Covariant & Contravariant Components

[Biffinator87]

Homework Statement

This is really 3 questions in one but I figure it can be grouped together:

1. The vector A = i xy + j (2y-z²) + k xz. is in rectangular coordinates (bold i,j,k denote unit vectors). Transform the vector to spherical coordinates in the unit vector basis.

2. Transform the vector in Problem 1 into its covariant components in spherical coordinates.

3. Transform the vector in Problem 1 into its contravariant components in spherical coordinates.

Homework Equations

Transform equations, relationship to between covariant and contravariant components.

The Attempt at a Solution

I feel really bad because this is like the 3rd time I have posted on here in an attempt to get help but I am out of options with my college professor. Last time I asked for help his only response was STUDY HARDER! It is really hard to when he doesn't provide decent examples and solid theory.

Anyways, I need help understanding some of what he is talking about above. I know how to transform A into spherical coordinates. does the unit vector basis referred to mean in r,θ,φ? If so I know what he is asking then and can solve that. My REAL problem, is figuring out how to come up with covariant and contravariant components. Will they end up being vectors? I have checked wikipedia and several questions on this forum but just can't seem to make the connection.

ANY help at all is greatly appreciated.

 

[Ray Vickson]

Homework Statement

This is really 3 questions in one but I figure it can be grouped together:

1. The vector A = i xy + j (2y-z²) + k xz. is in rectangular coordinates (bold i,j,k denote unit vectors). Transform the vector to spherical coordinates in the unit vector basis.

2. Transform the vector in Problem 1 into its covariant components in spherical coordinates.

3. Transform the vector in Problem 1 into its contravariant components in spherical coordinates.

Homework Equations

Transform equations, relationship to between covariant and contravariant components.

The Attempt at a Solution

I feel really bad because this is like the 3rd time I have posted on here in an attempt to get help but I am out of options with my college professor. Last time I asked for help his only response was STUDY HARDER! It is really hard to when he doesn't provide decent examples and solid theory.

Anyways, I need help understanding some of what he is talking about above. I know how to transform A into spherical coordinates. does the unit vector basis referred to mean in r,θ,φ? If so I know what he is asking then and can solve that. My REAL problem, is figuring out how to come up with covariant and contravariant components. Will they end up being vectors? I have checked wikipedia and several questions on this forum but just can't seem to make the connection.

ANY help at all is greatly appreciated.

A good place to start is by reading on-line sources; for example, the article
https://en.wikipedia.org/wiki/Covariance_and_contravariance_of_vectors
offers some pretty nice explanations.

 

[Biffinator87]

Thanks for the reply. I have already read that page and still cannot fully grasp the concept. I am having a lot of difficulty relating these concepts together.

 

[Chestermiller]

I think I can help you.

Homework Statement

This is really 3 questions in one but I figure it can be grouped together:

1. The vector A = i xy + j (2y-z²) + k xz. is in rectangular coordinates (bold i,j,k denote unit vectors). Transform the vector to spherical coordinates in the unit vector basis.

2. Transform the vector in Problem 1 into its covariant components in spherical coordinates.

3. Transform the vector in Problem 1 into its contravariant components in spherical coordinates.

Homework Equations

Transform equations, relationship to between covariant and contravariant components.

The Attempt at a Solution

I feel really bad because this is like the 3rd time I have posted on here in an attempt to get help but I am out of options with my college professor. Last time I asked for help his only response was STUDY HARDER! It is really hard to when he doesn't provide decent examples and solid theory.

Anyways, I need help understanding some of what he is talking about above. I know how to transform A into spherical coordinates. does the unit vector basis referred to mean in r,θ,φ? If so I know what he is asking then and can solve that. My REAL problem, is figuring out how to come up with covariant and contravariant components. Will they end up being vectors? I have checked wikipedia and several questions on this forum but just can't seem to make the connection.

ANY help at all is greatly appreciated.

You say you can solve problem 1. Let's see what you got.

Are you familiar with the term "coordinate basis vectors." If so, please define what they mean.

Chet

 

[Biffinator87]

Ok I will get back to you in a little bit I am at work. I will workout that first part. Thanks for the replies!

 

[Biffinator87]

Ok been a crazy few days. Haven't had a chance to work the first part. The coordinate basis vectors refer to the unit vectors right? So in this case he wants the vector transformed in spherical coordinates with respect to the r,θ,φ right? I feel like every since I started this class my understanding of vectors has regressed.

 

[Chestermiller]

Ok been a crazy few days. Haven't had a chance to work the first part. The coordinate basis vectors refer to the unit vectors right?

