Question

Let u = f(r, s), r = x + at, s = y + bt, where x, y, t are independent variables and, a and b are constants. Show that ∂u/∂t = a (∂u/∂x) + b (∂u/∂y)

Answer 1

$$\dfrac{\partial u}{\partial t}=\dfrac{\partial u}{\partial r}\dfrac{\partial r}{\partial t}+\dfrac{\partial u}{\partial s}\dfrac{\partial s}{\partial t}$$

$$=a\big(\dfrac{\partial u}{\partial r}\big)+b\big(\dfrac{\partial u}{\partial s}\big)$$