# How to find the dual basis vector for the following

• B
• LSMOG
In summary, the author is trying to explain the process of finding the dual basis vectors of a coordinate function using the gradient of the coordinate function. However, he does not mention what book he is using or what the conditions of a dual basis are. He also does not mention how many unknowns he has.

#### LSMOG

ei=i+j+2vk , how to find the dual basis vector if the above is a natural base?

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.

Orodruin 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.
Thak you. They are
e1=i+j+2vk
e2=i-j+2uk
e
3=k
How to find their dual basis?

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

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

Orodruin 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 base

LSMOG said:
Mmm, my book does not explain. It just says, the dual basis are...I can't figure it out
This is not helping much. Please do not refer to "my book" without stating which book you are using.

LSMOG said:
I got. I use the ∇ operator to find the gradient of each natural base
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.

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.

alejandromeira 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.
I think the product is Kronecker delta

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

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##?

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