Whakaoti i te varepsilon=pi/E(sigma_{1}-v(sigma_{2}+sigma_{3}))

$$\varepsilon = \frac { \pi } { E } ( \sigma _ { 1 } - v ( \sigma _ { 2 } + \sigma _ { 3 } ) )$$

Whakaoti mō E

$$\left\{\begin{matrix}E=\frac{\pi \left(\sigma _{1}-v\sigma _{3}-v\sigma _{2}\right)}{\epsilon }\text{, }&\sigma _{1}\neq v\left(\sigma _{2}+\sigma _{3}\right)\text{ and }\epsilon \neq 0\text{ and }\sigma _{1}\neq v\sigma _{2}+v\sigma _{3}\\E\neq 0\text{, }&\epsilon =0\text{ and }\sigma _{1}=v\left(\sigma _{2}+\sigma _{3}\right)\end{matrix}\right.$$

Ngā Upane mō te Whakaoti Whārite Rārangi

Tē taea kia ōrite te tāupe E ki 0 nā te kore tautuhi i te whakawehenga mā te kore. Whakareatia ngā taha e rua o te whārite ki te E.

$$\epsilon E=\pi \left(\sigma _{1}-v\left(\sigma _{2}+\sigma _{3}\right)\right)$$

Whakamahia te āhuatanga tohatoha hei whakarea te v ki te \sigma _{2}+\sigma _{3}.

$$\epsilon E=\pi \left(\sigma _{1}-\left(v\sigma _{2}+v\sigma _{3}\right)\right)$$

Hei kimi i te tauaro o v\sigma _{2}+v\sigma _{3}, kimihia te tauaro o ia taurangi.

$$\epsilon E=\pi \left(\sigma _{1}-v\sigma _{2}-v\sigma _{3}\right)$$

Whakamahia te āhuatanga tohatoha hei whakarea te \pi ki te \sigma _{1}-v\sigma _{2}-v\sigma _{3}.

$$\epsilon E=\pi \sigma _{1}-\pi v\sigma _{2}-\pi v\sigma _{3}$$

He hanga arowhānui tō te whārite.

$$\epsilon E=\pi \sigma _{1}-\pi v\sigma _{3}-\pi v\sigma _{2}$$

Whakawehea ngā taha e rua ki te \epsilon .

$$\frac{\epsilon E}{\epsilon }=\frac{\pi \left(\sigma _{1}-v\sigma _{3}-v\sigma _{2}\right)}{\epsilon }$$

Mā te whakawehe ki te \epsilon ka wetekia te whakareanga ki te \epsilon .

$$E=\frac{\pi \left(\sigma _{1}-v\sigma _{3}-v\sigma _{2}\right)}{\epsilon }$$

Tē taea kia ōrite te tāupe E ki 0.

$$E=\frac{\pi \left(\sigma _{1}-v\sigma _{3}-v\sigma _{2}\right)}{\epsilon }\text{, }E\neq 0$$

Whakaoti mō v

$$\left\{\begin{matrix}v=\frac{\pi \sigma _{1}-E\epsilon }{\pi \left(\sigma _{2}+\sigma _{3}\right)}\text{, }&E\neq 0\text{ and }\sigma _{2}\neq -\sigma _{3}\\v\in \mathrm{R}\text{, }&\sigma _{1}=\frac{E\epsilon }{\pi }\text{ and }\sigma _{2}=-\sigma _{3}\text{ and }E\neq 0\end{matrix}\right.$$

Ngā Upane mō te Whakaoti Whārite Rārangi

Whakareatia ngā taha e rua o te whārite ki te E.

$$\epsilon E=\pi \left(\sigma _{1}-v\left(\sigma _{2}+\sigma _{3}\right)\right)$$

Whakamahia te āhuatanga tohatoha hei whakarea te v ki te \sigma _{2}+\sigma _{3}.

$$\epsilon E=\pi \left(\sigma _{1}-\left(v\sigma _{2}+v\sigma _{3}\right)\right)$$

Hei kimi i te tauaro o v\sigma _{2}+v\sigma _{3}, kimihia te tauaro o ia taurangi.

$$\epsilon E=\pi \left(\sigma _{1}-v\sigma _{2}-v\sigma _{3}\right)$$

Whakamahia te āhuatanga tohatoha hei whakarea te \pi ki te \sigma _{1}-v\sigma _{2}-v\sigma _{3}.

$$\epsilon E=\pi \sigma _{1}-\pi v\sigma _{2}-\pi v\sigma _{3}$$

Whakawhitihia ngā taha kia puta ki te taha mauī ngā kīanga tau taurangi katoa.

$$\pi \sigma _{1}-\pi v\sigma _{2}-\pi v\sigma _{3}=\epsilon E$$

Tangohia te \pi \sigma _{1} mai i ngā taha e rua.

$$-\pi v\sigma _{2}-\pi v\sigma _{3}=\epsilon E-\pi \sigma _{1}$$

Whakaraupapatia anō ngā kīanga tau.

$$-\pi v\sigma _{2}-\pi v\sigma _{3}=E\epsilon -\pi \sigma _{1}$$

Pahekotia ngā kīanga tau katoa e whai ana i te v.

$$\left(-\pi \sigma _{2}-\pi \sigma _{3}\right)v=E\epsilon -\pi \sigma _{1}$$

Whakawehea ngā taha e rua ki te -\pi \sigma _{2}-\pi \sigma _{3}.

$$\frac{\left(-\pi \sigma _{2}-\pi \sigma _{3}\right)v}{-\pi \sigma _{2}-\pi \sigma _{3}}=\frac{E\epsilon -\pi \sigma _{1}}{-\pi \sigma _{2}-\pi \sigma _{3}}$$

Mā te whakawehe ki te -\pi \sigma _{2}-\pi \sigma _{3} ka wetekia te whakareanga ki te -\pi \sigma _{2}-\pi \sigma _{3}.

$$v=\frac{E\epsilon -\pi \sigma _{1}}{-\pi \sigma _{2}-\pi \sigma _{3}}$$

Whakawehe \epsilon E-\pi \sigma _{1} ki te -\pi \sigma _{2}-\pi \sigma _{3}.

$$v=\frac{E\epsilon -\pi \sigma _{1}}{-\pi \left(\sigma _{2}+\sigma _{3}\right)}$$