If u = f(x,y), where x+y=2eθcosφ and x-y= 2ieθsinφ, prove that ∂2u/∂θ2 + ∂2u/∂φ2 = 4xy ∂²u/∂x∂y. Please answer it soon !!!!

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Differential Calculus> Partial differentiation...

If u = f(x,y), where x+y=2e^θcosφ and x-y= 2ie^θsinφ, prove that

∂²u/∂θ² + ∂²u/∂φ² = 4xy ∂²u/∂x∂y.

Please answer it soon !!!!

Mainak Chakraborty , 13 Years ago

Grade Upto college level

1 Answers

Jitender Singh

Last Activity: 10 Years ago

Ans:

$x+y = 2e^{\theta }cos\phi$ …..........(1)

$x-y = 2e^{\theta }sin\phi$ .............(2)

(1) + (2)

$2x = 2e^{\theta }cos\phi + 2ie^{\theta }sin\phi$

$x = e^{\theta }(cos\phi + isin\phi)$

$x = e^{\theta }e^{i\phi } = e^{\theta +i\phi }$

(1) – (2)

$2y = 2e^{\theta }cos\phi - 2ie^{\theta }sin\phi$

$y = e^{\theta }(cos\phi -isin\phi)$

$x = e^{\theta }e^{-i\phi } = e^{\theta -i\phi }$

Whatever the function ‘f’ is in terms of x & y, you can put the value of x, y in that & simply differentiate w.r.t to theta & phi.

Thanks & Regards

Jitender Singh

IIT Delhi

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Differential Calculus> Partial differentiation...

If u = f(x,y), where x+y=2eθcosφ and x-y= 2ieθsinφ, prove that ∂2u/∂θ2 + ∂2u/∂φ2 = 4xy ∂²u/∂x∂y. Please answer it soon !!!!

Mainak Chakraborty , 13 Years ago

Jitender Singh

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