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\begin{document}

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\section*{The Geometry of Physics - An Introduction}

\hypertarget{context}{}\subsubsection*{{Context}}\label{context}

\hypertarget{differential_geometry}{}\paragraph*{{Differential geometry}}\label{differential_geometry}

[[!include synthetic differential geometry - contents]]

\hypertarget{physics}{}\paragraph*{{Physics}}\label{physics}

[[!include physicscontents]]

This page provides a hyperlinked index for the book

\begin{itemize}%
\item [[Theodore Frankel]],

\emph{The Geometry of Physics - An introduction}

Cambridge University Press, 1997, 2004, 2012

(\href{https://doi.org/10.1017/CBO9781139061377}{doi:10.1017/CBO9781139061377}, \href{http://www.math.ucsd.edu/~tfrankel/}{website} with errata and preface for 3rd edition)



\end{itemize}
on methods of [[differential geometry]] and their meaning and use in [[physics]], especially [[gravity]] and [[gauge theory]].

Among the nice aspects of the book are

\begin{itemize}%
\item it discusses [[pseudoforms]] on top of ordinary [[differential forms]], instead of just assuming that all [[manifolds]] are [[orientation|oriented]] as often done --- and what's more, it explains the physical meaning of this! (But it still uses [[Riemannian metric|metrics]] more than necessary.)

\end{itemize}
Related books are

\begin{itemize}%
\item [[Werner Greub]], [[Stephen Halperin]], [[Ray Vanstone]], \emph{[[Connections, Curvature, and Cohomology]]}


