consider an interest rate derivative whose value $V$ depends on $n$ interest rates $r_1, \dots, r_n$. Hence $V$ is a function in $n$ variables $V(r_1, \dots, r_n)$. My question concerns the gamma $\gamma$ of this derivative with respect to parallel shifts.
Does it "always" hold that $$ \gamma(r_1, \dots, r_n) = \sum_{i = 1}^n \frac{\partial^2 V}{\partial r_i^2}(r_1, \dots, r_n). $$
To clarify the notation: I define the delta $\delta$ of the derivative with respect to parallel shifts as $$ \delta(r_1, \dots, r_n) = \lim_{h \rightarrow 0} \frac{V(r_1 + h, \dots, r_n + h) - V(r_1, \dots, r_n)}{h}, $$ and $\gamma$ is then defined as $$ \gamma(r_1, \dots, r_n) = \lim_{h \rightarrow 0} \frac{\delta(r_1 + h, \dots, r_n + h) - \delta(r_1, \dots, r_n)}{h}. $$
From a mathematical standpoint, I think the identity should not hold in general since mixed partial derivatives (i.e. $\frac{\partial^2 V}{\partial r_i r_j}$) should be considered as well. I also assume that $V$ is sufficiently smooth function such that all second partial derivatives exist and are continuous. However, literature seems to suggest that the identity holds in general. What do you guys think?