Editing Exterior algebra (section)

=== Oriented volume in affine space ===
The natural setting for (oriented) <math>k</math>-dimensional volume and exterior algebra is [[affine space]]. This is also the intimate connection between exterior algebra and [[differential forms]], as to integrate we need a 'differential' object to measure infinitesimal volume. If <math> \mathbb{A}</math> is an affine space over the vector space {{tmath|V}}, and a ([[simplex]]) collection of ordered <math>k+1</math> points <math> A_0, A_1, ... , A_k</math>, we can define its oriented <math>k</math>-dimensional volume as the exterior product of vectors <math> A_0A_1\wedge A_0A_2\wedge \cdots\wedge A_0A_k ={}</math> <math>(-1)^jA_jA_0\wedge A_jA_1\wedge A_jA_2\wedge \cdots\wedge A_jA_k</math> (using concatenation <math>PQ</math> to mean the [[displacement vector]] from point <math>P</math> to <math>Q</math>); if the order of the points is changed, the oriented volume changes by a sign, according to the parity of the permutation. In {{tmath|n}}-dimensional space, the volume of any <math>n</math>-dimensional simplex is a scalar multiple of any other.

The sum of the <math>(k-1)</math>-dimensional oriented areas of the boundary simplexes of a {{tmath|k}}-dimensional simplex is zero, as for the sum of vectors around a triangle or the oriented triangles bounding the tetrahedron in the previous section.

The vector space structure on <math>{\textstyle\bigwedge}(V)</math> generalises addition of vectors in {{tmath|V}}: we have <math>(u_1 + u_2) \wedge v = u_1 \wedge v + u_2 \wedge v</math> and similarly a {{math|''k''}}-blade <math>v_1 \wedge \dots \wedge v_k</math> is linear in each factor.

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