Undergrad How Are Base Vectors Defined in Covariance and Contravariance?

[Somali_Physicist]

I'm confused at how the base vectors are found for both.

e⁽¹⁾ = ∂r/∂u
e₍₁₎= ∇u
where r = xi + yj+zk
x = x(u,v,w)
y=y""
z=z""

cant understand why.

 

[Orodruin]

Exactly what is it that confuses you?

 

[Somali_Physicist]

Exactly what is it that confuses you?

How the covariant and contravariant base vectors are found.
For instance the covariant base vectors is found through:
e₁ = ∇u , why is this?

 

[Orodruin]

It is a definition.

You choose those base vectors to be normal vectors to the coordinate functions. The normal vector of a function is given by its gradient.

The other set of base vectors is chosen to be the tangent vectors of the coordinate lines.

 

[Somali_Physicist]

It is a definition.

You choose those base vectors to be normal vectors to the coordinate functions. The normal vector of a function is given by its gradient.

The other set of base vectors is chosen to be the tangent vectors of the coordinate lines.

not to sound slow but just to clarify.

e₁ ⋅ (equivalent coordinate function base) = 0
e¹⋅(equivalent coordinate function base) = 0
e₁ ⋅ (equivalent coordinate function base) = e¹⋅(equivalent coordinate function base)
e₁ ⋅(1/e¹) = 1
∂u/∂x ⋅ ∂u/∂x = ε^2 ( magnitude)
=ε^2cosθ , if orthogonal , = 1
Is that the rational they took to get use the vectors?