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Let <math>f:M\rightarrow \mathbb{R}</math> be a [[smooth]] [[function]] on a [[manifold]]. The '''differential''' (or '''exterior derivative'''), <math>df</math>, is a [[covector field]] on ''M'' defined as follows: for ''v'' a [[tangent]] [[vector]] at a point <math>p</math>

Let <math>f:M\rightarrow \mathbb{R}</math> be a [[smooth]] [[function]] on a [[manifold]]. The '''differential''' (or '''exterior derivative'''), <math>df</math>, is a [[covector field]] on ''M'' defined as follows: for ''v'' a [[tangent]] [[vector]] at a point <math>p</math>

More generally, we define the differential <math>d\omega</math> by extending the above definition by [[linearity]].

More generally, we define the differential <math>d\omega</math> by extending the above definition by [[linearity]].

==Cohomological properties of the differential==

==Cosmological properties of the differential==

The [[operator]] ''d'' has the important property that <math>d\circ d = 0</math>. This essentially follows from the equality of mixed [[partial derivative]]s. The following simplest example illustrates the general proof: Let <math>f(x,y)</math> be a smooth function in two variables. Then  

The [[operator]] ''d'' has the important property that <math>d\circ d = 0</math>. This essentially follows from the equality of mixed [[partial derivative]]s. The following simplest example illustrates the general proof: Let <math>f(x,y)</math> be a smooth function in two variables. Then