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this article we generalize the classical gradient estimate for the minimal surface equation to higher codimension. We consider a vector-valued function $u : \Omega \subset \math{R}^n \to \math{R}^m$ that satisfies the minimal surface system, see equation (1.1) in §1. The graph of u is then a minimal submanifold of $\math{R}^{n+m}$. We prove an a priori gradient bound under the assumption that the Jacobian of $ du: \math{R}^n \to \math{R}^m$ on any two dimensional subspace of $\mtah{R}^n$ is less than or equal to one. This assumption is automatically satisfied when $du$ is of rank one and thus the estimate covers the case when m=1, i.e., the original minimal surface equation. This is applied to Bernstein type theorems for minimal submanifolds of higher codimension.

minimal surface system