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in Evans while proving the mean value property for heat equation $(u_t-\Delta u=0)$ where $u\in C^2_1(U_T)$ they define $$\phi (r)=\frac{1}{r^n}\int\int_{E(r)}u(y,s)\frac{|y|^2}{s^2}dyds.$$ where $E(r)=E(0,0,r)$ and $E(x,t,r)=\{(y,s)\in R^{n+1}:\mu(x-y,t-s)\geq\frac{1}{r^n}\}$ is the heat ball and $\mu$ is the fundamental solution of the heat equation.

after that they applied change of variable to say that $$\phi (r)= \int\int_{E(1)}u(ry,r^2s)\frac{|y|^2}{s^2}dyds,$$ Can anyone please explain me how this change of variable take place. Any time of help would be appreciated. Thanks in advance

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  • $\begingroup$ Well the change of variables is right there in the arguments to $u$ - send $(y,s)$ to $(ry, r^2 s)$. Work out what the Jacobian matrix of this transformation looks like and apply the change of variables formula for integration. $\endgroup$ – Anthony Carapetis Mar 25 '17 at 14:16
  • $\begingroup$ ok i got it. thanks for the help $\endgroup$ – bunny Mar 26 '17 at 18:22

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