
  1533. {-f}
  1534. \EQEDbrackets
  1535. {\left(}
  1536. {\EQEDplain
  1537. {x}
  1538. }
  1539. {\right)}
  1540. }
  1541. {\Delta \EQEDplain
  1542. {x}
  1543. }
  1544. }
  1545. @
  1546. pochodna cz╣stkowa
  1547. \frac
  1548. {\delta \EQEDplain
  1549. {u}
  1550. }
  1551. {\delta \EQEDplain
  1552. {x}
  1553. }
  1554. \EQEDplain
  1555. {=}
  1556. \EQEDlim
  1557. {\lim}
  1558. {\Delta \EQEDplain
  1559. {x}
  1560. \rightarrow \EQEDplain
  1561. {0}
  1562. }
  1563. {\frac
  1564. {\EQEDplain
  1565. {f}
  1566. \EQEDbrackets
  1567. {\left(}
  1568. {\EQEDplain
  1569. {x+}
  1570. \Delta \EQEDplain
  1571. {x}
  1572. \EQEDplain
  1573. {,y,z,...,t}
  1574. }
  1575. {\right)}
  1576. \EQEDplain
  1577. {-f}
  1578. \EQEDbrackets
  1579. {\left(}
  1580. {\EQEDplain
  1581. {x,y,z,...,t}
  1582. }
  1583. {\right)}
  1584. }
  1585. {\Delta \EQEDplain
  1586. {x}
  1587. }
  1588. }
  1589. @
  1590. r≤┐niczka zupe│na
  1591. \EQEDplain
  1592. {du=}
  1593. \frac
  1594. {\delta \EQEDplain
  1595. {u}
  1596. }
  1597. {\delta \EQEDplain
  1598. {x}
  1599. }
  1600. \EQEDplain
  1601. {dx+}
  1602. \frac
  1603. {\delta \EQEDplain
  1604. {u}
  1605. }
  1606. {\delta \EQEDplain
  1607. {y}
  1608. }
  1609. \EQEDplain
  1610. {dy+...+}
  1611. \frac
  1612. {\delta \EQEDplain
  1613. {u}
  1614. }
  1615. {\delta \EQEDplain
  1616. {t}
  1617. }
  1618. \EQEDplain
  1619. {dt}
  1620. @
  1621. wz≤r Leibnitza
  1622. \EQEDhi
  1623. {\EQEDplain
  1624. {n}
  1625. }
  1626. {\EQEDplain
  1627. {D}
  1628. }
  1629. \EQEDbrackets
  1630. {\left(}
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  1632. {uv}
  1633. }
  1634. {\right)}
  1635. \EQEDplain
  1636. {=u}
  1637. \EQEDhi
  1638. {\EQEDplain
  1639. {n}
  1640. }
  1641. {\EQEDplain
  1642. {D}
  1643. }
  1644. \EQEDplain
  1645. {v+}
  1646. \EQEDchoose
  1647. {\EQEDplain
  1648. {n}
  1649. }
  1650. {\EQEDplain
  1651. {1}
  1652. }
  1653. \EQEDplain
  1654. {Du}
  1655. \EQEDhi
  1656. {\EQEDplain
  1657. {n-1}
  1658. }
  1659. {\EQEDplain
  1660. {D}
  1661. }
  1662. \EQEDplain
  1663. {v+}
  1664. \EQEDchoose
  1665. {\EQEDplain
  1666. {n}
  1667. }
  1668. {\EQEDplain
  1669. {2}
  1670. }
  1671. \EQEDhi
  1672. {\EQEDplain
  1673. {2}
  1674. }
  1675. {\EQEDplain
  1676. {D}
  1677. }
  1678. \EQEDplain
  1679. {u}
  1680. \EQEDhi
  1681. {\EQEDplain
  1682. {n-2}
  1683. }
  1684. {\EQEDplain
  1685. {D}
  1686. }
  1687. \EQEDplain
  1688. {v+...+}
  1689. \EQEDchoose
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  1691. {n}
  1692. }
  1693. {\EQEDplain
  1694. {k}
  1695. }
  1696. \EQEDhi
  1697. {\EQEDplain
  1698. {k}
