CORE 009: Vector Analysis
Syllabus

The topic list for this project is: theory of curves, vector fields and operators, vector integration, theory of surfaces, and tensors and exterior forms.

Instructor: George E. Hrabovsky, george@madscitech.org, 608-276-6832.

Project Objectives:

• To learn the differential geometry of curves.
• To learn the theory of vector calculus.
• To learn about the vector operator nabla.
• To learn the basics of gradient, divergence, and curl.
• To learn about the standard theorems of vector integration.
• To learn the differential geometry of curves.
• To learn about tensors.

Modules:

1. Theory of curves.
2. Vector fields and operators.
3. Vector integration.
4. Theory of surfaces.
5. Tensors.

Tasks for Module #1: Theory of curves.

At one topic per day, this module can be completed in 2 weeks.

1. Begin a notebook for the project. This will be worth 1 point upon completion.
2. Representation of curves.
3. Arc length and tangent.
4. Curvature.
5. Plane curves.
6. Torsion.
7. Space curves.
8. Frenet formulas.
9. Natural equations.
10. Derived curves.

Tasks for Module #2: Vector fields and operators.

At one topic per day, this module can be completed in 2 weeks.

1. Representations of surfaces.
2. The nabla operator.
3. Gradient.
4. Scalar fields.
5. Divergence.
6. Vector fields.
7. Curl.
8. Properties of the nabla operator.
9. Coordinate systems.
10. Orthogonal curvilinear coordinate systems.

Tasks for Module #3: Vector integration.

At one topic per day, this module can be completed in 2 weeks.

1. Line integrals.
2. Surface integrals.
3. Volume integrals.
4. Green's theorem in the plane.
5. Green's theorem.
6. Stokes' theorem.
7. The divergence theorem.
8. Green's first identity.
9. Green's second identity.
10. Other integral theorems.

Tasks for Module #4: Theory of surfaces.

At one topic per day, this module can be completed in 2 weeks.

1. The first fundamental form.
2. Normal and tangent planes.
3. Developable surfaces.
4. Second fundamental form.
5. Euler's theorem.
6. Dupin's indicatrix.
7. The sphere and surfaces of revolution.
8. Asymptotic and curvature lines.
9. Conjugate directions.
10. Triply orthogonal systems of surfaces.

Tasks for Module #5: Tensors.

At one topic per day, this module can be completed in 2 weeks.

1. Cartesian tensors.
2. Algebra of tensors.
3. Kronecker delta and Levi-Civita density.
4. Integral theorems for tensors.
5. Curvilinear coordinates.
6. The metric tensor.
7. General coordinate transformations.
8. Tensor differentiation.
9. Covariant differentiation and differential operators.
10. Geodesics.

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