The exhaustive list of topics in Differential Geometry in which we provide Help with Homework Assignment and Help with Project is as follows:
- Space curves: Definition of space curves, arc length, natural equation of a line. Intersection of curves, intersection of curve and surface, order of contact. Tangent, normal, osculating plane, principal normal and binormal at a point of a curve.
- Curvature, torsion, Frenet – Serret formulae, plane curves, osculating circle and osculating sphere. Fundamental theorem for space curves.
- Helices, evolutes and involutes.
- Surfaces: Definition of a surface, tangent plane and normal at a point of a surface, surfaces of revolution, conicoids and helicoids.
- Envelopes and developable surfaces, ruled surfaces. First fundamental form(Metric), metric potentials, direction coefficients and angle between two curves on a surface.
- Second fundamental form, lines of curvatures, principal curvatures. Meusnier’s theorem, Euler’s theorem, Dupin’s indicatrix, Mean curvature, Gaussian curvature.
- Normal curvature, Minimal surface, developable surface,umbilical points, Geodesic curvature and geodesic, its differential equation, geodesic as curve of minimum length drawn on a surface. Weingarten equations, Mainardi-Codazzi equations, Bending invariant. Fundamental theorem on surfaces.
- Tensors: Tensors, metric tensor, Christoffel symbols, Covariant derivative and their uses in differential geometry of 3 or higher dimensions, Curvature tensor.