Math

From dimensional analysis, volume of a n-ball is proportional to $R^n$ $$ V_n=C_n R^n $$Then from $\int_0^R S_n dr=V_n$ $$ \frac{dV_n}{dR}=V_n=n C_n R^{n-1} $$...

Group Theory for Physicists 2.2 Group and its Multiplication Table Symmetric Transformation: A transformation is called the symmetric transformation of a system if it preserves the system invariant. Definition of a Group: A group $G$ is a set of symmetric transformation with a defined multiplication of elements, and satisfies the following rules: ...

A visual introduction to differential forms and calculus on manifolds Chapter 1 Background material 1.1 Review of vector spaces vector spaces $u,v,w\in V$,$c,d \in \mathbb R$ $v+w=w+v$ $(u+v)+w=u+(v+w)$ $v+0=0+v$ $v+(-v)=0$ $1\cdot v=v$ $c\cdot (d\cdot v)=(c\cdot d)\cdot v$ $c\cdot(v+w)=c\cdot v+c\cdot w$ $(c+d)\cdot v=c\cdot v+d\cdot v$ ...