Definition 0.0.0.1.
Level: Definition/convention
PreTeXt-style label:
definition
Interpretation
"For the purposes of There and Back Again, use this term/notation as specified."
Preface Conventions
How to Read This Book.
Epistemic Statements.
Remark 0.0.0.2.
Level: Observation
PreTeXt-style label:
remark
Interpretation: "I notice this recurring pattern."
Level: Empirical Law
PreTeXt-style label:
remark/principle
Interpretation: "Repeated observation supports this relation in a defined domain."
Note 0.0.0.3.
Level: Heuristic
PreTeXt-style label:
note
Interpretation
"This seems useful for reasoning, but is not established."
Principle 0.0.0.4.
Level: Working Principle
PreTeXt-style label:
principle
Interpretation: "Treat this as a provisional organizing principle."
Level: Empirical law
PreTeXt-style label:
remark/principle
Interpretation: "Repeated observation supports this relation in a defined domain."
Conjecture 0.0.0.5.
Level: Conjecture
PreTeXt-style label:
conjecture
Interpretation
"I suspect this may hold generally."
Proposition 0.0.0.6.
Level: Proposition
PreTeXt-style label:
proposition
Interpretation
"Given stated assumptions, this follows mathematically."
Hypothesis 0.0.0.7.
Level: Scientific Hypothesis
PreTeXt-style label:
hypothesis
Interpretation
"This is a claim about a natural system that could be tested."
Assumption 0.0.0.8.
Level: Assumption/Postulate
PreTeXt-style label:
assumption/axiom
Interpretation
"This is stipulated as foundational within the framework."
Axiom 0.0.0.9.
Level: Axiom/Postulate
PreTeXt-style label:
axiom
Interpretation
"This is conventionally stipulated as foundational within the framework."
Mathematical Notation.
Standard Sets & Constants.
Context 1 Standard Sets
Standard Number Sets.
- \(\Z\)
-
Integers\(\{ 0, \pm 1, \pm 2, \dots \}\)
- \(\N\) (or \(\mathbb{Z}^+\))
-
Natural numbers\(\{1, 2, 3, \dots \}\)
- \(\N_0\)
-
\(\mathbf{N} \cap \{0\}\)The set containing the natural numbers and zero
Original Cartesian Mechanics.
Context 1 General Mathematics
- \(x\)
-
scalar
- \(\mathbf{x}\)
-
An \(n\)-dimensional vector with scalar elements.For \(n=1\text{,}\) \(\mathbf{x}\cong x\text{.}\)
Context 2 Analytical & Classical Mechanics
Phase Space.
- \(\mathcal{P}\)
-
The phase space manifold parameterized by generalized coordinates and conjugate momenta.
- \((\mathbf{q}, \mathbf{p})\in\mathcal{P}\)
-
A point in phase space where
-
\(\mathbf{q} = (q_1, \dots, q_n)\) represents generalized coordinates.
-
\(\mathbf{p} = (p_1, \dots, p_n)\) represents conjugate momenta.
Scalar coordinate and momentum magnitudes are denoted by \(q = \|\mathbf{q}\|\) and \(p = \|\mathbf{p}\|\text{,}\) respectively. -
- \(\dot{\mathbf{q}} = \frac{d\mathbf{q}}{dt}\)
-
Generalized velocities expressed via Newton’s dot notation for time derivatives.
General Mathematics, Sets & Constants.
Linear Algebra & Vector Spaces.
Geometric Algebra & Multivector Systems.
Analytical & Classical Mechanics.
Calculus of Variations & Field Theory.
Quantum Mechanics & Quantum Information.
Computational Methods, Signal Processing & Machine Learning.
Probability, Statistics & Empirical Methods.
Abbreviations.
- GA
-
Geometric Algebra
- psd
-
positive semi-definite
- PSD
-
power spectral density
- PIML
-
Physics-Informed Machine Learning
- PINN
-
Physics-Informed Neural Networks
Icons.
- Proofmark
-
Use the Proofmark icon as a signature of authenticity or to mark verified, high-fidelity content that has undergone rigorous review.
- Eigenote
-
Use an Eigenote icon to capture “eigen-insights”—the core, irreducible components of a concept or fundamental patterns that emerge across different domains.
- Technical Observation
-
Use a Technical Observation icon for specific implementation details, low-level technical notes, or cross-references.
- Heuristic Warning
-
Use a Heuristic Warning icon to document rules of thumb, common pitfalls, and mental models that help navigate complex problem spaces.
