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Question Number 18243 by ajfour last updated on 17/Jul/17

Commented by ajfour last updated on 17/Jul/17

$$\mathrm{If}\mathrm{z}=\mid \mathrm{z}\mid ({\mathrm{e}}^{\mathrm{i}\mathit{\varphi }}\cos \theta +\mathrm{jsin}\theta )$$$${\mathrm{z}}_{\mathit{\varphi }+△\mathit{\varphi }}=\mid \mathrm{z}\mid ({\mathrm{e}}^{\mathrm{i}\mathit{\varphi }+\mathrm{i}△\mathit{\varphi }}\cos \theta +\mathrm{jsin}\theta )$$$${\mathrm{z}}^{\theta +△\theta }=\mid \mathrm{z}\mid [{\mathrm{e}}^{\mathrm{i}\mathit{\varphi }}\cos (\theta +△\theta )+\mathrm{jsin}(\theta +△\theta )]$$$$\mathrm{Then}\mathrm{prove}:$$$$\left(\mathrm{i}\right){\mathrm{z}}^{\theta +△\theta }=(\cos △\theta )\mathrm{z}+(\sin △\theta )\frac{\partial \mathrm{z}}{\partial \theta }$$$$\left(\mathrm{ii}\right){\mathrm{z}}_{\mathit{\varphi }+△\mathit{\varphi }}=(\cos △\mathit{\varphi })\mathrm{z}+$$$$(\sin △\mathit{\varphi })\frac{\partial \mathrm{z}}{\partial \mathit{\varphi }}+\mathrm{j}(1-\cos △\mathit{\varphi })\sin \theta$$$$\left(\mathrm{iii}\right){\mathrm{i}}^2=-1;\mathrm{find}{\mathrm{j}}^2$$$$\mathrm{Prove}{\mathrm{z}}_{\mathit{\varphi }+△\mathit{\varphi }}^{\theta +△\theta }=(\cos △\theta ){\mathrm{z}}_{\mathit{\varphi }+△\mathit{\varphi }}+$$$$(\sin △\theta ){\left(\frac{\partial \mathrm{z}}{\partial \theta }\right)}_{\mathit{\varphi }+△\mathit{\varphi }}.$$

Commented by ajfour last updated on 09/Jul/18

$$3\mathrm{D}-\mathrm{complex}numbers$$

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