\item [[Chris Isham]], \emph{[[Modern Differential Geometry for Physicists]]}



\end{itemize}
\hypertarget{contents}{}\section*{{Contents}}\label{contents}

\noindent\hyperlink{i_manifolds_tensors_and_exterior_forms}{I Manifolds, Tensors and Exterior Forms}\dotfill \pageref*{i_manifolds_tensors_and_exterior_forms} \linebreak
\noindent\hyperlink{1_manifolds_and_vector_fields}{1 Manifolds and Vector Fields}\dotfill \pageref*{1_manifolds_and_vector_fields} \linebreak
\noindent\hyperlink{11_submanifolds_of_euclidean_space}{1.1 Submanifolds of Euclidean space}\dotfill \pageref*{11_submanifolds_of_euclidean_space} \linebreak
\noindent\hyperlink{11a_submanifolds_of_}{1.1a Submanifolds of $\mathbb{R}^N$}\dotfill \pageref*{11a_submanifolds_of_} \linebreak
\noindent\hyperlink{11b_the_geometry_of_jacobian_matrices_the_differential}{1.1b The Geometry of Jacobian Matrices: The ``Differential''}\dotfill \pageref*{11b_the_geometry_of_jacobian_matrices_the_differential} \linebreak
\noindent\hyperlink{11c_the_main_theorem_on_submanifolds_of_}{1.1c The main theorem on submanifolds of $\mathbb{R}^N$}\dotfill \pageref*{11c_the_main_theorem_on_submanifolds_of_} \linebreak
\noindent\hyperlink{11d_a_nontrivial_example_the_configuration_space_of_a_rigid_body}{1.1d A Nontrivial Example: The Configuration Space of a Rigid Body}\dotfill \pageref*{11d_a_nontrivial_example_the_configuration_space_of_a_rigid_body} \linebreak
\noindent\hyperlink{12_manifolds}{1.2 Manifolds}\dotfill \pageref*{12_manifolds} \linebreak
\noindent\hyperlink{13_tangent_vectors_and_mappings}{1.3 Tangent Vectors and Mappings}\dotfill \pageref*{13_tangent_vectors_and_mappings} \linebreak
\noindent\hyperlink{14_vector_fields_and_flows}{1.4 Vector Fields and Flows}\dotfill \pageref*{14_vector_fields_and_flows} \linebreak
\noindent\hyperlink{2_tensors_and_exterior_forms}{2 Tensors and Exterior Forms}\dotfill \pageref*{2_tensors_and_exterior_forms} \linebreak
\noindent\hyperlink{21_covectors_and_riemannian_metrics}{2.1 Covectors and Riemannian Metrics}\dotfill \pageref*{21_covectors_and_riemannian_metrics} \linebreak
\noindent\hyperlink{22_the_tangent_bundle}{2.2 The Tangent Bundle}\dotfill \pageref*{22_the_tangent_bundle} \linebreak
\noindent\hyperlink{23_the_cotangent_bundle_and_phase_space}{2.3 The Cotangent Bundle and Phase Space}\dotfill \pageref*{23_the_cotangent_bundle_and_phase_space} \linebreak
\noindent\hyperlink{24_tensors}{2.4 Tensors}\dotfill \pageref*{24_tensors} \linebreak
\noindent\hyperlink{25_the_grassmann_or_exterior_algebra}{2.5 The Grassmann or Exterior Algebra}\dotfill \pageref*{25_the_grassmann_or_exterior_algebra} \linebreak
\noindent\hyperlink{26_exterior_differentiation}{2.6 Exterior Differentiation}\dotfill \pageref*{26_exterior_differentiation} \linebreak
\noindent\hyperlink{27_pullbacks}{2.7 Pull-Backs}\dotfill \pageref*{27_pullbacks} \linebreak
\noindent\hyperlink{28_orientation_and_pseudoforms}{2.8 Orientation and Pseudoforms}\dotfill \pageref*{28_orientation_and_pseudoforms} \linebreak
\noindent\hyperlink{29_interior_products_and_vector_analysis}{2.9 Interior Products and Vector Analysis}\dotfill \pageref*{29_interior_products_and_vector_analysis} \linebreak
\noindent\hyperlink{210_dictionary}{2.10 Dictionary}\dotfill \pageref*{210_dictionary} \linebreak
\noindent\hyperlink{3_integration_of_differential_forms}{3 Integration of Differential Forms}\dotfill \pageref*{3_integration_of_differential_forms} \linebreak
\noindent\hyperlink{31_integration_over_a_parameterized_subset}{3.1 Integration over a Parameterized Subset}\dotfill \pageref*{31_integration_over_a_parameterized_subset} \linebreak
\noindent\hyperlink{32_integration_over_manifolds_with_boundary}{3.2 Integration over Manifolds with Boundary}\dotfill \pageref*{32_integration_over_manifolds_with_boundary} \linebreak
\noindent\hyperlink{33_stokes_theorem}{3.3 Stokes' Theorem}\dotfill \pageref*{33_stokes_theorem} \linebreak
\noindent\hyperlink{34_integration_of_pseudoforms}{3.4 Integration of Pseudoforms}\dotfill \pageref*{34_integration_of_pseudoforms} \linebreak
\noindent\hyperlink{35_maxwells_equations}{3.5 Maxwell's Equations}\dotfill \pageref*{35_maxwells_equations} \linebreak
