Ngā Waehere me ā rātou Mahi
He ariā taketake ngā whārite i roto i te pāngarau me te ahupūngao, ā, he whānuitia te whakamahinga i roto i ngā momo mara o te pūtaiao me te hangarau. Ehara i te mea he mea nui noa te ariā o ngā whārite mō te mārama ki ngā wāhi āhuahanga engari he mea nui anō hoki ki te tātari raraunga, te arotau, tae atu ki te mātauranga horihori. Ka matapakihia e tēnei tuhinga te ariā o ngā whārite, ō rātou āhuatanga, me ngā mahi rerekē ka taea te mahi ki runga i a rātou.
Te Mārama ki ngā Wētere
I te nuinga o te wā, he rahinga te whārite e rua ōna āhuatanga matua: te rahi (te roa) me te ahunga. Kāore i rite ki ngā tauine, he rahi anake tō rātou, ka whakaratohia e ngā whārite he kōrero tāpiri mō te ahunga, ā, he tino whai hua i roto i ngā momo tono.
Whakaaturanga Wetereo
I roto i te āhuahanga, he maha ngā wā ka whakaatuhia he pere i te wāhi. Ko te pito o te pere e tohu ana i te ahunga o te pere, ko te roa o te pere e tohu ana i te rahi o te pere. I roto i te wāhi rua-ahu, he maha ngā wā ka tuhia he pere penei \( \mathbf{v} = (v_x, v_y) \), ko \( v_x \) me \( v_y \) ngā wāhanga o te pere i roto i ngā ahunga-x me te-y. I roto i te wāhi toru-ahu, ka tuhia he pere penei \( \mathbf{v} = (v_x, v_y, v_z) \).
Tohu Wēka
Ko ngā whākarite he tohu mātotoru pēnei i te \( \mathbf{v} \) he pere rānei kei runga ake pēnei i te \( \vec{v} \). I ngā horopaki, i ngā taiao rānei kua tuhia ā-ringa, kāore e taea te whakaatu mātotoru, ka taea te tohu i ngā whākarite mā te rārangi raro, mā te tohu whakarara rānei.
Ngā Momo o ngā Wetere
He maha ngā momo pūwero me mārama:
1. Kore Wēkere: He wēkere kāore he rahi, kāore hoki he ahunga motuhake, e tuhia ana ko \( \mathbf{0} \).
2. Wēka Waeine: He wēka he kotahi te rahi. Ka whakamahia ngā wēka waeine hei tohu i te ahunga me te kore e tohu i te rahi, ā, he maha ngā wā ka tohua ki tētahi pōtae pēnei i te \( \hat{i} \), \( \hat{j} \), me te \( \hat{k} \).
3. Wāhanga Tūnga: He wāhi e hono ana i te pūtake ki tētahi pūwāhi motuhake i te wāhi. I roto i ngā āhuatanga e rua, ko te wāhi mai i te pūwāhi \( A (x, y) \) ki te pūtake ko \( \mathbf{r} = (x, y) \).
4. Ngā Wētera Pou me te Rārangi: He maha ngā wā ka tuhia ngā wētera ki te āhua pou, ki te āhua rarangi rānei, inā koa i roto i te horopaki o te taurangi rārangi. Hei tauira, ko te wētera rarangi \( \mathbf{v} \) i roto i te āhua pou ko:
\[
\mathbf{v} = \begin{pmatrix} v_x \\ v_y \end{pmatrix}
\]
Ahakoa kei roto i te āhua rarangi ka tuhia penei \( \mathbf{v} = [v_x, v_y] \).
Ngā Mahi i runga i ngā Wetere
Muri iho, ka matapakihia e tātou ētahi mahi taketake ka taea te mahi i runga i ngā whārite:
Tāpiritanga Wetere
Ka taea te tāpiri i ngā whārite e rua mā te tāpiri i ō rāua wāhanga e rite ana. Hei tauira, mēnā he whārite e rua ā tātou \( \mathbf{u} = (u_x, u_y) \) me \( \mathbf{v} = (v_x, v_y) \), ko te tāpiritanga koia tēnei:
\[
\mathbf{u} + \mathbf{v} = (u_x + v_x, u_y + v_y)
\]
Te Tangohanga Wetereo
He rite tonu te tangohanga whārite ki te tāpiri engari mā te tango i ngā wāhanga e rite ana. Hei tauira, mēnā he whārite tā tātou \( \mathbf{u} = (u_x, u_y) \) me \( \mathbf{v} = (v_x, v_y) \), ko te tangohanga ko:
\[
\mathbf{u} – \mathbf{v} = (u_x – v_x, u_y – v_y)
\]
Whakareatanga Tauine
Ko te whakarea tauine te mahi whakarea i tētahi irahiko ki tētahi tauine. Mēnā he irahiko tā tātou \( \mathbf{v} = (v_x, v_y) \) me tētahi tauine \( k \), ko te hua o te whakarea ko:
\[
k \mathbf{v} = (k v_x, k v_y)
\]
Ka whakarerekē te whakarea tauine i te rahi o tētahi irahiko me te kore e whakarerekē i tōna ahunga.
