Difference between revisions of "Exterior derivative"

Revision as of 02:31, January 25, 2010

The exterior derivative is a generalization of the concept of derivatives which allows for succinct statements of many of the most fundamental and powerful theorems in calculus, topology, and geometry.

For more on exterior derivatives, see the article on this topic at MathWorld.

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-{{math-a}}
+The '''exterior derivative''' is a generalization of the concept of derivatives which allows for succinct statements of many of the most fundamental and powerful theorems in calculus, topology, and geometry.
-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 <!--Don't wikilink this unless you know what you are doing-->tangent [[vector]] at a point <math>p</math>
+For more on exterior derivatives, see the article on this topic at [http://mathworld.wolfram.com/ExteriorDerivative.html MathWorld.]
-<math>
-df(v) = D_{v}(f)
-</math>
-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]]:
-<math>
-df = g_1 dx_1 +...+ g_n dx_n
-</math>
-In fact, since <math>dx_i(\frac{\partial}{\partial x_j}) = \delta^i_j</math>, we get that:
-<math>
-df = \frac{\partial f}{\partial x_1} dx_1 + ... + \frac{\partial f}{\partial x_n} dx_n
-</math>
-==Exterior derivative of differential forms==
-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
-<math>
-f_{i_1\cdots i_k} dx_{i_1}\wedge\cdots\wedge dx_{i_k}
-</math>
-then in this coordinate system, <math>d\omega</math> equals
-<math>
-df_{i_1\cdots i_k}\wedge dx_{i_1}\wedge\cdots\wedge dx_{i_k}
-</math>
-More generally, we define the differential <math>d\omega</math> by extending the above definition by [[linearity]].
-<!--"Cohomological" is the correct spelling!  Please don't keep breaking it-->
-==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 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>
-df = \frac{\partial f}{\partial x} dx + \frac{\partial f}{\partial y} dy
-</math>
-Thus
-<math>
-d^2 f = \frac{\partial^2 f}{\partial y\partial x}dy\wedge dx + \frac{\partial^2 f}{\partial x\partial y}dx\wedge dy
-</math>
-Since <math>dx\wedge dy = -dy\wedge dx</math>, the equality of mixed partials shows that <math>d^2 f = 0</math>.
 [[Category:Mathematics]]

Difference between revisions of "Exterior derivative"

Revision as of 02:31, January 25, 2010

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