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The high school exterior angle theorem (HSEAT) says that the size of an exterior angle at a vertex of a triangle equals the sum of the sizes of the interior angles at the other two vertices of the triangle (remote interior angles). So, in the picture, the size of angle ACD equals the size of angle ABC plus the size of angle CAB.
Because the exterior derivative d has the property that d 2 = 0, it can be used as the differential (coboundary) to define de Rham cohomology on a manifold. The k -th de Rham cohomology (group) is the vector space of closed k -forms modulo the exact k -forms; as noted in the previous section, the Poincaré lemma states that these vector spaces ...
On the exterior algebra of differential forms over a smooth manifold, the exterior derivative is the unique linear map which satisfies a graded version of the Leibniz law and squares to zero. It is a grade 1 derivation on the exterior algebra. In R 3, the gradient, curl, and divergence are special cases of the exterior derivative. An intuitive ...
where {e 1 ∧ e 2, e 3 ∧ e 1, e 2 ∧ e 3} is the basis for the three-dimensional space ⋀ 2 (R 3). The coefficients above are the same as those in the usual definition of the cross product of vectors in three dimensions, the only difference being that the exterior product is not an ordinary vector, but instead is a bivector .
In mathematics, the discrete exterior calculus (DEC) is the extension of the exterior calculus to discrete spaces including graphs, finite element meshes, and lately also general polygonal meshes [1] (non-flat and non-convex). DEC methods have proved to be very powerful in improving and analyzing finite element methods: for instance, DEC-based ...
Then angle APB is the arithmetic mean of the angles AOB and COD. This is a direct consequence of the inscribed angle theorem and the exterior angle theorem. There are no cyclic quadrilaterals with rational area and with unequal rational sides in either arithmetic or geometric progression. [26]
For a polyhedron, the defect at a vertex equals 2π minus the sum of all the angles at the vertex (all the faces at the vertex are included). If a polyhedron is convex, then the defect of each vertex is always positive. If the sum of the angles exceeds a full turn, as occurs in some vertices of many non-convex polyhedra, then the defect is ...
denote the tangent bundle and cotangent bundle, respectively, of the smooth manifold . , denote the tangent spaces of , at the points , , respectively. denotes the cotangent space of at the point .
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