### 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.

Corbett – thank you for sharing your work on mathematical research.

And thank you for visiting, Dr. Slunt.

Nice material! Glad you had a chance to look at this with Dr. Chiang. Best wishes,

Thanks, Corbett, for sharing the work you did with Dr. C.