  1699. }
  1700. {\EQEDplain
  1701. {D}
  1702. }
  1703. \EQEDplain
  1704. {u}
  1705. \EQEDhi
  1706. {\EQEDplain
  1707. {n-k}
  1708. }
  1709. {\EQEDplain
  1710. {D}
  1711. }
  1712. \EQEDplain
  1713. {v+...+}
  1714. \EQEDhi
  1715. {\EQEDplain
  1716. {n}
  1717. }
  1718. {\EQEDplain
  1719. {D}
  1720. }
  1721. \EQEDplain
  1722. {uv}
  1723. @
  1724. wz≤r prostok╣t≤w
  1725. \EQEDint
  1726. {\EQEDplain
  1727. {a}
  1728. }
  1729. {\EQEDplain
  1730. {b}
  1731. }
  1732. {\EQEDplain
  1733. {ydx}
  1734. \approx \EQEDplain
  1735. {h}
  1736. \EQEDbrackets
  1737. {\left(}
  1738. {\EQEDlo
  1739. {\EQEDplain
  1740. {0}
  1741. }
  1742. {\EQEDplain
  1743. {y}
  1744. }
  1745. \EQEDplain
  1746. {+}
  1747. \EQEDlo
  1748. {\EQEDplain
  1749. {1}
  1750. }
  1751. {\EQEDplain
  1752. {y}
  1753. }
  1754. \EQEDplain
  1755. {+...+}
  1756. \EQEDlo
  1757. {\EQEDplain
  1758. {n-1}
  1759. }
  1760. {\EQEDplain
  1761. {y}
  1762. }
  1763. }
  1764. {\right)}
  1765. }
  1766. @
  1767. wz≤r trapez≤w
  1768. \EQEDint
  1769. {\EQEDplain
  1770. {a}
  1771. }
  1772. {\EQEDplain
  1773. {b}
  1774. }
  1775. {\EQEDplain
  1776. {ydx}
  1777. \approx \frac
  1778. {\EQEDplain
  1779. {1}
  1780. }
  1781. {\EQEDplain
  1782. {2}
  1783. }
  1784. \EQEDplain
  1785. {h}
  1786. \EQEDbrackets
  1787. {\left(}
  1788. {\EQEDlo
  1789. {\EQEDplain
  1790. {0}
  1791. }
  1792. {\EQEDplain
  1793. {y}
  1794. }
  1795. \EQEDplain
  1796. {+2}
  1797. \EQEDlo
  1798. {\EQEDplain
  1799. {1}
  1800. }
  1801. {\EQEDplain
  1802. {y}
  1803. }
  1804. \EQEDplain
  1805. {+2}
  1806. \EQEDlo
  1807. {\EQEDplain
  1808. {2}
  1809. }
  1810. {\EQEDplain
  1811. {y}
  1812. }
  1813. \EQEDplain
  1814. {+...+2}
  1815. \EQEDlo
  1816. {\EQEDplain
  1817. {n-1}
  1818. }
  1819. {\EQEDplain
  1820. {y}
  1821. }
  1822. \EQEDplain
  1823. {+}
  1824. \EQEDlo
  1825. {\EQEDplain
  1826. {n}
  1827. }
  1828. {\EQEDplain
  1829. {y}
  1830. }
  1831. }
  1832. {\right)}
  1833. }
  1834. @
  1835. wz≤r parabol (Simpsona)
  1836. \EQEDint
  1837. {\EQEDplain
  1838. {a}
  1839. }
  1840. {\EQEDplain
  1841. {b}
  1842. }
  1843. {\EQEDplain
  1844. {ydx}
  1845. \approx \frac
  1846. {\EQEDplain
  1847. {1}
  1848. }
  1849. {\EQEDplain
  1850. {3}
  1851. }
  1852. \EQEDplain
  1853. {h}
  1854. \EQEDbrackets
  1855. {\left(}
  1856. {\EQEDlo
  1857. {\EQEDplain
  1858. {0}
  1859. }
  1860. {\EQEDplain
  1861. {y}
  1862. }
  1863. \EQEDplain
  1864. {+4}
  1865. \EQEDlo
  1866. {\EQEDplain
  1867. {1}
  1868. }
  1869. {\EQEDplain
  1870. {y}
  1871. }
  1872. \EQEDplain
  1873. {+2}
  1874. \EQEDlo
  1875. {\EQEDplain
  1876. {2}
