# Homework Help: Covariant & Contravariant Components

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1. Sep 17, 2015

### Biffinator87

1. The problem statement, all variables and given/known data
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.

2. Relevant equations

Transform equations, relationship to between covariant and contravariant components.

3. 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.

2. Sep 17, 2015

### Ray Vickson

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.

3. Sep 18, 2015

### 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.

4. Sep 18, 2015

### Staff: Mentor

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

5. Sep 18, 2015

### 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!

6. Sep 21, 2015

### 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.

7. Sep 21, 2015

### Staff: Mentor

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?
Yes, for part 1 of your problem.

Chet

8. Sep 21, 2015

### Biffinator87

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

#### Attached Files:

• ###### Solution to Problem 3.pdf
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9. Sep 22, 2015

### Staff: Mentor

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

10. Sep 22, 2015

### 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!

#### Attached Files:

• ###### Problem 3 Second Attempt.pdf
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11. Sep 22, 2015

### Staff: Mentor

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

12. Sep 27, 2015

### 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?

13. Sep 27, 2015

### Staff: Mentor

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

14. Sep 27, 2015

### Biffinator87

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

15. Oct 7, 2015

### Biffinator87

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

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16. Oct 7, 2015

### Staff: Mentor

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

17. Oct 8, 2015

### Biffinator87

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

18. Oct 8, 2015

### Staff: Mentor

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

19. Oct 8, 2015

### 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!

20. Oct 9, 2015

### Staff: Mentor

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

21. Oct 12, 2015

### Biffinator87

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

V=Vrir+Vθriθ+Vφrsinθiφ

or in terms of the components Vr=V(r)/1 ; Vθ=V(θ)/r ; Vφ=V(φ)/rsinθ

22. Oct 12, 2015

### Staff: Mentor

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.)

23. Oct 19, 2015

### Biffinator87

Would I have: ar= 1 / ir aθ= 1 / iθ*r aφ= 1 / iφ* rsinθ

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

24. Oct 20, 2015

### Staff: Mentor

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

25. Oct 20, 2015

### 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?