- #1

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**e**

_{i}=

**i**+

**j**+2v

**k**, how to find the dual basis vector if the above is a natural base?

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- Thread starter LSMOG
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- #1

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- #2

- 18,533

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- #3

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Thak you. They are

e

How to find their dual basis?

- #4

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What should the dual basis satisfy? How can you find vectors that satisfy these conditions?

- #5

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Mmm, my book does not explain. It just says, the dual basis are....I can't figure it outWhat should the dual basis satisfy? How can you find vectors that satisfy these conditions?

- #6

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I got. I use the ∇ operator to find the gradient of each natural baseWhat should the dual basis satisfy? How can you find vectors that satisfy these conditions?

- #7

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This is not helping much. Please do not refer to "my book" without stating which book you are using.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. (I got. I use the ∇ operator to find the gradient of each natural base

- #8

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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.

- #9

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I think the product is Kronecker delta

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.

- #10

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So how many relations does that give you? How many unknowns do you have?I think the product is Kronecker delta

Edit: And more importantly, what does it tell you about, for example, the relation between ##\vec e^1## and ##\vec e_2## and ##\vec e_3##?

- #11

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Yes.LSMOG said:I think the product is Kronecker delta

Dot product, and then Kronecker delta has relevant consecuences in the orientation and the length of the vectors of the dual basis.

You can compute the dual basis without the metric like I think Orodruin suggest you, based in geometry. But raise and lower indices, is also a routinary task with the metric.

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