Advanced Analysis II: Demonstration of Functions and Tangent Planes

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Cones and Cylinders: Geometry Descriptive

Explores cones and cylinders in descriptive geometry, covering ruling surfaces and tangent planes.

Shells I

Covers linear pressure vessels, thin shells, and critical buckling pressure, emphasizing the dimensional reduction from 3D to 2D.

Ellipses and Conics

Explores ellipses, conic sections, Kepler's laws, and geometric reconstruction in architectural modeling.

Calculus Applications: Lengths and Surfaces of Revolution

Discusses the applications of calculus in calculating lengths and surfaces of revolution, emphasizing integral calculus and geometric interpretations.

Differentiable Functions: Definitions and Interpretation

Covers the definitions of differentiable and derivable functions and their geometric interpretation.

Surface Normals: Vector Analysis

Explains surface normals for parametric and implicit surfaces, focusing on vector analysis and examples with spheres.

Differential Forms Integration

Covers the integration of differential forms on smooth manifolds, including the concepts of closed and exact forms.

Tangent Vectors: Definition and Applications

Explores the definition and applications of tangent vectors in determining curve directions on surfaces.

Surface Integrals: Implicit Surfaces

Covers implicit surfaces, parametric descriptions, regular parametrization, normal vectors, and orientable surfaces.

Green's Theorem: Surface Integrals

Explores Green's Theorem applied to surface integrals, emphasizing regular surfaces and coordinate transformations.