has to be orientable, and the form has to be compactly supported in order to give a well-defined integral. The two points and form the boundary of the closed interval. More generally, Stokes' theorem applies to oriented manifolds with boundary. The boundary of is itself a manifold and inherits a natural orientation from that of . For example, the natural orientation of the interval gives an orientation of the two boundary points. Intuitively, inherits the opposite orientation as , as they are at opposite ends of the interval. So, "integrating" over two boundary points , is taking the difference . In even simpler terms, one can consider the points as boundaries of curves, that is as 0-dimensional boundaries of 1-dimensional manifolds. So, just as one can find the value of an integral () over a 1-dimensional manifold () by considering the anti-derivative () at the 0-dimensional boundaries (), one can generalize the fundamental theorem of calculus, with a few additional caveats, to deal with the value of integrals () over -dimensional manifolds () by considering the antiderivative () at the -dimensional boundaries () of the manifold. So the fundamental theorem reads: Formulation for smooth manifolds with boundary Let be an oriented smooth manifold with boundary of dimension and let be a smooth -differential form that is compactly supported on . First, suppose that is compactly supported in the domain of a single, oriented coordinate chart . In this case, we define the integral of over as i.e., via the pullback of to . More generally, the integral of over is defined as follows: Let be a partition of unity associated with a locally finite cover of (consistently oriented) coordinate charts, then define the integral where each term in the sum is evaluated by pulling back to as described above. This quantity is well-defined; that is, it does not depend on the choice of the coordinate charts, nor the partition of unity. The generalized Stokes theorem reads: Here is the exterior derivative, which is defined using the manifold structure only. The right-hand side is sometimes written as to stress the fact that the -manifold has no boundary. (This fact is also an implication of Stokes' theorem, since for a given smooth -dimensional manifold , application of the theorem twice gives for any -form , which implies that .) The right-hand side of the equation is often used to formulate integral laws;