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In several high school treatments of geometry, the term "exterior angle theorem" has been applied to a different result, [1] namely the portion of Proposition 1.32 which states that the measure of an exterior angle of a triangle is equal to the sum of the measures of the remote interior angles. This result, which depends upon Euclid's parallel ...
The exterior derivative was first described in its current form by Élie Cartan in 1899. The resulting calculus, known as exterior calculus, allows for a natural, metric-independent generalization of Stokes' theorem, Gauss's theorem, and Green's theorem from vector calculus.
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 ...
The exterior algebra is the direct sum of the -th exterior powers of , and this makes the exterior algebra a graded algebra. The exterior algebra is universal in the sense that every equation that relates elements of V {\displaystyle V} in the exterior algebra is also valid in every associative algebra that contains V {\displaystyle V} and in ...
This allows expressing the fundamental theorem of calculus, the divergence theorem, Green's theorem, and Stokes' theorem as special cases of a single general result, the generalized Stokes theorem. Differential 1 -forms are naturally dual to vector fields on a differentiable manifold , and the pairing between vector fields and 1 -forms is ...
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 .
The development of analytical geometry and rigorous integral calculus in the 17th-19th centuries subsumed the method of exhaustion so that it is no longer explicitly used to solve problems. An important alternative approach was Cavalieri's principle , also termed the method of indivisibles which eventually evolved into the infinitesimal ...
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