Multivariable Chain rule

Let F(x,y), m(a,b), and n(a,b) be differentiable functions.

If f(a,b) = F(m(a,b), n(a,b)), find ∂²f/∂a² and ∂²f/∂a∂b.

I'm having a bit of trouble finding these two. What's the proper way to take the second partial derivative with respect to the same variable, versus with respect to two?

By the Chain Rule,
∂f/∂a = ∂F/∂m ∂m/∂a + ∂F/∂n ∂n/∂a

∂²f/∂a²
= (∂/∂a) ∂f/∂a
= (∂/∂a) [∂F/∂m ∂m/∂a + ∂F/∂n ∂n/∂a]
= [(∂/∂a)(∂F/∂m) * ∂m/∂a + ∂F/∂m * ∂²m/∂a²] + [(∂/∂a)(∂F/∂n) * ∂n/∂a + ∂F/∂n * ∂²n/∂a²], product rule
= [(∂²F/∂m² ∂m/∂a + ∂²F/∂n∂m ∂n/∂a) * ∂m/∂a + ∂F/∂m * ∂²m/∂a²] +
[(∂²F/∂m∂n ∂m/∂a + ∂²F/∂n² ∂n/∂a) * ∂n/∂a + ∂F/∂n * ∂²n/∂a²], chain rule again!

= ∂²F/∂m² (∂m/∂a)² + 2 ∂²F/∂m∂n ∂m/∂a ∂n/∂a + ∂²F/∂n² (∂n/∂a)² + ∂F/∂m ∂²m/∂a² + ∂F/∂n ∂²n/∂a²]
-----------------------
Similarly for ∂²f/∂a∂b = ∂²f/∂b∂a:
∂²f/∂a∂b = ∂²f/∂b∂a
..............= (∂/∂b)(∂f/∂a)
..............= (∂/∂b)[∂F/∂m ∂m/∂a + ∂F/∂n ∂n/∂a].

See if you can imitate what I did above from here...

I hope this helps!