- #1
mmpstudent
- 16
- 0
prove the identity $$\nabla\times(f\cdot\vec{v})=(\nabla f) \times \vec{v} + f \cdot \nabla \times \vec{v}$$
I can do the proof with normal vector calculus, but I am in a tensor intensive course and would like to do this with
einstein summation notation, but am having some trouble since I am brand new to this.
my attempt
left side
[tex] \epsilon_{ijk} \partial_{j} (\nabla f \cdot \vec{v})_{k}=\epsilon_{ijk}\partial_{j} f \vec{v}_{k} [/tex]
I didn't really know where to go from here so I moved onto the right side and expressed it in einstein notation
[tex]\epsilon_{ijk} (\nabla f)_{j} v_{k} + f \epsilon_{ijk} \partial_{j} v_{k} [/tex]
[tex]\epsilon_{ijk} \partial_{j} f v_{k} + f \epsilon_{ijk} \partial_{j} v_{k} [/tex]
which I don't see how I can rearrange this to get what is on the left. I see how it could be twice what I have on the left, but that obviously is incorrect. Did I do something wrong in expressing these? Do I have to express the right side with different sets of indicees?
I can do the proof with normal vector calculus, but I am in a tensor intensive course and would like to do this with
einstein summation notation, but am having some trouble since I am brand new to this.
my attempt
left side
[tex] \epsilon_{ijk} \partial_{j} (\nabla f \cdot \vec{v})_{k}=\epsilon_{ijk}\partial_{j} f \vec{v}_{k} [/tex]
I didn't really know where to go from here so I moved onto the right side and expressed it in einstein notation
[tex]\epsilon_{ijk} (\nabla f)_{j} v_{k} + f \epsilon_{ijk} \partial_{j} v_{k} [/tex]
[tex]\epsilon_{ijk} \partial_{j} f v_{k} + f \epsilon_{ijk} \partial_{j} v_{k} [/tex]
which I don't see how I can rearrange this to get what is on the left. I see how it could be twice what I have on the left, but that obviously is incorrect. Did I do something wrong in expressing these? Do I have to express the right side with different sets of indicees?
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