Defining a Covariant Tensor

Question

We usually define covariant tensors as those which rotate in the same sense w.r.t the basis vector. It is often also stated that if $P$ is covariant, $P'_{\alpha}=\frac{\partial{x^{\beta}}}{\partial{x'^{\alpha}}} P_{\beta}$. However I am not able to make sense of whether these two ways of defining are the same in spirit. Can I have some more insight into this?

Answer 1

I do not like the "rotation" formulation, it is misleadingly geometrical. What is true is that with a change of basis or more often of the coordinate functions (and their implied basis of the tangent space as dual to the direct basis of linear forms of the co-tangent space) ... with such a change of fundamentals the coordinate tuples of linear functionals or forms transforms the same way as the tuple of basis vectors.

Vectors are represented as $e_i\,x^i$ while linear forms are represented as $c_i\,\theta^i$. In differential geometry $e_i=\frac\partial{\partial x^i}$ and $\theta^i=dx^i$. So the distribution of vectorial and scalar objects is switched between vectors and covectors. Basis tuple and coordinate tuple are different classes of objects, not really comparable.

Then there is the difference of rotation of the basis tuple as in rotation of the vectors themselves and linear combination of the basis vectors taking the coefficients from an orthogonal matrix, $e'_i=Re_i=e_jR^j_{\;i}$. Of course this is all connected, but not always intuitively.

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In the end, the $P_\beta$ are the coordinates of the linear functional $P_\beta\, dx^\beta$. In a different coordinate system the same linear form has a representation $P'_\gamma\,dx'^\gamma$. Now insert the tangent vectors $\frac\partial{\partial x'^\alpha}$ to get $$ P_\beta\,\frac{\partial x^\beta}{\partial x'^\alpha} =P'_\gamma\,\frac{\partial x'^\gamma}{\partial x'^\alpha}=P'_\alpha. $$