Total differential equations

Question

if x = s cos alpha - t sin alpha , y = s sin alpha+t cos alpha Then show that (del^2 u/del x^2)+(del^2u/del y^2) = (del^2u/del s^2)+(del^2u/del t^2)

Answer 1

Use Chain-Rule formulas like

∂/∂x = (∂s/∂x)∂/∂s + (∂t/∂x)∂/∂t

that implies

∂²/∂x² = (∂²s/∂x²)∂/∂s + (∂²t/∂x²)∂/∂t + (∂s/∂x)²∂²/∂s² + (∂t/∂x)²∂²/∂t².