Fig. 3 shows the free energy for the R = 4 ξ and D = 6 ξ rod. The 3D view of the Cooper pair density, | ψ | $^{2}$, shows curved top-to-side vortices, that is, vortices that enter from the top surface and exit perpen- dicularly in the side surface. We call them N-fold states, where N describes the number of vortices. Along the energy curve vs µ in Fig. 3 the states are labeled according to this definition. The N- fold states are not clearly seen in Fig. 3 because their free energy lines is superposed to the Meiss- ner line from where is their onset and disappear- ance. However these states are clearly seen to exist by analyzing the | ψ | 2 isosurface, as shown in Fig. 4. In the two biggest rods considered here we observed the nucleation of N-fold states fol- lowed by Meissner and GVS. This features are justified by the growth of a normal region closer to the the magnetic dot position. When this nor- mal region occupies a significant part of the rod the remaining superconducting part behaves as a thin rod, i.e., it does not exhibits N-fold states. So, we retrieve the Meissner state. This feature is in Table 1 and correspond to the second Meissner state. - 2 , 0 - 1 , 6 - 1 , 2 - 0 , 8 - 0 , 4 , 0 - 1 , 0 - 0 , 8 - 0 , 6 - 0 , 4 - 0 , 2 , 0 / R R R R , 0 , 8 , 6 , 4 , 2 , 0 i s l d l d i s Figure 3: (Color online) Free energy F/F$_{0}$ vs µ/µ$_{0}$ for a rod with R = 4 ξ and D = 8 ξ . For increasing magnetic moment the vortex pattern evolves continuously through the N-fold configurations, as labeled in the inset. See, Fig. 4 for the corresponding | ψ | 2 isosurface views of these N-fold states. Figure 2: (Color online) Free energy F/F$_{0}$ and magnetiza- tion M/M$_{0}$ versus magnetic moment µ/µ$_{0}$ for mesoscopic rods with thickness D = 2 ξ and several radii. Giant vor- tices with increasing maximum vorticity are observed by increasing the radius. 3 The inhomogeneity of the field becomes impor- tant for the thicker rods D = 6 ξ and 8 ξ , as shown here. Even for small radii, the rod already dis- plays behavior that differs significantly from the homogeneous field case. We find smooth free en- ergy lines, with no first order transition occa- sioned by the entrance of quantized vortex lines. weaker at distances away from the center. Conse- quently the average magnetic magnetic field does not grow with the radius. This effect is equivalent to that studied by Ovchinnikov sometime ago [12]. A sawtooth behavior in the magnetization with bigger H$_{c}$$_{2}$ is clearly established for bigger radius. This behavior is observed for R = 3 ξ and D = 2 ξ . In Fig. 2 we show the vortex states obtained for different thin rods, i.e., disks. Their magnetiza- tion curves are similar to those obtained in pres- ence of homogeneous field. In table 1 we give the complete description of the states through the set of parameters ( R, D, µ/µ$_{0}$ ).