I have been trying to evaluate the following double integral:
\(\displaystyle \displaystyle \frac{\partial}{\partial \theta_1 \partial \theta_2} \int_{\theta_1-\theta_2}^{\theta_1+\theta_2} \int_{\theta_1 -\theta_2}^{x} u(y,x) (x-y)^{n-2} dydx \)
where \(\displaystyle u(.)\) is a differentiable function.
I know that I can make use of Leibniz's rule but I think something goes wrong along the way.The rather long result I get is:
\(\displaystyle -u(\theta_1-\theta_2,\theta_1+\theta_2)*(2\theta_2)^n +u\prime_{\theta_2} (y,\theta_1+\theta_2) *(\theta_1+\theta_2-y)^{n-2}\)
. . . . .\(\displaystyle +(n-2)*u(y,\theta_1+\theta_2)*(\theta_1+\theta_2-y)^{n-3}-u(\theta_1-\theta_2,\theta_1+\theta_2)*(2\theta_2)^n \)
. . . . . . . . . .\(\displaystyle +u\prime _{\theta_2} (\theta_1-\theta_2,x)*(x+\theta_2-\theta_1)^{n-2}-(n-2)u(\theta_1-\theta_2,x)*(x+\theta_2-\theta_1)^{n-3} \)
Could somebody do it the long way so that I can see where I went wrong and how to correctly apply Leibniz rule?
Thank you in advance.
\(\displaystyle \displaystyle \frac{\partial}{\partial \theta_1 \partial \theta_2} \int_{\theta_1-\theta_2}^{\theta_1+\theta_2} \int_{\theta_1 -\theta_2}^{x} u(y,x) (x-y)^{n-2} dydx \)
where \(\displaystyle u(.)\) is a differentiable function.
I know that I can make use of Leibniz's rule but I think something goes wrong along the way.The rather long result I get is:
\(\displaystyle -u(\theta_1-\theta_2,\theta_1+\theta_2)*(2\theta_2)^n +u\prime_{\theta_2} (y,\theta_1+\theta_2) *(\theta_1+\theta_2-y)^{n-2}\)
. . . . .\(\displaystyle +(n-2)*u(y,\theta_1+\theta_2)*(\theta_1+\theta_2-y)^{n-3}-u(\theta_1-\theta_2,\theta_1+\theta_2)*(2\theta_2)^n \)
. . . . . . . . . .\(\displaystyle +u\prime _{\theta_2} (\theta_1-\theta_2,x)*(x+\theta_2-\theta_1)^{n-2}-(n-2)u(\theta_1-\theta_2,x)*(x+\theta_2-\theta_1)^{n-3} \)
Could somebody do it the long way so that I can see where I went wrong and how to correctly apply Leibniz rule?
Thank you in advance.
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