No. The coordinate basis vectors are defined for spherical coordinates by:
$$d\vec{r}=\vec{a_r}dr+\vec{a_θ}dθ+\vec{a_φ}dφ$$
where $\vec{r}$ is a position vector from the origin. You need express your vector A in terms of the three coordinate basis vectors in order to determine its contravariant components. Now, how are the coordinate basis vectors related to the unit vectors in the r, θ, and φ directions?

So in this case he wants the vector transformed in spherical coordinates with respect to the r,θ,φ right?

Yes, for part 1 of your problem.

Chet

 

[Biffinator87]

Ok. I have attached my solution to problem 3. I ended up with some pretty long terms.

 

[Chestermiller]

Ok. I have attached my solution to problem 3. I ended up with some pretty long terms.

Your methodology is correct, but not your algebra. I found at least 1 error: in one of the j terms, you lost an exponent of 2 on r in going from the first equation to the second equation. There may be other algebraic errors. Please check it over.

Also, I can't read your handwriting with regard to distinguishing between the φ's and the θ's. I'm wondering whether you can.

I don't want to proceed further until you're sure that your algebra is correct.

Chet

 

[Biffinator87]

Ok I attempted to write bigger and used Φ instead of φ to hopefully make it more visibly different. Again, thank you for all the help so far!

 

[Chestermiller]

I haven't checked your "arithmetic," but that's not really necessary for what we want to do next. To get us started on the contravariant components, I need you to try to answer the question I asked in post #7.

Chet

 

[Biffinator87]

Ok. So is there some more transformation stuff I need to do or do I simply need to take what I have already transformed and group stuff together differently?

 

[Chestermiller]

Ok. So is there some more transformation stuff I need to do or do I simply need to take what I have already transformed and group stuff together differently?

For an orthogonal coordinate system like this, it is just a matter of grouping stuff differently (particularly in getting the contravariant components). The key to this lies in post #7.

Chet

 

[Biffinator87]

Ok cool. I will work this some more and try to have another post by tomorrow. Thanks!

 

[Biffinator87]

How's this? Sorry been busy trying to buy a house and with work.

 

[Chestermiller]

How's this? Sorry been busy trying to buy a house and with work.

Sorry, no. It's much simpler than that. In terms of the unit vectors:
$$\vec{dr}=\vec{i}_rdr+r\vec{i}_θdθ+r\sinθ\vec{i}_φdφ$$
Now you have two equations for dr, one in terms of the unit vectors and the other in terms of the coordinate basis vectors. So, how are the coordinate basis vectors related to the unit vectors?

Chet

 

[Biffinator87]

Would that be a relation of the unit vector dr=∂r/∂x*A_r?

 

[Chestermiller]

Would that be a relation of the unit vector dr=∂r/∂x*A_r?

No. The relationships would be
$$\vec{a}_r=\frac{\partial \vec{r}}{\partial r}=\vec{i}_r$$
$$\vec{a}_θ=\frac{\partial \vec{r}}{\partial θ}=r\vec{i}_θ$$
$$\vec{a}_φ=\frac{\partial \vec{r}}{\partial φ}=r\sinθ\vec{i}_φ$$
Can you see by comparing the equations in posts #16 and #7 where these equations come from?

Chet

 

[Biffinator87]

Oh yes I can now! Sorry I didn't realize I was comparing those two. But after you said it I realize I was being dumb!

 

[Chestermiller]

Oh yes I can now! Sorry I didn't realize I was comparing those two. But after you said it I realize I was being dumb!

Suppose I have an arbitrary vector $\vec{V}$. I can express it in component form in terms of the spherical coordinate unit vectors by writing:
$$\vec{V}=V_{(r)}\vec{i}_r+V_{(θ)}\vec{i}_θ+V_{(φ)}\vec{i}_φ$$
where the parentheses in the subscripts signify that I am talking about the components with respect to the unit vectors. I can also represent the exact same vector $\vec{V}$ in terms of the coordinate basis vectors by writing:
$$\vec{V}=V^r\vec{a_r}+V^θ\vec{a_θ}+V^φ\vec{a_φ}$$
The superscripted components in this equation are called "the contravariant components of the vector $\vec{V}$." Given the relationships between the unit vectors and the coordinate basis vectors that I presented in post #18, how are the contravariant components related to the components expressed in terms of the unit vectors?

Chet

 

[Biffinator87]

So by using what you have given me from the previous post I have:

V=V^ri_r+V^θri_θ+V^φrsinθi_φ

or in terms of the components V^r=V_(r)/1 ; V^θ=V_(θ)/r ; V^φ=V_(φ)/rsinθ

 

[Chestermiller]

So by using what you have given me from the previous post I have:

V=V^ri_r+V^θri_θ+V^φrsinθi_φ

or in terms of the components V^r=V_(r)/1 ; V^θ=V_(θ)/r ; V^φ=V_(φ)/rsinθ

Good job. These are the contravariant components of the vector.

