I was trying to intuitively understand the covariant and contravariant bases for a coordinate system and I came across this image on Wikipedia:
Based on this image, it seems that the covariant basis vectors $\hat{e}_i$ are a basis of unit vectors that point in a direction tangent to the lines of the coordinate grid, while the contravariant basis vectors $\hat{e}^i$ are a basis of unit vectors that point in a direction normal to the lines of the coordinate grid.
Is this a correct interpretation of this image and what the covariant vs. contravariant basis vectors are? Even if it is correct, if there is a better way to intuitively understand covariant and contravariant vectors let me know.
Next, if that interpretation is correct, then would an expression like $\dfrac{\partial \hat{e}_i}{\partial x^j}$ mean the vector displacement of the covariant basis vector $\hat{e}_i$ if you move an infinitesimal distance along the coordinate grid lines in the direction $x^j$?
In other words, in general, does the symbol $x^j$ represent a coordinate grid line direction, the symbol $\hat{e}_i$ a covariant basis vector, and the symbol $\hat{e}^k$ a contravariant basis vector?
Finally, does the expression $\hat{e}^i \cdot \hat{e}_j = \delta^{i}_{j}$ mean that the covariant and contravariant basis vectors are defined so that the projection of the contravariant basis vectors onto the covariant basis vectors is only nonzero for corresponding covariant and contravariant basis vectors?