Dynamic Graphs and Quaternion Analysis

• Outline: Dynamic Graphs of Complex Numbers
• Mathematical Physics
• Dynamic Graphs of Complex Numbers
• Origin Is Unique
• Around the Origin
• Big Circle
• Many Frequencies
• Seeing Uncertainty
• Summary: Dynamic Graphs of Complex Numbers
• Outline: Dynamic Graphs of Quaternions
• Dynamic Graphs of Quaternions
• Key for Dynamic Graphs
• Time Reflection
• Space Reflection
• Tilted Big Circle
• Circles and Reflections
• Frequencies and Amplitudes
• Cosine
• Sine
• Sine Versus Cosine from Zero to Pi
• Summary: Dynamic Graphs of Quaternions
• Sine
• Cosine
• Tangent
• Sine, Cosine, and Tangent
• Inverse Sine (Arcsine)
• Inverse Cosine
• Inverse Tangent
• Inverse Sine, Cosine, and Tangent
• Hyberbolic Sine, Cosine, and Tangent
• Inverse Hyberbolic Sine, Cosine, and Tangent
• Division: Normalize by All Histories between Conjugates
• Derivative: Division in a Limit
• Better Derivative: Division in a Two-Step Limit Process
• Causal Order of Limits: Timelike, Spacelike, or Lightlike (1/2)
• Causal Order of Limits: Timelike, Spacelike, or Lightlike (2/2)
• General Balanced Basis Vectors
• Length of Basis Vectors
• Span of Automorphisms
• Representing Components with Automorphisms