Now for the covariant components. To get the covariant components, we need to introduce another set of basis vectors. These are called the "reciprocal basis vectors," and are defined as follows:
$$\vec{a}^r\centerdot \vec{a}_r=1$$
$$\vec{a}^θ\centerdot \vec{a}_θ=1$$
$$\vec{a}^φ\centerdot \vec{a}_φ=1$$
The dot product of $\vec{a}^r$ with the other two coordinate basis vectors is zero. The same goes for $\vec{a}^θ$ and $\vec{a}^φ$.

We are going to express the vector $\vec{V}$ in terms of the reciprocal basis vectors and the covariant (subscripted) components of $\vec{V}$ as follows:
$$\vec{V}=V_r\vec{a}^r+V_θ\vec{a}^θ+V_φ\vec{a}^φ$$

So tell me what the reciprocal basis vectors are in terms of the unit vectors and also in terms of the coordinate basis vectors.

(Note that the analysis we have done so far is strictly for an orthogonal coordinate system, in this case, spherical coordinates). For a non-orthogonal coordinate system, the relationships are similar, but a little more complicated.)

 

[Biffinator87]

Would I have: a^r= 1 / i_r a^θ= 1 / i_θ*r a^φ= 1 / i_φ* rsinθ

or in other words. Divide V_r,θ,φ by their scale factors?

 

[Chestermiller]

Would I have: a^r= 1 / i_r a^θ= 1 / i_θ*r a^φ= 1 / i_φ* rsinθ

or in other words. Divide V_r,θ,φ by their scale factors?

No. In vector analysis, we never have a vector appearing in the denominator of any equation (unless it's dotted there with something else). The three reciprocal basis vectors are:

$\vec{a}^r=\vec{i}_r$, $\vec{a}^θ=\frac{\vec{i}_θ}{r}$, and $\vec{a}^φ=\frac{\vec{i}_φ}{r\sinθ}$

Tell me if this makes sense.

Chet

 

[Biffinator87]

Yes it does. So now that I know what these relationships are I need to apply them to the transformation I solved for correct?

 

[Chestermiller]

Yes it does. So now that I know what these relationships are I need to apply them to the transformation I solved for correct?

Yes, but the transformation you solved for is peripheral to what we are trying to establish here. What we've tried to show is how the covariant and contravariant components of vectors are defined in terms of the coordinate basis vectors and the reciprocal basis vectors for our coordinate system. This is much more general than any specific problem. Also, please note that what we have done here does not handle all possible coordinate systems. It only applies to orthogonal coordinate systems. Although correct for orthogonal coordinate systems, it needs to be generalized a little further to handle non-orthogonal systems of coordinates (which you are likely to encounter in your further studies).

Chet

 

[Biffinator87]

Ok. That makes sense. So when I use the covariant and contravariant I don't directly use that transformation I just did?

 

[Chestermiller]

Ok. That makes sense. So when I use the covariant and contravariant I don't directly use that transformation I just did?

I don't quite understand this question. Please specify exactly which transformation you are talking about.

 

[Biffinator87]

Sorry, the original questions called for first transforming to spherical polar coordinates then finding the contravariant and covariant components of the vector.

 

[Chestermiller]

Sorry, the original questions called for first transforming to spherical polar coordinates then finding the contravariant and covariant components of the vector.

My motivation here was to show you the simplest method of getting the covariant and contravariant components of the vector for the special case of an orthogonal coordinate system (in this case spherical coordinates). I also wanted to introduce you to the coordinate basis vectors and the reciprocal basis vectors, so that you could have a better idea of how a vector (which is an entity independent of coordinate system) is expressed in component form in terms of (a) the contravariant components combined with the coordinate basis vectors or (b) the covariant components combined with the reciprocal basis vectors. I wanted you to have a feel for all this.

For more general coordinate systems that could possibly involve curvilinear coordinates that are non-orthogonal, the starting point is still the coordinate basis vectors and the reciprocal basis vectors. However, in these more advanced developments, the transformation equations for going directly from the components in one coordinate system to another have been worked out in advance for us, without going through the coordinate basis vectors as intermediates. Of course, the starting point of these developments is still the coordinate basis vectors.

With regard to your specific problem, you have now worked out all the equations for the covariant and contravariant components. Yes, all you have to do is combine these with your previous transformation in terms of the unit vectors.

Chet

 

[Biffinator87]

Ok do you mind if I try and solve those and then show them too you to see if I completely understand now?

 

[Chestermiller]

Ok do you mind if I try and solve those and then show them too you to see if I completely understand now?

No problem. But the hard part was over once you found the components in terms of the unit vectors. That was before we even started talking about covariant and contravariant.

Chet