  1877. }
  1878. {\EQEDplain
  1879. {y}
  1880. }
  1881. \EQEDplain
  1882. {+4}
  1883. \EQEDlo
  1884. {\EQEDplain
  1885. {3}
  1886. }
  1887. {\EQEDplain
  1888. {y}
  1889. }
  1890. \EQEDplain
  1891. {+...+2}
  1892. \EQEDlo
  1893. {\EQEDplain
  1894. {n-2}
  1895. }
  1896. {\EQEDplain
  1897. {y}
  1898. }
  1899. \EQEDplain
  1900. {+4}
  1901. \EQEDlo
  1902. {\EQEDplain
  1903. {n-1}
  1904. }
  1905. {\EQEDplain
  1906. {y}
  1907. }
  1908. \EQEDplain
  1909. {+}
  1910. \EQEDlo
  1911. {\EQEDplain
  1912. {n}
  1913. }
  1914. {\EQEDplain
  1915. {y}
  1916. }
  1917. }
  1918. {\right)}
  1919. }
  1920. @
  1921. ca│ka niew│a£ciwa
  1922. \EQEDint
  1923. {\EQEDplain
  1924. {a}
  1925. }
  1926. {\infty }
  1927. {\EQEDplain
  1928. {f}
  1929. \EQEDbrackets
  1930. {\left(}
  1931. {\EQEDplain
  1932. {x}
  1933. }
  1934. {\right)}
  1935. \EQEDplain
  1936. {dx=}
  1937. \EQEDlim
  1938. {\lim}
  1939. {\EQEDplain
  1940. {B}
  1941. \rightarrow \infty }
  1942. {\EQEDint
  1943. {\EQEDplain
  1944. {a}
  1945. }
  1946. {\EQEDplain
  1947. {B}
  1948. }
  1949. {\EQEDplain
  1950. {f}
  1951. \EQEDbrackets
  1952. {\left(}
  1953. {\EQEDplain
  1954. {x}
  1955. }
  1956. {\right)}
  1957. \EQEDplain
  1958. {dx}
  1959. }
  1960. }
  1961. }
  1962. @
  1963. ca│ka krzywoliniowa
  1964. \EQEDint
  1965. {\EQEDplain
  1966. {K}
  1967. }
  1968. {\EQEDplain
  1969. {}
  1970. }
  1971. {\EQEDplain
  1972. {f}
  1973. \EQEDbrackets
  1974. {\left(}
  1975. {\EQEDplain
  1976. {x,y}
  1977. }
  1978. {\right)}
  1979. \EQEDplain
  1980. {ds=}
  1981. \EQEDlim
  1982. {\lim}
  1983. {\Delta \EQEDlo
  1984. {\EQEDplain
  1985. {i}
  1986. }
  1987. {\EQEDplain
  1988. {s}
  1989. }
  1990. \rightarrow \EQEDplain
  1991. {0,n}
  1992. \rightarrow \infty }
  1993. {\EQEDsum
  1994. {\EQEDplain
  1995. {i=1}
  1996. }
  1997. {\EQEDplain
  1998. {n}
  1999. }
  2000. {\EQEDplain
  2001. {f}
  2002. \EQEDbrackets
  2003. {\left(}
  2004. {\EQEDlo
  2005. {\EQEDplain
  2006. {i}
  2007. }
  2008. {\xi }
  2009. \EQEDplain
  2010. {,}
  2011. \EQEDlo
  2012. {\EQEDplain
  2013. {i}
  2014. }
  2015. {\eta }
  2016. }
  2017. {\right)}
  2018. \Delta \EQEDlo
  2019. {\EQEDplain
  2020. {i-1}
  2021. }
  2022. {\EQEDplain
  2023. {s}
  2024. }
  2025. }
  2026. }
  2027. }
  2028. @
  2029. ca│ka podw≤jna
  2030. \EQEDint
  2031. {\EQEDplain
  2032. {S}
  2033. }
  2034. {\EQEDplain
  2035. {}
  2036. }
  2037. {\EQEDplain
  2038. {f}
  2039. \EQEDbrackets
  2040. {\left(}
  2041. {\EQEDplain
  2042. {x,y}
  2043. }
  2044. {\right)}
  2045. \EQEDplain
  2046. {dS=}
  2047. \EQEDlim
  2048. {\lim}
  2049. {\Delta \EQEDlo
