Ngā Wetere me ngā Pūnaha Taunga: Te Pūtake o te Pāngarau Hou
Pendahuluan
I roto i te pāngarau me te pūtaiao, ko ngā ariā o ngā whārite me ngā pūnaha taunga he tūāpapa nui e āhei ai te mārama me te whakaoti rapanga i roto i ngā mara pēnei i te ahupūngao, te hangarau, me te pūtaiao rorohiko. Ka arotakehia e tēnei tuhinga ngā ariā taketake o ngā whārite me ngā pūnaha taunga, me ō rātou tono i roto i ngā momo marautanga.
Ngā Wētere: Te Whakamāramatanga me te Whakarōpūtanga
I te whakarapopototanga, he mea pāngarau te whārite he rahi, he ahunga hoki. Mā tēnei ka rerekē ai i te tauine, he rahi anake tōna engari kāore he ahunga. I roto i te pāngarau, he maha ngā wā ka tohuhia ngā whārite mā te whakamahi i ngā pere i roto i te wāhi rua-ahu (2D) toru-ahu rānei (3D), ko te roa o te pere e tohu ana i te rahi, ā, ko te ahunga o te pere e tohu ana i te ahunga.
Ngā Momo o ngā Wetere
1. Wēka Tūnga: He wēka e whakaatu ana i te taunga o tētahi pūwāhi i te wāhi e pā ana ki te pūtake.
2. Te Tere o te Wā: E whakaatu ana i te tere o te panoni o te tūnga o tētahi mea i roto i te wā.
3. Te Wāhanga Āhua: He wēhanga e whakaatu ana i te rahi o te kaha me te ahunga e pā ai te kaha ki tētahi mea.
4. Wēka Wāhanga: He wēka he kotahi te roa o te kotahi wae e tohu ana i tētahi ahunga i te wāhi.
Te Tuhituhi me ngā Mahi o te Wetereo
Te Māngai
I roto i te wāhi ahu-rua, ka tuhia ngā whārite i te āhua \( \mathbf{v} = (v_1, v_2) \), ā, i roto i te wāhi ahu-toru, ka tuhia ēnei hei \( \mathbf{v} = (v_1, v_2, v_3) \). Hei tauira, ko te whārite \( \mathbf{v} = (3, 4) \) he wāhanga o te 3 i runga i te tuaka-x me te wāhanga o te 4 i runga i te tuaka-y.
Te Tāpiri me te Tangohanga o te Wetere
Ko te tāpiri i ngā whārite e rua ka mahia mā te tāpiri i ō rāua wāhanga. Hei tauira, mēnā \( \mathbf{u} = (u_1, u_2) \) me \( \mathbf{v} = (v_1, v_2) \), kātahi \( \mathbf{u} + \mathbf{v} = (u_1 + v_1, u_2 + v_2) \). Ka mahia te tangohanga i te ara kotahi: \( \mathbf{u} – \mathbf{v} = (u_1 – v_1, u_2 – v_2) \).
Whakareatanga Tauine
Ko te whakarea tauine ko te whakarea i tētahi tauine ki tētahi tau tūturu. Mēnā ko \( \mathbf{v} = (v_1, v_2) \) ā, he tauine tauine a k, ko \( k\mathbf{v} = (kv_1, kv_2) \).
Hua Ira me te Hua Whakawhiti
I roto i te wāhi ahu-toru, e rua ngā mahi nui e uru ana ngā whārite e rua: te hua ira me te hua whakawhiti.
Hua Ira: \( \mathbf{u} \cdot \mathbf{v} = u_1v_1 + u_2v_2 + u_3v_3 \). Ko te hua o te hua ira he tauine, ā, he ine i te putanga mahi o tētahi whārite i te ahunga kotahi ki tētahi atu.
Hua Whakawhiti: Ka puta he whārite hou e tūhonohono ana (poutū) ki ngā whārite taketake e rua. He uaua ake tōna whakaaturanga taurangi, engari he mea tino nui i roto i te ahupūngao, inā koa i te whakatau i te taipana, i te wā rānei o te kaha.
Pūnaha Taunga: Ariā me ngā Momo
He anga te pūnaha taunga hei whakatau i te tūranga o tētahi pūwāhi i te wāhi. He maha ngā momo pūnaha taunga, engari ko ngā mea tino noa ko ngā pūnaha taunga Cartesian, polar, me te cylindrical.
