User:MarSch

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The Invisible Pink Unicorn logo, which is used to represent atheism.
The Invisible Pink Unicorn logo, which is used to represent atheism.

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user:MarSch/math article structure

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If e is a bivector, then we have e^{ix} = -e^{xi}\; or equivalently e^{ij} = -e^{ji}\;. In quaternions much the same formula is true. We have \mathrm{e}^{\mathrm{i}\mathrm{j}} = \mathrm{e}^{-\mathrm{j}\mathrm{i}} = \mathrm{e}^\mathrm{k}\;, but \mathrm{e}^{\mathrm{i}x} = \mathrm{e}^{-x\mathrm{i}}\; holds only if x is purely quaternionic: x = j\mathrm{j} + k\mathrm{k}\; for real j and k.

Now let x, p, d and μ be real numbers and e a bivector on a compact 2-manifold R. Then we can calculate the integral \int_R expd\mu \in \mathbf{R} and it is a real number. On the other hand it is much easier to simply calculate \int_\mathbf{R} \exp \; \mathrm{d\mu} = \infty, because it is infinity. Unfortunately Greek letters are autoitalicized.

This stuff is all pretty simple. Let's get into some deeper waters.

Let i := ({}^0_\mathrm{i} {}^{-\mathrm{i}}_0)\;. Then i^2 = 1\; so we have \mathrm{e}^{ix} = \cosh x + i \sinh x\;. On the other hand \mathrm{e}^{\mathrm{i}x} = \cos x + \mathrm{i} \sin x\;.


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Mathematics, manifold, topological manifold, differentiable manifold, Laplace operator, scalar, scalar field, bijection, injection and surjection

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