- #1
JonnyMaddox
- 74
- 1
Hi, 2-forms are defined as
du^{j} \wedge du^{k}(v,w) = v^{j}w^{k}-v^{k}w^{j} = \begin{vmatrix} du^{j}(v) & du^{j}(w) \\ du^{k}(v) & du^{k}(w) \end{vmatrix}
But what if I have two concret 1-forms in R^{3} like (2dx-3dy+dz)\wedge (dx+2dy-dz) and then I calculte (2dx-3dy+dz)\wedge (dx+2dy-dz)=-7dy \wedge dx +3dz \wedge dx - dy \wedge dz= 7 dx \wedge dy + 3 dz \wedge dx + dy \wedge dz (now a 2-form)
I know this is the same as the cross product between (2,-3,1)^{T} and (1,2,-1)^{T}
What has this to do with the determinant? If I calculate this for 1-forms in R^{2} like (2dx+4dx)\wedge (3dx+9dy) = -18 dx\wedge dy +12 dx \wedge dy = 6 dx \wedge dy which equals the determinant of the vectors (2,4) and (3,9). But this is not like 1-forms in R^{3}. And is the geometric interpretation right that 7 dx \wedge dy + 3 dz \wedge dx + dy \wedge dz means that 7 is the part of the area projected to the xy axis, 3 the part projecte to the zx axis and 1 the part projected onto the yz axis? Or is it a different coordinate system like dxdy, dzdx and dydz ?
Now actually these are all functions of vectors like v or w as in the definition of 2-forms. What happens if they come into this picture?
du^{j} \wedge du^{k}(v,w) = v^{j}w^{k}-v^{k}w^{j} = \begin{vmatrix} du^{j}(v) & du^{j}(w) \\ du^{k}(v) & du^{k}(w) \end{vmatrix}
But what if I have two concret 1-forms in R^{3} like (2dx-3dy+dz)\wedge (dx+2dy-dz) and then I calculte (2dx-3dy+dz)\wedge (dx+2dy-dz)=-7dy \wedge dx +3dz \wedge dx - dy \wedge dz= 7 dx \wedge dy + 3 dz \wedge dx + dy \wedge dz (now a 2-form)
I know this is the same as the cross product between (2,-3,1)^{T} and (1,2,-1)^{T}
What has this to do with the determinant? If I calculate this for 1-forms in R^{2} like (2dx+4dx)\wedge (3dx+9dy) = -18 dx\wedge dy +12 dx \wedge dy = 6 dx \wedge dy which equals the determinant of the vectors (2,4) and (3,9). But this is not like 1-forms in R^{3}. And is the geometric interpretation right that 7 dx \wedge dy + 3 dz \wedge dx + dy \wedge dz means that 7 is the part of the area projected to the xy axis, 3 the part projecte to the zx axis and 1 the part projected onto the yz axis? Or is it a different coordinate system like dxdy, dzdx and dydz ?
Now actually these are all functions of vectors like v or w as in the definition of 2-forms. What happens if they come into this picture?
