(Solved): Analogously to the transformation from polar coordinates by the equations x=rcos,y=rsin, consid ...

Analogously to the transformation from polar coordinates by the equations $x=r\cos ?,y=r\sin ?$, consider the coordinates $(?,?),?>0$ defined by $x=??,y=\frac{1}{2}\left({?}^2?{?}^2\right).$ (A) Calculate, accurate to four decimal places, the $?,?$ coordinates of the point with cartesian coordinates $x=1.528,y=1.330$, and use that to calculate the laplacian of the function $f=x^2{y}^2$. (B) The laplacian in $?,?$ coordinates has the expression $\frac{1}{{?}^2+{?}^2}\left(\frac{{?}^{2}f}{?{?}^2}+\frac{{?}^{2}f}{?{?}^2}\right)$. Substitute $x=??$ and $y=\frac{1}{2}\left({?}^2?{?}^2\right)$ into $f=x^2{y}^2$, and, using that expression, calculate again the laplacian at the $?,?$ which you found in (A). (C) As was done in the lecture for polar coordinates, find expressions for $\frac{?f}{?x}$ and $\frac{?f}{?y}$ in terms $?,?,\frac{?f}{??}$, and $\frac{?f}{??}$. (D) As was done in the lecture for polar coordinates, derive the formula for the laplacian in $?,?$ coordinates, for an arbitrary function $f(?,?)$. This is a long calculation; indiscriminate expansion will result in expression swell. At one juncture there are 20 terms in my solution, but in the end the required laplacian has only two terms. You must provide a complete and organized solution which convinces that your work correctly arrives at the correct expression. Your solution will be spot checked. If you cannot provide a solution that can be easily verified then I will assign an grade of zero for this part.