Question

How that, for the standard orientation of the Gauss map N, the components of the second fundamental form are given by e = ⟨σu, σv, σuu⟩/|σu × σv| , f =⟨σu, σv, σuv⟩/|σu × σv| and g =⟨σu, σv, σvv⟩/|σu × σv| where ⟨x, y, z⟩= ⟨(x × y), z ⟩is the triple product.

Answer 1

The question discusses the components of the second fundamental form in Differential Geometry, which are related to surface curvature and are calculated using the triple product of the surface's tangent vectors and their partial derivatives.

Step-by-step explanation:

The student's question pertains to the mathematical subject of Differential Geometry, specifically understanding the second fundamental form of a surface in three-dimensional space. The second fundamental form relates to the curvature of a surface at a point and is defined in terms of the Gauss map N. The formulas provided describe how the coefficients of the second fundamental form, denoted as e, f, and g, can be determined by taking the triple product between tangent vectors σu and σv of the surface and their respective partial derivatives, normalized by the area element |σu × σv|. The triple product ⟩x, y, z⟫ equals the dot product ⟩(x × y), z⟫, providing a way to calculate the components in terms of unit vectors of the axes.