Itô's formula for a $\mathcal{C}^2$ function of two variables F reads: \begin{align} F(X_t, Y_t) &= F(X_0, Y_0) + \int_0^t \frac{\partial F}{dx}(X_s, Y_s) \, dY_s + \int_0^t \frac{\partial F}{dy}(X_s, Y_s) dX_s \\ &+ \frac{1}{2}\int_0^t\frac{\partial^2 F}{dx^2}(X_s, Y_s) d\langle Y, Y\rangle_s + \frac{1}{2}\int_0^t \frac{\partial^2 F}{dy^2}(X_s, Y_s) d\langle X, X\rangle_s \\ &+ \frac{1}{2}\int_0^t \frac{\partial^2 F}{dx \,dy}(X_s, Y_s) d\langle X, Y\rangle_s \end{align}
However, it seems that when applied to 1) $X_t = t$ and $Y_t = B_t$ (Brownian motion in 1D) , or 2) $X_t$ local martingale and $Y_t = \langle X, X \rangle_t$, the last two integrals disappear in the formulas I've seen.
Why is that so ?