I'm having trouble understanding what this scalar field in spherical coordinates represents:
$\vec{a}(r, \theta, \phi) = 4\theta \hat{e_r} + \hat{e_{\theta}}$
Without converting the spherical basis vectors to cartesian, is it possible to visualize the vector field? How would you plot the vector at a specific point $P$ without transformation?
I understand that the spherical basis vectors change for different $(r, \theta, \phi)$, but they are still orthogonal for all $(r, \theta, \phi)$, and each basis vector points in the direction of an increasing coordinate. I also understand that you can transform to cartesian with $e_r = (cos\theta sin\phi \hat{i}, sin\theta sin\phi \hat{y}, cos\phi \hat{z})$ and so on for $e_{\theta}, e_{\phi}$.
EDIT:
Turloc's answer helped me visualize the field.
To find the basis spherical basis vectors and interpret the equation algebraically, I find myself using the transform matrix to cartesian system. Is this necessary?
To picture a vector field, try vector projections against the orthonormal basis. This isolates the vector's coordinate. This also varies according to the exact definition of the basis, some bases are not normalized adding a layer of complexity.
In this case $\vec{a}\cdot\hat{e_r}=4\theta$. This means that the r component points directly away from the origin starts out very small and increases as the coordinate moves counter clockwise, like a spiral.
$\vec{a}\cdot\hat{e_\theta}=1$ everywhere. This vector is perpendicular to the vector $\hat{e_r}$ with constant unit magnitude everywhere.
So in the end, you have counter-clockwise spiral with vectors point slightly counter clockwise to the vector pointing directly from the origin to the location.