Hua Ira
Ka puta he tauine ira i te hua ira o ngā whārite e rua, ā, ka tatauhia mā te tāpiri i ngā hua o ō rāua wāhanga e rite ana. Mēnā he whārite \( \mathbf{u} = (u_x, u_y) \) me \( \mathbf{v} = (v_x, v_y) \), ko tā rāua hua ira ko:
\[
\mathbf{u} \cdot \mathbf{v} = u_x v_x + u_y v_y
\]
Ka homai e te hua ira he mōhiohio e pā ana ki te nui o te whakarara o ngā whārite e rua.
Hua Whakawhiti
Ka tautuhia te hua whakawhiti i roto i te wāhi toru-ahu anake, ā, ka puta he whārite hou e poutū ana ki ngā whārite taketake e rua. Mena he whārite tā tātou \( \mathbf{u} = (u_x, u_y, u_z) \) me \( \mathbf{v} = (v_x, v_y, v_z) \), ko te hua whakawhiti ko:
\[
\mathbf{u} \times \mathbf{v} = \begin{vmatrix}
\hat{i} me \hat{j} me \hat{k} \\
u_x me u_y me u_z \\
v_x me v_y me v_z \\
\end{vmatrix}
\]
Ka puta i te hua whakawhiti he whārite he ahunga poutū ki te papa i hangaia e \( \mathbf{u} \) me \( \mathbf{v} \), me te rahi e ōrite ana ki te horahanga o te whakarara i hangaia e ngā whārite e rua.
Whakaōritenga Wetereo
Ko te whakataurite i te whārite ko te tukanga huri i tētahi whārite ki tētahi whārite kotahi e rite ana te ahunga ki te whārite taketake. Ka mahia te whakataurite mā te wehewehe i te whārite ki tōna rahi. Mena he whārite tā tātou \( \mathbf{v} = (v_x, v_y) \), ko te rahi o \( |\mathbf{v}| \) ko:
\[
|\mathbf{v}| = \sqrt{v_x^2 + v_y^2}
\]
Kātahi ko te wetere kotahi ko:
\[
\hat{v} = \frac{\mathbf{v}}{|\mathbf{v}|} = \left( \frac{v_x}{|\mathbf{v}|}, \frac{v_y}{|\mathbf{v}|} \right)
\]
Ngā Taupānga Wetereo
He whānuitia ngā whakamahinga o ngā wetereo me ā rātou mahi i te ao tūturu. Ko ētahi o ngā whakamahinga nui ko:
1. Ahupūngao: Ka whakamahia ngā whākatere hei tohu i ngā rahinga pēnei i te tere, te kaha, me te nekehanga. Ka whakamahia te tāpiri me te tango whākatere hei whakakotahi i ngā kaha, i ngā nekehanga rānei.
2. Whakairoiro Rorohiko: Ka whakamahia ngā wetere i roto i ngā panonitanga ā-ira, pērā i te hurihanga me te nekehanga o ngā mea. Ka whakamahia ngā hua ira me te whakawhiti hei whakatau i ngā tirohanga me te māramatanga.
3. Mātauranga Horihori: E whakamahia ana ngā whārite i roto i ngā whatunga io horihori, e whakaatuhia ana ngā taumaha me ngā whakaaro o te whatunga hei whārite.
4. Arotautanga: Ka whakamahia ngā whārite i roto i te tikanga whakahekenga rōnaki hei kimi i te iti rawa o tētahi mahi.
5. Te Tukatuka Tohu: Ka whakamahia ngā wetere hei tohu i ngā tohu i roto i te tātari me te tukatuka mamati, pērā i te panoni Fourier.
Whakamutunga
He mea nui te mahi a ngā whārite me ā rātou mahi i roto i te whānuitanga o ngā kaupapa pūtaiao. Mā te mārama ki ngā ariā taketake me ngā mahi rerekē i runga i ngā whārite, ka taea e tātou te whai taputapu kaha mō te tātari me te whakaoti rapanga i roto i te ahupūngao, te pāngarau, te hangarau, me te pūtaiao rorohiko. Mai i te tāpiri ngāwari ki ngā hua ira me te whakawhiti, he tono motuhake tā ia momo mahi hei āwhina i a tātou ki te mārama me te whakahaere i te ao e karapoti nei i a tātou.