  2050. {\EQEDplain
  2051. {i}
  2052. }
  2053. {\EQEDplain
  2054. {S}
  2055. }
  2056. \rightarrow \EQEDplain
  2057. {0,n}
  2058. \rightarrow \infty }
  2059. {\EQEDsum
  2060. {\EQEDplain
  2061. {i=1}
  2062. }
  2063. {\EQEDplain
  2064. {n}
  2065. }
  2066. {\EQEDplain
  2067. {f}
  2068. \EQEDbrackets
  2069. {\left(}
  2070. {\EQEDlo
  2071. {\EQEDplain
  2072. {i}
  2073. }
  2074. {\EQEDplain
  2075. {x}
  2076. }
  2077. \EQEDplain
  2078. {,}
  2079. \EQEDlo
  2080. {\EQEDplain
  2081. {i}
  2082. }
  2083. {\EQEDplain
  2084. {y}
  2085. }
  2086. }
  2087. {\right)}
  2088. \EQEDplain
  2089. {d}
  2090. \EQEDlo
  2091. {\EQEDplain
  2092. {i}
  2093. }
  2094. {\EQEDplain
  2095. {S}
  2096. }
  2097. }
  2098. }
  2099. }
  2100. @
  2101. ca│ka potr≤jna
  2102. \EQEDint
  2103. {\EQEDplain
  2104. {V}
  2105. }
  2106. {\EQEDplain
  2107. {}
  2108. }
  2109. {\EQEDplain
  2110. {f}
  2111. \EQEDbrackets
  2112. {\left(}
  2113. {\EQEDplain
  2114. {x,y,z}
  2115. }
  2116. {\right)}
  2117. \EQEDplain
  2118. {dV=}
  2119. \EQEDlim
  2120. {\lim}
  2121. {\Delta \EQEDlo
  2122. {\EQEDplain
  2123. {i}
  2124. }
  2125. {\EQEDplain
  2126. {V}
  2127. }
  2128. \rightarrow \EQEDplain
  2129. {0,n}
  2130. \rightarrow \infty }
  2131. {\EQEDsum
  2132. {\EQEDplain
  2133. {i=1}
  2134. }
  2135. {\EQEDplain
  2136. {n}
  2137. }
  2138. {\EQEDplain
  2139. {f}
  2140. \EQEDbrackets
  2141. {\left(}
  2142. {\EQEDlo
  2143. {\EQEDplain
  2144. {i}
  2145. }
  2146. {\EQEDplain
  2147. {x}
  2148. }
  2149. \EQEDplain
  2150. {,}
  2151. \EQEDlo
  2152. {\EQEDplain
  2153. {i}
  2154. }
  2155. {\EQEDplain
  2156. {y}
  2157. }
  2158. \EQEDplain
  2159. {,}
  2160. \EQEDlo
  2161. {\EQEDplain
  2162. {i}
  2163. }
  2164. {\EQEDplain
  2165. {z}
  2166. }
  2167. }
  2168. {\right)}
  2169. \EQEDplain
  2170. {d}
  2171. \EQEDlo
  2172. {\EQEDplain
  2173. {i}
  2174. }
  2175. {\EQEDplain
  2176. {V}
  2177. }
  2178. }
  2179. }
  2180. }
  2181. @
  2182. wz≤r Stokesa
  2183. \EQEDint
  2184. {\EQEDplain
  2185. {K}
  2186. }
  2187. {\EQEDplain
  2188. {}
  2189. }
  2190. {\EQEDplain
  2191. {Pdx+Qdy+Rdz=}
  2192. \EQEDint
  2193. {\EQEDplain
  2194. {S}
  2195. }
  2196. {\EQEDplain
  2197. {}
  2198. }
  2199. {\EQEDbrackets
  2200. {\left(}
  2201. {\frac
  2202. {\delta \EQEDplain
  2203. {Q}
  2204. }
  2205. {\delta \EQEDplain
  2206. {x}
  2207. }
  2208. \EQEDplain
  2209. {-}
  2210. \frac
  2211. {\delta \EQEDplain
  2212. {P}
  2213. }
  2214. {\delta \EQEDplain
  2215. {y}
  2216. }
  2217. }
  2218. {\right)}
  2219. \EQEDplain
  2220. {dxdy+}
  2221. \EQEDbrackets
  2222. {\left(}
  2223. {\frac
