Equality of directional derivatives.

Question

I am not sure how to prove: $$d_{u}d_{v}f=d_{v}d_{u}f$$ if $f$ is $C^{2}$ for any vectors $u$ and $v$. A theorem says $d_{v}f(y)=df(y)v$ but I am not sure how to apply it to solve this problem. Any help is appreciated.

Answer 1

We use Einstein summation notation. Let $u = u^{i} e_{i}$ and $v = v^{i} e_{i}$. Then we have

$$ d_u d_v f = u^i D_i (v^j D_j f) = u^i v^j D_{ij} f = v^j D_j (u^i D_i f) = d_v d_u f, $$

proving the claim. In a more familiar notation, let $u, v \in \mathbb{R}^{d}$ be $d$-dimensional vectors and $f : \mathbb{R}^d \to \mathbb{R}$ be of the class $C^2$. If we expand $u$ and $v$ with respect to the standard basis $\{ e_1, \cdots, e_n \}$

$$ u = \sum_{i = 1}^{d} u^i e_i \quad \text{and} \quad v = \sum_{i = 1}^{d} v^i e_i,$$

then we have

$$ d_u f = \sum_{i=1}^{d} u^i \frac{\partial f}{\partial x^i}.$$

Thus we have

$$ d_u d_v f = \sum_{i=1}^{d} u^i \frac{\partial}{\partial x^i} \left( \sum_{j=1}^{d} v^j \frac{\partial f}{\partial x^j} \right) = \sum_{i=1}^{d} \sum_{j=1}^{d} u^i v^j \frac{\partial^2 f}{\partial x^i \partial x^j}. $$

Since the Hessian of $f$ is symmetric, or in other words

$$ \frac{\partial^2 f}{\partial x^i \partial x^j} = \frac{\partial^2 f}{\partial x^j \partial x^i},$$

it follows that

$$ d_u d_v f = d_v d_u f$$

as desired.