We need to prove $\int_{-C}\vec{F}d\vec{R}=-\int_{C} \vec{F}d\vec{R}$
Usual attempt in Textbook: Suppose $C:\vec{R(t)},a\leq t\leq b$. Then $-C:\vec{R(-t)},-b\leq t\leq -a$. I am able to verify the result. Can I define as $-C:\vec{R(a+b-t)},a\leq t\leq b?$
$\int_{-C}\vec{F}d\vec{R}= \int_{a...
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I wanted to try to prove the commutative property of addition before reading too much about it and "spoiling" things for myself. So I am curious how close I got.
First, some axioms (statements/relationships we take to be true):
$$\begin{align}
0 &\in \mathbb{N} \tag{$0$ is a natural number} \\
...
In mathematics, triality is a relationship among three vector spaces, analogous to the duality relation between dual vector spaces. Most commonly, it describes those special features of the Dynkin diagram D4 and the associated Lie group Spin(8), the double cover of 8-dimensional rotation group SO(8), arising because the group has an outer automorphism of order three. There is a geometrical version of triality, analogous to duality in projective geometry.
Of all simple Lie groups, Spin(8) has the most symmetrical Dynkin diagram, D4. The diagram has four nodes with one node located at the center...
10 hours ago, by Rick
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