  2224. {\delta \EQEDplain
  2225. {R}
  2226. }
  2227. {\delta \EQEDplain
  2228. {y}
  2229. }
  2230. \EQEDplain
  2231. {}
  2232. \frac
  2233. {\delta \EQEDplain
  2234. {Q}
  2235. }
  2236. {\delta \EQEDplain
  2237. {z}
  2238. }
  2239. }
  2240. {\right)}
  2241. \EQEDplain
  2242. {dydz+}
  2243. \EQEDbrackets
  2244. {\left(}
  2245. {\frac
  2246. {\delta \EQEDplain
  2247. {P}
  2248. }
  2249. {\delta \EQEDplain
  2250. {z}
  2251. }
  2252. \EQEDplain
  2253. {}
  2254. \frac
  2255. {\delta \EQEDplain
  2256. {R}
  2257. }
  2258. {\delta \EQEDplain
  2259. {x}
  2260. }
  2261. }
  2262. {\right)}
  2263. \EQEDplain
  2264. {dzdx}
  2265. }
  2266. }
  2267. @
  2268. wz≤r Ostrogradzkiego-Gaussa
  2269. \EQEDint
  2270. {\EQEDplain
  2271. {}
  2272. }
  2273. {\EQEDplain
  2274. {}
  2275. }
  2276. {\EQEDint
  2277. {\EQEDplain
  2278. {V}
  2279. }
  2280. {\EQEDplain
  2281. {}
  2282. }
  2283. {\EQEDint
  2284. {\EQEDplain
  2285. {}
  2286. }
  2287. {\EQEDplain
  2288. {}
  2289. }
  2290. {\EQEDbrackets
  2291. {\left(}
  2292. {\frac
  2293. {\delta \EQEDplain
  2294. {P}
  2295. }
  2296. {\delta \EQEDplain
  2297. {x}
  2298. }
  2299. \EQEDplain
  2300. {+=}
  2301. \frac
  2302. {\delta \EQEDplain
  2303. {Q}
  2304. }
  2305. {\delta \EQEDplain
  2306. {y}
  2307. }
  2308. \EQEDplain
  2309. {+}
  2310. \frac
  2311. {\delta \EQEDplain
  2312. {R}
  2313. }
  2314. {\delta \EQEDplain
  2315. {z}
  2316. }
  2317. }
  2318. {\right)}
  2319. \EQEDplain
  2320. {dV=}
  2321. \EQEDint
  2322. {\EQEDplain
  2323. {}
  2324. }
  2325. {\EQEDplain
  2326. {}
  2327. }
  2328. {\EQEDint
  2329. {\EQEDplain
  2330. {S}
  2331. }
  2332. {\EQEDplain
  2333. {}
  2334. }
  2335. {\EQEDplain
  2336. {Pdydz+Qdzdx+Rdxdy}
  2337. }
  2338. }
  2339. }
  2340. }
  2341. }
  2342. @
  2343. r≤wnanie Bernoulliego
  2344. \EQEDhi
  2345. {\EQEDplain
  2346. {,}
  2347. }
  2348. {\EQEDplain
  2349. {y}
  2350. }
  2351. \EQEDplain
  2352. {+P}
  2353. \EQEDbrackets
  2354. {\left(}
  2355. {\EQEDplain
  2356. {x}
  2357. }
  2358. {\right)}
  2359. \EQEDplain
  2360. {y=Q}
  2361. \EQEDbrackets
  2362. {\left(}
  2363. {\EQEDplain
  2364. {x}
  2365. }
  2366. {\right)}
  2367. \EQEDhi
  2368. {\EQEDplain
  2369. {n}
  2370. }
  2371. {\EQEDplain
  2372. {y}
  2373. }
  2374. @
  2375. r≤wnanie Riccatiego
  2376. \EQEDhi
  2377. {\EQEDplain
  2378. {,}
  2379. }
  2380. {\EQEDplain
  2381. {y}
  2382. }
  2383. \EQEDplain
  2384. {=P}
  2385. \EQEDbrackets
  2386. {\left(}
  2387. {\EQEDplain
  2388. {x}
  2389. }
  2390. {\right)}
  2391. \EQEDhi
  2392. {\EQEDplain
  2393. {2}
  2394. }
  2395. {\EQEDplain
  2396. {y}
