Advanced Analysis II: Demonstration of Functions and Tangent Planes

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Implicit Functions Theorem

Explores the implicit functions theorem and the equation of the tangent plane in a graph.

Ruled Surfaces: Generators and Applications

Explores ruled surfaces, surfaces generated by moving lines in space, including cones, cylinders, and hyperboloids, with applications in architecture.

Envelope of Curves

Explores the concept of envelope of curves within families of lines, including implicit equations, super-ellipses, and curvature.

Mathematical Analysis: Functions and Composition

Covers the analysis of functions, composition, and mathematical induction.

Tangent Planes and Implicit Functions

Explores tangent planes, implicit functions, and directional derivatives in relation to graphs of functions.

Gaussian Curvature and Geodesics

Explores the derivative of curve lengths, fixed-end deformations, geodesics, surface point typologies, and sphere parametrization.

Closed Surfaces and Integrals

Explains closed surfaces like spheres, cubes, and cones without covers, and their traversal and removal of edges.

Curves and Surfaces

Covers the representation of plane curves and the nature of obtained curves.

Differentiability and Tangent Planes in Multivariable Functions

Covers differentiability in multivariable functions and the existence of tangent planes, emphasizing geometric interpretations and practical applications.

Tangent to Graph of a Function

Explores finding the equation of the tangent line to a function's graph at a point.