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Question Number 84624 by subhankar10 last updated on 14/Mar/20

$$\mathrm{find}\mathrm{grad}{\mathrm{r}}^{\mathrm{m}}\mathrm{where}\mathrm{r}={\mathrm{x}}^2+{\mathrm{y}}^2+{\mathrm{z}}^2$$

Answered by TANMAY PANACEA last updated on 14/Mar/20

$$\overrightarrow{▽}=i\frac{\partial }{\partial x}+j\frac{\partial }{\partial y}+k\frac{\partial }{\partial z}isgrade$$$$tofind\overrightarrow{▽}{(x^2+y^2+z^2)}^m$$$$(i\frac{\partial }{\partial x}+j\frac{\partial }{\partial y}+k\frac{\partial }{\partial z}){(x^2+y^2+z^2)}^m$$$$=m{(x^2+y^2+z^2)}^{m-1}(i2x+j2y+k2z)$$$$={mr}^{m-1}\times 2\left(\overrightarrow{r}\right)=2{mr}^{m-1}.\overrightarrow{r}$$$$orm{(x^2+y^2+z^2)}^{m-1}\times 2(ix+jy+kz)$$

Commented by subhankar10 last updated on 15/Mar/20

$$\mathrm{thank}\mathrm{you}\mathrm{sir}$$

Commented by TANMAY PANACEA last updated on 15/Mar/20

$$mostwelcome$$

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