  2397. }
  2398. \EQEDplain
  2399. {+Q}
  2400. \EQEDbrackets
  2401. {\left(}
  2402. {\EQEDplain
  2403. {x}
  2404. }
  2405. {\right)}
  2406. \EQEDplain
  2407. {y+R}
  2408. \EQEDbrackets
  2409. {\left(}
  2410. {\EQEDplain
  2411. {x}
  2412. }
  2413. {\right)}
  2414. @
  2415. r≤wnanie Lagrange'a
  2416. \EQEDplain
  2417. {a}
  2418. \EQEDbrackets
  2419. {\left(}
  2420. {\EQEDhi
  2421. {\EQEDplain
  2422. {,}
  2423. }
  2424. {\EQEDplain
  2425. {y}
  2426. }
  2427. }
  2428. {\right)}
  2429. \EQEDplain
  2430. {x+b}
  2431. \EQEDbrackets
  2432. {\left(}
  2433. {\EQEDhi
  2434. {\EQEDplain
  2435. {,}
  2436. }
  2437. {\EQEDplain
  2438. {y}
  2439. }
  2440. }
  2441. {\right)}
  2442. \EQEDplain
  2443. {y+c}
  2444. \EQEDbrackets
  2445. {\left(}
  2446. {\EQEDhi
  2447. {\EQEDplain
  2448. {,}
  2449. }
  2450. {\EQEDplain
  2451. {y}
  2452. }
  2453. }
  2454. {\right)}
  2455. \EQEDplain
  2456. {=0}
  2457. @
  2458. wz≤r Eulera 1
  2459. \sin \EQEDplain
  2460. {z=}
  2461. \frac
  2462. {\EQEDhi
  2463. {\EQEDplain
  2464. {iz}
  2465. }
  2466. {\EQEDplain
  2467. {e}
  2468. }
  2469. \EQEDplain
  2470. {-}
  2471. \EQEDhi
  2472. {\EQEDplain
  2473. {-iz}
  2474. }
  2475. {\EQEDplain
  2476. {e}
  2477. }
  2478. }
  2479. {\EQEDplain
  2480. {2i}
  2481. }
  2482. @
  2483. wz≤r Eulera 2
  2484. \cos \EQEDplain
  2485. {z=}
  2486. \frac
  2487. {\EQEDhi
  2488. {\EQEDplain
  2489. {iz}
  2490. }
  2491. {\EQEDplain
  2492. {e}
  2493. }
  2494. \EQEDplain
  2495. {+}
  2496. \EQEDhi
  2497. {\EQEDplain
  2498. {-iz}
  2499. }
  2500. {\EQEDplain
  2501. {e}
  2502. }
  2503. }
  2504. {\EQEDplain
  2505. {2}
  2506. }
  2507. @
  2508. wz≤r ca│kowy Cauchy'ego
  2509. \EQEDhi
  2510. {\EQEDbrackets
  2511. {\left(}
  2512. {\EQEDplain
  2513. {n}
  2514. }
  2515. {\right)}
  2516. }
  2517. {\EQEDplain
  2518. {f}
  2519. }
  2520. \EQEDbrackets
  2521. {\left(}
  2522. {\EQEDplain
  2523. {z}
  2524. }
  2525. {\right)}
  2526. \EQEDplain
  2527. {=}
  2528. \frac
  2529. {\EQEDplain
  2530. {n!}
  2531. }
  2532. {\EQEDplain
  2533. {2}
  2534. \Pi \EQEDplain
  2535. {i}
  2536. }
  2537. \EQEDint
  2538. {\EQEDplain
  2539. {C}
  2540. }
  2541. {\EQEDplain
  2542. {}
  2543. }
  2544. {\frac
  2545. {\EQEDplain
  2546. {f}
  2547. \EQEDbrackets
  2548. {\left(}
  2549. {\zeta }
  2550. {\right)}
  2551. }
  2552. {\EQEDhi
  2553. {\EQEDplain
  2554. {n+1}
  2555. }
  2556. {\EQEDbrackets
  2557. {\left(}
  2558. {\zeta \EQEDplain
  2559. {-z}
  2560. }
  2561. {\right)}
  2562. }
  2563. }
  2564. \EQEDplain
  2565. {d}
  2566. \zeta }
  2567. @
  2568. residua
  2569. \EQEDplain
  2570. {resf}
