We consider a parameterization of a surface, then the second derivatives of the parameterization can be expressed in terms of the first derivatives and the unit normal vector. The coefficients of the normal vector in the linear expansion are the coefficients of the second fundamental form. The coefficients in terms of the first derivatives can be expressed by the Christoffel symbols. We prove the Gauss relationships for these symbols in terms of derivatives of the coefficients of the first fundamental form.
#mikethemathematician, #mikedabkowski, #profdabkowski, #differentialgeometry
The Gauss Equations for the Christoffel Symbols on a Surface
Теги
mike the mathematicianmike dabkowski mathsecond fundamental formdifferential geometrysurface theorycurvaturenormal vector of surfaceChristoffel SymbolsChristoffel symbols on a surfacesurface Christoffel symbolssolving for Christoffel symbolsChristoffel symbols derivative of metric tensorChristoffel symbols derivative of first fundamental formGauss equations for Christoffel symbolspressley solutionsTheorema EgregiumGauss Theorema EgregiumGauss
