2.4 Higher order derivatives:

\frac{∂^{2}f}{∂x_{i_{1}} ∂x_{i_{2}}} = \frac{∂}{∂x_{i_{1}}} (\frac{∂}{∂x_{i_{2}}} (f))

Ex: Let f(x, y, z) = x^{2}ln(yz)

\frac{∂f}{∂x} =  \frac{∂f}{∂y} =  \frac{∂f}{∂z} =
\frac{∂^{2}f}{∂x^{2}} =  \frac{∂^{2}f}{∂x∂y} =  \frac{∂^{2}f}{∂z∂y} =
\frac{∂^{3}f}{∂x^{3}} =  \frac{∂^{3}f}{∂x^{2}∂z} =  \frac{∂^{3}f}{∂x∂y∂z} =

Defn: Let V be a nonempty open subset of R^{n}, f : V → R^{m}, p ∈ N.

i.) f is C^{p} on V is each partial derivative of order k ≤ p exists and is continuous on V.

ii.) f is C^{∞} on V if f is C^{p} on V for all p ∈ N (f is smooth).

Ex: g(x, y) = (x + y, x)

Cor 1.7 If f ∈ C^{r} on U, then \frac{∂^{k}f}{∂x_{i_{1}} ∂x_{i_{2}} ...∂x_{i_{k}}} = \frac{∂^{k}f}{∂x_{j_{1}} ∂x_{j_{2}} ...∂x_{j_{k}}} where (j_{1}, j_{2}, ..., j_{k}) is a permutation of (i_{1}, i_{2}, ..., i_{k})