  Let $\{(X(t), Y(s)): t\in T, s\in S\}$ be an $\mathbb{R}^2$-valued, centered,
unit-variance smooth Gaussian vector field, where $T$ and $S$ are compact
  rectangles in $\mathbb{R}^N$. It is shown that, as $u\to \infty$, the joint
excursion probability $\mathbb{P} \{\sup_{t\in T} X(t) \geq u, \sup_{s\in S}
Y(s) \geq u \}$
  can be approximated by $\mathbb{E}\{\chi(A_u)\}$, the expected Euler
characteristic of the excursion set $A_u=\{(t,s)\in T\times S: X(t) \ge u, Y(s)
\ge u\}$,
  such that the error is super-exponentially small. This verifies the expected
Euler characteristic heuristic (cf. Taylor, Takemura and Alder (2005), Alder
and
  Taylor (2007)) for a large class of smooth Gaussian vector fields.
