Trouble understanding contravariant transformations for vectors

In summary, the conversation discusses the confusion surrounding the transformation of contravariant components under a change of coordinate system. The formula for transforming these components involves the partial derivatives of the new coordinates with respect to the old coordinates. However, it is important to note that in this transformation, the basis vectors also change, leading to a factor of 1/2 in the transformation of the components. This is what is meant by contravariant transformation.
  • #1
whisperzone
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TL;DR Summary
Don't understand equation of contravariant transformation for vectors
Hey, so I've been studying some math on my own and I'm really confused by this one bit. I understand what contravariant components of a vector are, but I don't understand the ways in which they transform under a change of coordinate system.

For instance, let's say we have two coordinate systems ##(x^1, x^2, ..., x^n)## and ##(\overline{x}^1, \overline{x}^2, ..., \overline{x}^n)## and a coordinate transformation relating the two systems: ##\overline{x}^j = \overline{x}^j (x^1, x^2, ..., x^n)##. Everything I've read says that

$$\overline{a}^j = \sum_{k = 1}^{n} \frac{\partial \overline{x}^j}{\partial x^k} a^k, \text{ where } j = 1, 2, ... n$$

but intuitively this makes no sense to me at all. For instance, consider a coordinate transformation ##(x^1, x^2, ..., x^n) \rightarrow (2x^1, 2x^2, ..., 2x^n)##. Then the partial derivative of ##\overline{x}^j## with respect to ##x^k## should be 2, right (or am I missing something here)? And if the partial derivative is 2, is that really a contravariant transformation, since whatever you're applying the transformation to would also get multiplied by 2?

Apologies if anything was too unclear, I've just been struggling with this for a while.
 
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  • #2
Contravariant refers to transforming the opposite way of the tangent basis vectors, which are defined as
$$
\vec E_i = \frac{\partial \vec x}{\partial x^i}.
$$
This means that
$$
\vec E_i ' = \frac{\partial \vec x}{\partial x^j} \frac{\partial x^j}{\partial x'^i}.
$$
In the case of your transformation, you therefore have ##\vec E_i' = \vec E_i / 2## so there is a factor of 1/2 in the transformation of the basis vectors. The components therefore (which are multiplied by 2) transform in the opposite fashion, i.e., contravariantly.
 

1. What are contravariant transformations for vectors?

Contravariant transformations for vectors are transformations that involve changing the coordinate system of a vector while keeping its direction and magnitude the same. This is done by multiplying the vector's components by a transformation matrix.

2. Why are contravariant transformations important in science?

Contravariant transformations are important because they allow scientists to work with vectors in different coordinate systems, making it easier to analyze and understand physical phenomena. It also helps to simplify calculations and equations in various fields such as physics, engineering, and mathematics.

3. How do contravariant transformations differ from covariant transformations?

Contravariant transformations involve changing the coordinate system of a vector, while covariant transformations involve changing the vector itself. In other words, contravariant transformations keep the direction and magnitude of the vector the same, while covariant transformations change these properties.

4. Can you give an example of a contravariant transformation for a vector?

One example of a contravariant transformation for a vector is converting from Cartesian coordinates to polar coordinates. In this transformation, the x and y components of the vector are multiplied by the transformation matrix, while the direction and magnitude of the vector remain the same.

5. How can I better understand contravariant transformations for vectors?

To better understand contravariant transformations for vectors, it is important to have a strong understanding of vector operations, coordinate systems, and transformation matrices. It may also be helpful to practice solving problems and working with vectors in different coordinate systems. Seeking clarification from a teacher or mentor can also aid in understanding this concept.

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