Surface Integrals: Orientation and Computation

This lecture covers the concept of orientable surfaces, defining them based on the continuity of the unit normals field. Examples of orientable and non-orientable surfaces are provided, such as the Möbius strip and Klein bottle. The computation of surface integrals for scalar fields is explained, emphasizing the analogy with line integrals. The process involves parametrizing the surface, defining a continuous scalar field, and calculating the integral over the surface using the normal vector. Practical applications, like computing surface area and mass, are also discussed. Detailed examples with a sphere and a cone illustrate the step-by-step process of parameterization, normal vector computation, and integral evaluation.