Pūnaha Taunga Kāretiana
Ko te pūnaha taunga Cartesian te pūnaha e whakamahia whānuitia ana, inā koa i roto i te pāngarau me te ahupūngao taketake. I roto i tēnei pūnaha, ka whakatauhia te tūranga o ia pūwāhi i te wāhi mā tōna tawhiti mai i ngā papa tohutoro poutū e rua, e toru rānei.
– 2D: I roto i te wāhi rua-ahu, ka whakatauhia ia pūwāhi \( (x, y) \) e tōna tawhiti mai i te tuaka-x me te tuaka-y.
– Ahu-3: I roto i te wāhi ahu-toru, ka whakamahia e te pūwāhi \( (x, y, z) \) tētahi tuaka-z tāpiri hei whakatau i te tūranga.
Ngā Pūnaha Taunga Pōro me Porotakaroa
Ngā Taunga Pōro: Ka whakamahia nuitia tēnei pūnaha i roto i ngā raruraru e pā ana ki te ōritetanga radial. I roto i ngā taunga pōro, ka tautuhia ia pūwāhi e tōna tawhiti radial (r) mai i te pūtake me te koki \( \theta \) i inehia mai i te tuaka-x pai.
\[ (r, \theta) \]
Ngā Taunga Porotaka: He huinga o ngā taunga Cartesian me ngā taunga polar, e whakamahi ana i te \( (r, \theta) \) hei tohu i te tūranga i roto i te papa me te z hei tohu i te teitei. He mea whakamahi noa i roto i ngā raruraru ahupūngao e pā ana ki ngā mea hurihuri pēnei i te rere o te wai i roto i ngā paipa.
Ngā Taupānga Wetereo me ngā Pūnaha Taunga
Ahupūngao
He mea nui ngā whākatere ki te ahupūngao. Ko te tere, te whakateretere, me te kaha he ariā ā-tinana katoa e tohuhia ana e ngā whākatere. Hei tauira, ka taea te whakaatu i te ture tuarua a Newton i roto i te āhua whākatere: \( \mathbf{F} = m\mathbf{a} \), ko \( \mathbf{F} \) te kaha, ko \( m \) te papatipu, ā, ko \( \mathbf{a} \) te whakateretere.
Te Hangarau me te Hangarau
I roto i ngā momo kaupapa hangarau, ka whakamahia te tātaritanga whārite hei whakahaere i ngā tātaitanga uaua. Hei tauira, ko te tātaritanga hanganga i roto i te hangarau ā-iwi ko te tāpiri i ngā whārite kaha e pā ana ki tētahi pūnaha hei whakatau i ngā ahotea me ngā whakarerekētanga.
Pūtaiao Rorohiko me ngā Whakairoiro
I roto i ngā whakairoiro rorohiko, ka whakamahia ngā pūnaha taunga hei tautuhi i te tūranga o ngā pika i runga i te mata. Ko ngā panonitanga wekita hoki te pūtake o te pakiwaituhi 3D, e neke ai, e hurihuri ai, e whakapohehetia ai ngā mea mā roto i ngā mahi wekita me ngā mahi matihiko.
Whakawhitinga Taunga
Ko te whakawhiti taunga ko te neke i tētahi pūwāhi mai i tētahi pūnaha taunga ki tētahi atu. He mea whai hua tēnei i roto i ngā āhuatanga maha, pērā i te whakarerekē i te pūtake i roto i te arapūrangi rārangi, te hurihuri rānei i tētahi mea i roto i ngā whakairoiro 3D.
Whakamutunga
He mea nui ngā pūnaha taunga me ngā pūnaha weketere ki te pāngarau me ngā momo marautanga pūtaiao. Mā te mārama ki ēnei ka taea te whakaoti i te whānuitanga o ngā raruraru rorohiko me te tātari uaua. Mai i te whakatau i te tūranga o ngā mea i te wāhi ki te whakaahua i ngā āhuatanga ā-tinana, he taputapu tino nui ēnei i roto i te pūnaha pāngarau o ēnei rā. Mā te ako hohonu ake, ka whānui haere tonu ngā tono o ngā pūnaha weketere me ngā pūnaha taunga, ka panaia ake ngā rohe o te mātauranga tangata.
