- #1

fab13

- 318

- 6

1) Firstly, in the context of a dot product with Einstein notation :

$$\text{d}(\vec{V}\cdot\vec{n} )=\text{d}(v_{i}\dfrac{\text{d}y^{i}}{\text{d}s})$$

with ##\vec{n}## representing the cosine directions vectors, ##v_{i}## the covariant components of ##\vec{V}## vector, ##y^{i}## the curvilinear coordinates and ##s## an affine parameter (

For example, in polar coordinates, How can I express the 2 cosine directions with the 2 vectors curvilinear basis ##\vec{e_{r}}## and ##\vec{e_{\theta}}## ?

On the following figure, I don't know exactly how to represent the vector ##\vec{n}## :

2) Secondly, if ##\dfrac{\text{d}y^{i}}{\text{d}s}## are cosine directions,

$$\text{d}(\vec{V}\cdot\vec{n} )=\text{d}(v^{i}\dfrac{\text{d}y^{i}}{\text{d}s})$$

with ##v^{i}## the contravariant components of ##\vec{V}##.

Indeed,both (##v_{i}## and ##v^{i}##) are related by :

$$v_{i}=v^{i}\,\cos(\theta_i)$$ with ##\cos(\theta_i)## the cosine direction for the angle ##\theta_i## between ##\vec{V}## and ##\vec{n}## vectors.

I hope you will understand these 2 questions.

Any help is welcome, Thanks

$$\text{d}(\vec{V}\cdot\vec{n} )=\text{d}(v_{i}\dfrac{\text{d}y^{i}}{\text{d}s})$$

with ##\vec{n}## representing the cosine directions vectors, ##v_{i}## the covariant components of ##\vec{V}## vector, ##y^{i}## the curvilinear coordinates and ##s## an affine parameter (

**it may be the length along the line, isn't it ?**)**Author says that ##\dfrac{\text{d}y^{i}}{\text{d}s}## (with ##y^{i}## curvilinear coordinates) represent the cosine directions and I would like to prove it : how to do it ?**For example, in polar coordinates, How can I express the 2 cosine directions with the 2 vectors curvilinear basis ##\vec{e_{r}}## and ##\vec{e_{\theta}}## ?

On the following figure, I don't know exactly how to represent the vector ##\vec{n}## :

2) Secondly, if ##\dfrac{\text{d}y^{i}}{\text{d}s}## are cosine directions,

**why does above scalar product is not equal rather to :**$$\text{d}(\vec{V}\cdot\vec{n} )=\text{d}(v^{i}\dfrac{\text{d}y^{i}}{\text{d}s})$$

with ##v^{i}## the contravariant components of ##\vec{V}##.

**?**Indeed,both (##v_{i}## and ##v^{i}##) are related by :

$$v_{i}=v^{i}\,\cos(\theta_i)$$ with ##\cos(\theta_i)## the cosine direction for the angle ##\theta_i## between ##\vec{V}## and ##\vec{n}## vectors.

I hope you will understand these 2 questions.

Any help is welcome, Thanks