# Defining a Covariant Tensor

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?

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.
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 $$P_\beta\,\frac{\partial x^\beta}{\partial x'^\alpha} =P'_\gamma\,\frac{\partial x'^\gamma}{\partial x'^\alpha}=P'_\alpha.$$