Akademik Dersler - İçerik

Explain the differential geometry of curves.
Explain the concept of parametrization of regular curves in Euclidean space and compare different types of parametrizations.
Define the concepts of curvature and torsion of a curve, compute these quantities, and interpret their geometric meanings.
Construct the Frenet frame and analyze the behavior of space curves using the Frenet–Serret formulas.
Recognize and distinguish special families of curves such as spherical curves, helices, and Bertrand curves.
State the fundamental theorems of curve theory and apply them to examples.
Solve problems related to curves using symbolic computation tools.
Explain the differential geometry of surfaces.
Define the concept of a regular surface and compute the tangent plane of a surface.
Obtain the first fundamental form of a surface and interpret its properties.
Define the Gauss map, the second fundamental form, and the shape operator, and explain the relationships among them.
Compute normal curvature, principal curvatures, and principal directions, and interpret their geometric meanings.
Explain the concepts of Gaussian curvature and mean curvature and apply them to different surfaces.
Analyze lines of curvature, asymptotic lines, and the Dupin indicatrix.
State Meusnier’s Theorem and apply it to practical problems.
Recognize special families of surfaces and compare their fundamental properties.
Effectively use symbolic computation methods to solve problems in differential geometry.