if-u-f-x-y-where-x-rcos-y-r-sin-prove-u-x-2-u-y-2-u-r-2-1-r-u-2-

Tinku Tara June 4, 2023 Differentiation 0 Comments

Question Number 88015 by M±th+et£s last updated on 07/Apr/20

if u=f(x,y) where x=rcos(θ) , y=r sin(θ) prove ((∂u/∂x))^2 +((∂u/∂y))^2 =((∂u/∂r))^2 +(1/r)((∂u/∂θ))^2

$${if}\:{u}={f}\left({x},{y}\right)\:{where}\:{x}={rcos}\left(\theta\right)\:\:,\:{y}={r}\:{sin}\left(\theta\right) \\ $$$${prove}\: \\ $$$$\left(\frac{\partial{u}}{\partial{x}}\right)^{\mathrm{2}} +\left(\frac{\partial{u}}{\partial{y}}\right)^{\mathrm{2}} =\left(\frac{\partial{u}}{\partial{r}}\right)^{\mathrm{2}} +\frac{\mathrm{1}}{{r}}\left(\frac{\partial{u}}{\partial\theta}\right)^{\mathrm{2}} \\ $$

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