Daily Archives: 2012/09/13

origin of differential forms in Euclidean-flat R^4


read carefully:

aid

 

errata:

\left\{\frac{\partial x}{\partial x}=1,\frac{\partial x}{\partial y }=0,\frac{\partial x}{\partial z}=0,\frac{\partial x}{\partial t}=0\right\}  \longrightarrow dx=[1,0,0,0] ={\rm{grad}}(x)
\left\{\frac{\partial y}{\partial x}=0,\frac{\partial y}{\partial y}=1,\frac{\partial y}{\partial z}=0,\frac{\partial y}{\partial t}=0\right\}  \longrightarrow dy=[0,1,0,0] ={\rm{grad}}(y)
\left\{\frac{\partial z}{\partial x}=0,\frac{\partial z}{\partial y }=0,\frac{\partial z}{\partial z}=1,\frac{\partial z}{\partial t}=0\right\}  \longrightarrow dz=[0,0,1,0] ={\rm{grad}}(z)
\left\{\frac{\partial t}{\partial x}=0,\frac{\partial t}{\partial y }=0,\frac{\partial t}{\partial z}=0,\frac{\partial t}{\partial t}=1\right\}  \longrightarrow dt=[0,0,0,1] ={\rm{grad}}(t)

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