The base manifold M must be a differentiable manifold ; by convention, it is either Minkowski space in particle physics applications, flat two-dimensional Euclidean space for condensed matter applications, or a Riemann surface, the worldsheet in string theory. The ∂ ∂ μ μ ϕ ϕ = ∂ ∂ ϕ ϕ / ∂ ∂ x μ μ is just the plain-old covariant derivative on the base spacetime manifold M . When M is flat, ∂ ∂ μ μ ϕ ϕ = ∇ ∇ ϕ ϕ is just the ordinary gradient of a scalar function (as ϕ ϕ is a scalar field, from the point of view of M itself.) In more precise language, ∂ ∂ μ μ ϕ ϕ is a section of the jet bundle of M × × Φ Φ .