Contravariant vs. Covariant vector

겨울방학때 텐서를 조금 이해하려고 마음 먹었었는데 생각보다 잘 안되었다.

하나 하나의 새로운 개념을 만나는 일이 생각보다 쉽게 넘어가지 못했기 때문이다.

책의 첫번째 chapter에서 아직도 헤매고 있으니, 이번 학기에 저 책을 다 보기는 쉽지 않을 듯 싶다. 책의 제목을 참고로 적어두면 Tensors, Differential Forms, and Variational Principles 이다.

처음 chapter를 읽다가 contravariant vector와 covariant vector의 정의를 보고 이해가 안되 한참 헤매다가 멋진 설명을 wikipedia에서 보고 여기 올려본다. 짤막하게 이해한 바에 따르면 contravariant vector는 우리가 기존에 이해하는 벡터의 의미(기하학적인 두점을 잇는 벡터)를 뜻하는 벡터인 것 같고, covariant vector는 조금 다른 의미의 벡터인 것 같다. 설명에 따르면 contravariant vector에 작용하여 좌표계의 선택에 무관한 scale invariance를 만들어 내는 선형함수(벡터에 작용하는 선형함수, 즉 f(v1+v2)=f(v1)+f(v2)인 f가 선형함수)들을 가리키는 듯하다. 넓은 의미에서 벡터는 벡터끼리의 합의 구조와 스칼라의 벡터에 대한 곱이 정의되는 집합을 벡터라 하므로 앞서 설명한 함수들의 집합도 벡터의 구조를 지니게 되고 그 원소들이 covariant vector가 되는 것 같다. 그리고 어떠한 벡터공간도 그에 대응하는 covariant vector공간을 만들수 있고, 이 covariant vector space가 dual space라는 구조를 만들어 내는 것 같다. 자세한 공부는 이번 학기에 계속해봐야 알게될 듯하다. 재밌는 구조인 듯하고, 수학의 세계가 조금은 더욱 넓어지는 계기가 될 듯하다. 자 이번학기도 힘좀 내볼까^^

밑의 설명에서 contravariant(역의 관계로 변하는 정도로 해석될 듯)와 covariant vector를 여행하는 중의 속도와 온도의 변화를 가지고 설명한 것은 정말 멋진 설명이다. 아무리 책을 봐도 이해가 안되던 부분을 한칼에 이해시키는 저런 능력은 어디서 오는 것일까? 암튼 강추다.

General relativity:Contravariant and Covariant Indices
From Wikibooks, the open-content textbooks collection
Jump to: navigation, search
< General relativity

Contents
[hide]
1 Rank and Dimension
2 Contravariant and Covariant Vectors
3 Scale Invariance
4 Vector Spaces and Basis Vectors



[edit]
Rank and Dimension
Now that we have talked about tensors, we need to figure out how to classify them. One import!ant characteristic is the rank of a tensor, which is the number of indices needed to specify the tensor. An ordinary matrix is a rank 2 tensor, a vector is a rank 1 tensor, and a scalar is rank 0. Tensors can, in general, have rank greater than 2, and often do.

Another characteristic of a tensor is the dimension of the tensor, which is the count of each index. For example, if we have a matrix consisting of 3 rows, with 4 elements in each row (columns), then the matrix is a tensor of dimension (3,4), or equivalently, dimension 12.

The import!ant thing about rank and dimension is that they are invariant to changes in the coordinate system. You can change the coordinate system all you want, and the rank and the dimensions don't change. This brings up the import!ant question of how tensors do change when you change the coordinate system. One thing we shall find when we look at the question is that in reality there are two different types of vectors.

[edit]
Contravariant and Covariant Vectors
Imagine that you are driving a car at 100 kilometers per hour toward the northwest. Lets call this vector v. Suddenly you realize that you are in a meter-ish mood and so we want to figure out how fast you are going using meters instead of kilometers. Quickly changing your coordinate system, you find that you are travelling 100 * 1000 meters per hour toward the northwest. No problem.

Now you are in the rolling countryside, and you notice the temperature changing. We then draw a map of how the temperature changes as we move across the countryside. We then travel along the path of steepest descent. We notice at a given point that the temperature is changing at 10 Celsius degrees per kilometer toward the southwest. Let's call this vector w. Again you go into a meter-ish mood. Doing a quick calculation you figure out that the gradient of the temperature change is downward at 10/1000 Celsius degrees per meter.

Did you notice something interesting?

