### by K. Corbett

### Faculty mentor: Dr. Yuan-Jen Chiang

We first introduce the concepts of surface theory including the coordinate patch, coordinate transformation, normal vector, tangent plane, etc. We next compute the first fundamental form of a surface: the matrix (gij) of metric coefficients, gij = xi x xj , the inner product with respect to the basis {x1, x2} of the tangent space of the surface. We then discuss the second fundamental form, Weingarten map (i.e., shape operator), Christoffel symbols, the geodesic, geodesic curvature, principal curvature, Gauss curvature, mean curvature, normal curvature, parallelism, etc. We will apply the proceeding terms to a few concrete examples by different calculations. We will utilize the software Mathematica to sketch various surfaces and their geometric properties.