# Tag Archives: differential forms in linear manifolds

## origin of differential forms in Euclidean-flat R^4

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