Even though we are talking about two vectors we are treating them very differently when we change our coordinates. In the first case, the vector reacted to the coordinate change by a multiplication. In the second case, we did a division. The first case we were changing a vector that was distance per something, while in the second case, the vector was something per distance. These are two very different types of vectors.

The mathematical term for the first type of vector is called a contravariant vector. The second type of vector is called a covariant vector. Sometimes a covariant vector is called a one form.

Attempting a fuller explanation
It is easy to see why w is called covariant. Covariant simply means that the characteristic that w measures, change in temperature, increases in magnitude with an increase in displacement along the coordinate system. In other words, the further you travel from a fixed point, the more the temperature changes, or equivalently, change in termperature covaries with change in displacement.
Although it is a bit more difficult to see, v is called contravariant for precisely the opposite reason. Since v represents a velocity, or distance per unit time, we can think of v as the inverse of time per unit distance, meaning the amount of time that passes in traveling a certain fixed amount of distance. Time per unit distance is clearly covariant, because as you travel further and further from a fixed point, more and more time elapses. In other words, time covaries with displacement. Since velocity is the inverse of time per unit distance, than it follows that velocity must be contravariant.
The difference is also evident in the units of measure. The units of measure for v are meters per hour, whereas the units for w are degrees Celsius per meter. The coordinate system is position in space, measured in units of meters. So again, we see that the coordinate system appears in the numerator of v, which suggests that v is contravariant (with inverse time in this case), wherease the coordinate system appears in the denominator of w, which indicates that w is covariant (with change in temperature).

These are, of course, just fancy mathematical names. As we can see contravariant vectors and covariant vectors are very different from each other and we want to avoid confusing them with each other. To do this mathematicians have come up with a clever notation. The components of a contravariant vector are represented by superscripts, while the components of a covariant vector are represented by subscripts. So the components of vector v are v1 and v2 while the components of vector w are w1 and w2. Now we can draw two pictures that schematically illustrate the difference between contravariant vectors (figure 1) and covariant vectors (figure 2).

[edit]
Scale Invariance
Now that we have contravariant vectors and covariant vectors, we can do something very interesting and combine them. We have a contravariant vector that describes the direction and speed at which we are going. We have covariant vector that describes the rate and direction at which the temperature changes. If we combine them using the dot product


we get the rate at which the temperature changes, f, as we move in a certain direction, with units of degrees Celsius per second. The interesting thing about the units of f is that they do not include any units of distance, such as meters or kilometers. So now suppose we change the coordinate system from meters to kilometers. How does f change? It doesn't. We call this characteristic scale invariance, and we say that f is a scale invariant quantity. The value of f is invariant with changes in the scale of the coordinate system.

Now so far we have been treating w as if it were just an odd type of vector. But there is a another more powerful way of thinking about w. Look at what we just did. We took v, combined it with w and got something that doesn't change when you change the coordinate system. Now one way of thinking about it is to say that w is a function, that takes v and converts it into a scale invariant value, f.




[edit]
Vector Spaces and Basis Vectors
This section is a bit heavy on mathematical jargon. Perhaps someone can rewrite it so that it is a bit more accessible to the mathematically challenged (like me).
Let V be a vector space. Recall that the dual space V * of V is defined as the set of all linear functionals on V. Also recall that V * is a vector space in its own right. Vectors in V are called contravariant vectors, and vectors in V * are called covariant vectors (note: covariant vectors are sometimes called "1-forms").


Components of contravariant vectors are written with superscript ("upper") indices. If the set {} is a basis for V, then is written as the linear combination ( we are using Einstein summation notation, see the next section).


Before moving on to covariant vectors, we must define the notion of a dual basis. Remember that elements of V * are linear functionals on V. So we can "apply" covariant vectors to contravariant vectors to get a scalar. For example, if and , then returns a scalar. Now, the dual basis is defined as follows: if {} is a basis for V, then the dual basis is a basis {} for V * which satisfies (where is the Kronecker delta) for every μ and ν. Note that the dual basis for the canonical basis is usually written as {}, for reasons we will not go into in this section.


Now, the components of covariant vectors are written with subscript ("lower") indices. As {} is a basis for V * , we can write a covariant vector as .


We can now eval!uate any functional (covariant vector) applied to any vector (contravariant vector). If and , then by linearity . Finally, if we define , we see that .

Retrieved from "http://en.wikibooks.org/wiki/General_relativity:Contravariant_and_Covariant_Indices