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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>

<math>

<math>

</math>

</math>

i.e., <math>df(v)</math> is the directional [[derivative]] of ''f'' in the direction ''v''.

i.e., <math>df(v)</math> is the [[directional derivative]] of ''f'' in the direction ''v''.

Note that if <math>x_1,...,x_n</math> are a local coordinate system for ''M'' at ''p'', then <math>dx_1,...,dx_n</math> define a local co-frame near ''p''. Thus, near ''p'', we may write the differential of ''f'' as a linear combination:

Note that if <math>x_1,...,x_n</math> are a local [[coordinate system]] for ''M'' at ''p'', then <math>dx_1,...,dx_n</math> define a local co-frame near ''p''. Thus, near ''p'', we may write the differential of ''f'' as a [[linear combination]]:

<math>

<math>

==Exterior derivative of differential forms==

==Exterior derivative of differential forms==

If <math>\omega</math> is a differential k-form (i.e., a smooth section of <math>\Lambda^k T^*M</math>), the exterior derivative <math>d\omega</math> is a differential (k+1)-form defined as follows:

If <math>\omega</math> is a [[differential form|differential k-form]] (i.e., a smooth section of <math>\Lambda^k T^*M</math>), the exterior derivative <math>d\omega</math> is a differential (k+1)-form defined as follows:

If we can write <math>\omega</math> in local coordinates as

If we can write <math>\omega</math> in local coordinates as

</math>

</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==

==Cohomological 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 derivatives. 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  

<math>

<math>

Since <math>dx\wedge dy = -dy\wedge dx</math>, the equality of mixed partials shows that <math>d^2 f = 0</math>.

Since <math>dx\wedge dy = -dy\wedge dx</math>, the equality of mixed partials shows that <math>d^2 f = 0</math>.