How can I use the transformations:

 x_1^* = \log(\frac{S_1}{X_1}) and  x_2^* = \log(\frac{S_2}{X_2})

and using other transformations analogous to those used in the one-dimensional case.

To show that the two-dimensional Black-Scholes equation can be reduced to the two-dimensional heat equation i.e.

<br />
\frac{\partial V}{\partial t} + \frac{1}{2}\sum_{i=1}^{2}\sum_{j=1}^{2}\sigma_i\si  gma_jS_iS_j\rho_{ij}\frac{\partial^2 V}{\partial^2 S_iS_j} + \sum_{i=1}^{2}(r - D_i) S_i \frac{\partial V}{\partial S_i} - rV = 0<br />

can be reduced to the two-dimensional heat equation i.e.

<br />
\frac{\partial u}{\partial \tau} = \frac{\partial^2 u}{\partial x_1^2} + \frac{\partial^2 u}{\partial x_2^2}<br />

I am really stuck with it, and don't know how to begin, any help is appreciated.

If any additional details are needed, feel free to ask.