  2571. \EQEDlo
  2572. {\EQEDplain
  2573. {z=a}
  2574. }
  2575. {\EQEDbrackets
  2576. {\left(}
  2577. {\EQEDplain
  2578. {z}
  2579. }
  2580. {\right)}
  2581. }
  2582. \EQEDplain
  2583. {=}
  2584. \frac
  2585. {\EQEDplain
  2586. {1}
  2587. }
  2588. {\EQEDplain
  2589. {2}
  2590. \Pi \EQEDplain
  2591. {i}
  2592. }
  2593. \EQEDint
  2594. {\EQEDplain
  2595. {C}
  2596. }
  2597. {\EQEDplain
  2598. {}
  2599. }
  2600. {\EQEDplain
  2601. {f}
  2602. \EQEDbrackets
  2603. {\left(}
  2604. {\zeta }
  2605. {\right)}
  2606. \EQEDplain
  2607. {d}
  2608. \zeta }
  2609. @
  2610. wz≤r Bayesa
  2611. \EQEDplain
  2612. {P}
  2613. \EQEDbrackets
  2614. {\left(}
  2615. {\EQEDlo
  2616. {\EQEDplain
  2617. {i}
  2618. }
  2619. {\EQEDplain
  2620. {A}
  2621. }
  2622. \EQEDbrackets
  2623. {\left\|}
  2624. {\EQEDplain
  2625. {B}
  2626. }
  2627. {\right.}
  2628. }
  2629. {\right)}
  2630. \EQEDplain
  2631. {=}
  2632. \frac
  2633. {\EQEDplain
  2634. {P}
  2635. \EQEDbrackets
  2636. {\left(}
  2637. {\EQEDplain
  2638. {B}
  2639. \EQEDbrackets
  2640. {\left\|}
  2641. {\EQEDlo
  2642. {\EQEDplain
  2643. {i}
  2644. }
  2645. {\EQEDplain
  2646. {A}
  2647. }
  2648. }
  2649. {\right.}
  2650. }
  2651. {\right)}
  2652. \EQEDplain
  2653. {P}
  2654. \EQEDbrackets
  2655. {\left(}
  2656. {\EQEDlo
  2657. {\EQEDplain
  2658. {i}
  2659. }
  2660. {\EQEDplain
  2661. {A}
  2662. }
  2663. }
  2664. {\right)}
  2665. }
  2666. {\EQEDsum
  2667. {\EQEDplain
  2668. {j=1}
  2669. }
  2670. {\EQEDplain
  2671. {N}
  2672. }
  2673. {\EQEDplain
  2674. {P}
  2675. \EQEDbrackets
  2676. {\left(}
  2677. {\EQEDplain
  2678. {B}
  2679. \EQEDbrackets
  2680. {\left\|}
  2681. {\EQEDlo
  2682. {\EQEDplain
  2683. {j}
  2684. }
  2685. {\EQEDplain
  2686. {A}
  2687. }
  2688. }
  2689. {\right.}
  2690. }
  2691. {\right)}
  2692. \EQEDplain
  2693. {P}
  2694. \EQEDbrackets
  2695. {\left(}
  2696. {\EQEDlo
  2697. {\EQEDplain
  2698. {j}
  2699. }
  2700. {\EQEDplain
  2701. {A}
  2702. }
  2703. }
  2704. {\right)}
  2705. }
  2706. }
  2707. @
  2708. rozk│ad Bernoulliego
  2709. \EQEDplain
  2710. {P}
  2711. \EQEDbrackets
  2712. {\left(}
  2713. {\EQEDplain
  2714. {A}
  2715. }
  2716. {\right)}
  2717. \EQEDplain
  2718. {=}
  2719. \EQEDchoose
  2720. {\EQEDplain
  2721. {n}
  2722. }
  2723. {\EQEDplain
  2724. {k}
  2725. }
  2726. \EQEDhi
  2727. {\EQEDplain
  2728. {k}
  2729. }
  2730. {\EQEDplain
  2731. {p}
  2732. }
  2733. \EQEDhi
  2734. {\EQEDplain
  2735. {n-k}
  2736. }
  2737. {\EQEDbrackets
  2738. {\left(}
  2739. {\EQEDplain
  2740. {1-p}
  2741. }
  2742. {\right)}
  2743. }
  2744. @
  2745.