geordief said:
from
https://en.wikipedia.org/wiki/Covariance_and_contravariance_of_vectors
"Examples of vectors with contravariant components include the position of an object relative to an observer"Can anyone help me with this at all ?(maybe just a guide as to how to follow it up)Also at the top of that Wiki page (first diagram),are those two orthogonal coordinate systems (the Tangent and the Dual ) aligned wrt each other as a direct function of the curve of the surface where they coincide? (there is a surface there isn't there?)
I think it's best not to worry at first about things being covariant or contravariant, but instead to get a clear idea of what a tangent vector and a dual vector (one-form, or covector) are.
There are two mathematical objects that are the most fundamental in studying manifold (space or spacetime).
- A parametrized path. This is a curve through the space, which can be mathematically represented as a continuous function ##\mathcal{P}(s)## from real numbers to points in space. The parameter ##s## is basically any real-valued quantity that increases smoothly along the path. For instance, if your space is 3D space, then a location as a function of time would be an example of a parametrized path. In that case ##s## would be time. But a curve drawn on a flat piece of paper is also a parametrized path, where the parameter is the distance traveled along the curve.
- A scalar field. This is a function ##\phi(\mathcal{P})## on the space that gives a scalar value (which could be a real number, or a complex number, or maybe something even more exotic, like quaternions, maybe) at each point in the space. So an example is the altitude (height above or below sea level) as a function of position in the 2D space that is the surface of the Earth. The requirement is that the value changes smoothly as you move from point to point.
Note: You don't need coordinates to make sense of these objects. A curve on a piece of paper is the same, regardless of whether you set up a coordinate system on the paper, and regardless of whether you use cartesian or polar coordinates. Altitude makes sense regardless of whether you describe your location using latitude and longitude.
Now, if you have a parametrized path ##\mathcal{P}(s)## and you also have a scalar field ##\phi(\mathcal{P})##, then you can combine them to get a function from reals to reals (or reals to complex numbers): ##F(s) = \phi(\mathcal{P}(s))##.
So where do vectors come in? Well, simply speaking, a tangent vector is a linear approximation to a parametrized path, and a one-form (or covector or dual vector) is a linear approximation to a scalar field. A tangent vector to a path ##\mathcal{P}(s)## is sometimes suggestively denoted as ##\frac{D\mathcal{P}}{Ds}##, and within a coordinate system has the components ##(\frac{D\mathcal{P}}{Ds})^j = \frac{dx^j}{ds}##, where ##x^j## is the value of coordinate number ##j## (viewed as a function of ##s## as one travels along the path ##\mathcal{P}(s)##).
The one-form ##d\phi##, which is a linear approximation to the scalar field ##\psi## has within a coordinate system the components ##(d\phi)_j = \frac{\partial \phi}{\partial x^j}##.
The fact that tangent vectors transform differently than one-forms or dual vectors follows from their definitions as linear approximations to paths or scalar functions.