if-x-cos-u-y-sin-u-and-z-f-x-y-then-show-that-2-z-x-2-2-z-y-2-u-4-2-z-u-2-u-3-z-u-u-4-2-z-2-

Question Number 197317 by universe last updated on 13/Sep/23

if x = ((cos θ)/u) , y = ((sin θ)/u) and z = f(x,y) then show that (∂^2 z/∂x^2 ) + (∂^2 z/∂y^2 ) = u^4 (∂^2 z/∂u^2 ) + u^3 (∂z/∂u) + u^4 (∂^2 z/∂θ^2 )

$$\:\mathrm{if}\:\:\:\mathrm{x}\:\:=\:\:\frac{\mathrm{cos}\:\theta}{\mathrm{u}}\:\:,\:\mathrm{y}\:\:=\:\frac{\mathrm{sin}\:\theta}{\mathrm{u}}\:\:{and}\:\mathrm{z}\:\:=\:\:\mathrm{f}\left(\mathrm{x},\mathrm{y}\right) \\ $$$$\mathrm{then}\:\mathrm{show}\:\mathrm{that}\: \\ $$$$\:\:\:\frac{\partial^{\mathrm{2}} \mathrm{z}}{\partial\mathrm{x}^{\mathrm{2}} }\:+\:\frac{\partial^{\mathrm{2}} \mathrm{z}}{\partial\mathrm{y}^{\mathrm{2}} }\:=\:\mathrm{u}^{\mathrm{4}} \:\frac{\partial^{\mathrm{2}} \mathrm{z}}{\partial\mathrm{u}^{\mathrm{2}} }\:+\:\mathrm{u}^{\mathrm{3}} \:\frac{\partial\mathrm{z}}{\partial\mathrm{u}}\:+\:\mathrm{u}^{\mathrm{4}} \:\frac{\partial^{\mathrm{2}} \mathrm{z}}{\partial\theta^{\mathrm{2}} } \\ $$

Facebook Tweet Pin

if-x-cos-u-y-sin-u-and-z-f-x-y-then-show-that-2-z-x-2-2-z-y-2-u-4-2-z-u-2-u-3-z-u-u-4-2-z-2-

Leave a Reply Cancel reply