I am not able to match the $d$ and $f$ orbitals in atomic structure worksheet given as homework (see the attached image). I was able to match the $p_{x}$ 
, $p_{y}$ 
, and $p_{z}$ 
orbitals. But I am facing difficulty with the $d$ and $f$ orbitals. Is it possible to define the shapes and orientations of all the orbitals mathematically in cartesian coordinate system (and graph it on Mathematica, etc.) so that it would be easier for me to understand the structure of orbitals, nodes, shells, etc., for example, the $s$ orbital can be defined as $(x - x_0 )^2 + (y - y_0 )^2 + ( z - z_0 )^2 = r^2$ which is a sphere
. I am not sure about other orbitals which are very complicated (I am not introduced to spherical harmonics).
In the given worksheet I have to match 
, 
, 
, 
, and 
with $d_{x^2-y^2}$, $d_{zx}$, $d_{3z^2-r^2}$, $d_{yz}$, and $d_{xy}$.
I am having trouble understanding the structure of orbitals, nodes, shells, subshells, etc. Is it possible to define the shapes and orientations of all the orbitals mathematically in cartesian coordinate system (and graph it on Mathematica, etc.). for example, the s orbital can be defined as a sphere.