Find Tangent Vector to Curve in 2D Cartesian Coordinates

[Apashanka]

In 2-D Cartesian coordinate system let's there exist a scaler field Φ(x₁,x₂) ,now we want to find how Φ changes with a curve which is described by the parameter(arc length) s
dΦ/ds=(∂Φ/∂xⁱ)dxⁱ/ds
Can we say for Cartesian coordinate system that along the curve at any s dxⁱ always points in the same direction and hence dxⁱ/ds
But for curvilinear coordinate system ,dxⁱ points out in different direction for different s as the unit vector has no longer fixed direction for xⁱ axis,
But how is dxⁱ/ds pointing out tangentially to the curve at any s??

 

[Orodruin]

Components do not point. They are just numbers. Given a coordinate basis $dx^i/ds$ tells you the component of the tangent in the $x^i$ direction at that point. Whether the basis is different at a different point is completely irelevant.

 

[Apashanka]

Components do not point. They are just numbers. Given a coordinate basis $dx^i/ds$ tells you the component of the tangent in the $x^i$ direction at that point. Whether the basis is different at a different point is completely irelevant.

For 2-D Cartesian coordinate system if the slope at any s be tanθ, then dx¹/ds gives cosθ and dx²/ds gives sinθ.

Will you please explain how these cosθ and sinθ are related to components of tangent as you said??
It remains a confusion for me the '' components of tangent''

 

[Orodruin]

They are the components.

 

[Apashanka]

They are the components.

If ds=1 then they are the components along x^ith direction
that ds we are calling tangent at a point s having magnitude 1 ??

 

[Orodruin]

No, ds is not a vector.

 

[Apashanka]

No, ds is not a vector.

Sir what actually mean component of tangent??
What here '''tangent'' mean??
As you said earlier

 

[PeroK]

Sir what actually mean component of tangent??
What here '''tangent'' mean??
As you said earlier

It seems to me that you may be confusing the terms tangent to a curve and tangent space associated with a given coordinate system.

 

[Apashanka]

It seems to me that you may be confusing the terms tangent to a curve and tangent space associated with a given coordinate system.

I want to use the general term ''components of tangent'' dxⁱ/ds to 2-D Cartesian coordinate system (in post#3) and tried to understand what it actually is but I can't conclude what it actually is ??

 

[PeroK]

I want to use the general term ''components of tangent'' dxⁱ/ds to 2-D Cartesian coordinate system (in post#3) and tried to understand what it actually is but I can't conclude what it actually is ??

It's an inner product space, essentially defined by its dimension and the inner product, which is determined by the inner product of its basis vectors.

In the simplest terms it is a local coordinate system based At the point in question with two basis vectors pointing in directions determined by the original coordinate system at that point.

Two examples are a Cartesian system, where the tangent space is just a change of origin; and, polar coordinates where the tangent space has a rotated orthogonal basis.

 

[Apashanka]

It's an inner product space, essentially defined by its dimension and the inner product, which is determined by the inner product of its basis vectors.

Ok then in this 2-D Cartesian coordinate system if tanθ be the slope at any s ,then dx¹/ds and dx²/ds are coming to be cosθ and sinθ .
Then what to call these cosθ and sinθ in this inner product space

 

[PeroK]

Ok then in this 2-D Cartesian coordinate system if tanθ be the slope at any s ,then dx¹/ds and dx²/ds are coming to be cosθ and sinθ .
Then what to call these cosθ and sinθ in this inner product space

The tangent space has nothing to do with the tangent to the curve. The tangent space depends only on the point and the coordinate system. The curve is irrelevant. That's the tangent to the curve you are thinking about.