I am finding the electric field from a spherical shell at a point on the z-axis outside the shell. The shell is centered at the origin,and I am only allowed to use coulomb's law. I want to find dE in spherical coordinates first then transform it to Cartesian before integrating to get E. So I choose the spherical coordinates of the point at which I want to find E to be (z , 0 , ø), and the coordinates of a piece of charge on the surface of the shell to be (R, Θ , ø), where R is the radius of the shell and z is the distance on the z axis above the origin. When I transform to Cartesian coordinates to get E_z, I use R'_z = R'_r cos(Θ) - R'_Θ sin(Θ). I get R'_z = (z-R)cos(Θ) + Θsin(Θ), Which is incorrect. The correct expression is R'_z = z-Rcos(Θ) . R' is the distance vector from the source to the point. I believe I am doing something wrong with choosing the coordinates, but I can't really tell where. Can somebody help me ?