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e_{i}=i+j+2vk , how to find the dual basis vector if the above is a natural base?
Thak you. They areOrodruin said:It is not sufficient to know one of the basis vectors to deduce the corresponding dual basis. You must know the entire basis to find its dual.
Mmm, my book does not explain. It just says, the dual basis are...I can't figure it outOrodruin said:What should the dual basis satisfy? How can you find vectors that satisfy these conditions?
I got. I use the ∇ operator to find the gradient of each natural baseOrodruin said:What should the dual basis satisfy? How can you find vectors that satisfy these conditions?
This is not helping much. Please do not refer to "my book" without stating which book you are using.LSMOG said:Mmm, my book does not explain. It just says, the dual basis are...I can't figure it out
The gradient of a coordinate function gives you the corresponding dual basis vector in the same way as the tangent vector basis are the tangent vectors of the coordinate lines. However, if you are just given the tangent vector basis and not the actual coordinate functions, this is not a viable way forward since you do not have the coordinate functions to take the gradient of. (Do you have the coordinate functions? Please reproduce the entire problem exactly as stated.) However, there are certain relations between the dual basis vectors and the tangent vector basis that need to be satisfied and that you can use to deduce the dual basis from the tangent vector basis.LSMOG said:I got. I use the ∇ operator to find the gradient of each natural base
I think the product is Kronecker deltaalejandromeira said:Do you know the conditions of a dual basis?
Have you built the metric and the inverse of the metic?
what is the dot product between the vectors of the covariant basis and the vectors of the contravariant basis?
I believe that these questions are relevants for your project.
Ok, I'm not an expert, then if I say something wrong I hope someone corrects me.
So how many relations does that give you? How many unknowns do you have?LSMOG said:I think the product is Kronecker delta
Yes.LSMOG said:I think the product is Kronecker delta