Heat Equation Asymptotics of Elliptic Operators with Non-scalar Leading Symbol

Let M be a compact smoot Riemannian manifold without boundary and let D = ad₀δ₀ + bδ₁d₁ – E on the space of smooth sections of the cotangent bundle where a and b are positive constants and where E is an endomorphism. We use functorial methods and the pseudo-differential operator calculus to compute the quadratic term a₄(D) in the asymptotic expansion of the heat equation trace.