| 1 | Learn the concept of manifold. |
| 2 | Recognize Euclidean space as a manifold. |
| 3 | Learn topological and differentiable manifolds. |
| 4 | Learn the concepts of tangent vector, tangent space, vector field, and space of vector fields in Euclidean space. |
| 5 | Learning the concepts of directional derivative, cotangent space, and a form. |
| 6 | Learning the differential and Jacobian of a transformation. |