\noindent\hyperlink{4_the_lie_derivative}{4 The Lie Derivative}\dotfill \pageref*{4_the_lie_derivative} \linebreak
\noindent\hyperlink{41_the_lie_derivative_of_a_vector_field}{4.1 The Lie Derivative of a Vector Field}\dotfill \pageref*{41_the_lie_derivative_of_a_vector_field} \linebreak
\noindent\hyperlink{42_the_lie_derivative_of_a_form}{4.2 The Lie Derivative of a Form}\dotfill \pageref*{42_the_lie_derivative_of_a_form} \linebreak
\noindent\hyperlink{43_differentiation_of_integrals}{4.3 Differentiation of Integrals}\dotfill \pageref*{43_differentiation_of_integrals} \linebreak
\noindent\hyperlink{44_a_problem_set_on_hamiltonian_mechanics}{4.4 A Problem Set on Hamiltonian Mechanics}\dotfill \pageref*{44_a_problem_set_on_hamiltonian_mechanics} \linebreak
\noindent\hyperlink{5_the_poincar_lemma_and_potentials}{5 The Poincar\'e{} Lemma and Potentials}\dotfill \pageref*{5_the_poincar_lemma_and_potentials} \linebreak
\noindent\hyperlink{6_holonomic_and_nonholonomic_constraints}{6 Holonomic and Nonholonomic Constraints}\dotfill \pageref*{6_holonomic_and_nonholonomic_constraints} \linebreak
\noindent\hyperlink{ii_geometry_and_topology}{II Geometry and Topology}\dotfill \pageref*{ii_geometry_and_topology} \linebreak
\noindent\hyperlink{7__and_minkowski_space}{7 $\mathbb{R}^3$ and Minkowski Space}\dotfill \pageref*{7__and_minkowski_space} \linebreak
\noindent\hyperlink{71_curvature_and_special_relativity}{7.1 Curvature and Special Relativity}\dotfill \pageref*{71_curvature_and_special_relativity} \linebreak
\noindent\hyperlink{72_electromagnetism_in_minkowski_space}{7.2 Electromagnetism in Minkowski Space}\dotfill \pageref*{72_electromagnetism_in_minkowski_space} \linebreak
\noindent\hyperlink{8_the_geometry_of_surfaces_in_}{8 The Geometry of Surfaces in $\mathbb{R}^3$}\dotfill \pageref*{8_the_geometry_of_surfaces_in_} \linebreak
\noindent\hyperlink{81_the_first_and_second_fundamental_form}{8.1 The First and Second Fundamental Form}\dotfill \pageref*{81_the_first_and_second_fundamental_form} \linebreak
\noindent\hyperlink{82_gaussian_and_mean_curvature}{8.2 Gaussian and Mean Curvature}\dotfill \pageref*{82_gaussian_and_mean_curvature} \linebreak
\noindent\hyperlink{83_the_brouwer_degree_of_a_map_a_problem_set}{8.3 The Brouwer Degree of a Map: A Problem Set}\dotfill \pageref*{83_the_brouwer_degree_of_a_map_a_problem_set} \linebreak
\noindent\hyperlink{84_area_mean_curvature_and_soap_bubbles}{8.4 Area, Mean Curvature, and Soap Bubbles}\dotfill \pageref*{84_area_mean_curvature_and_soap_bubbles} \linebreak
\noindent\hyperlink{85_gauss_theorema_egregium}{8.5 Gauss' \emph{Theorema Egregium}}\dotfill \pageref*{85_gauss_theorema_egregium} \linebreak
\noindent\hyperlink{86_geodesics}{8.6 Geodesics}\dotfill \pageref*{86_geodesics} \linebreak
\noindent\hyperlink{87_the_parallel_displacement_of_levicivita}{8.7 The Parallel Displacement of Levi-Civita}\dotfill \pageref*{87_the_parallel_displacement_of_levicivita} \linebreak
\noindent\hyperlink{9_covariant_differentiation_and_curvature}{9 Covariant Differentiation and Curvature}\dotfill \pageref*{9_covariant_differentiation_and_curvature} \linebreak
\noindent\hyperlink{91_covariant_differentiation}{9.1 Covariant Differentiation}\dotfill \pageref*{91_covariant_differentiation} \linebreak
\noindent\hyperlink{92_the_riemannian_connection}{9.2 The Riemannian Connection}\dotfill \pageref*{92_the_riemannian_connection} \linebreak
\noindent\hyperlink{93_cartans_exterior_covariant_differential}{9.3 Cartan's Exterior Covariant Differential}\dotfill \pageref*{93_cartans_exterior_covariant_differential} \linebreak
\noindent\hyperlink{94_change_of_basis_anf_gauge_transformations}{9.4 Change of Basis anf Gauge Transformations}\dotfill \pageref*{94_change_of_basis_anf_gauge_transformations} \linebreak
\noindent\hyperlink{95_the_curvature_forms_in_a_riemannian_manifold}{9.5 The Curvature Forms in a Riemannian Manifold}\dotfill \pageref*{95_the_curvature_forms_in_a_riemannian_manifold} \linebreak
\noindent\hyperlink{96_parallel_displacement_and_curvature_on_a_surface}{9.6 Parallel Displacement and Curvature on a Surface}\dotfill \pageref*{96_parallel_displacement_and_curvature_on_a_surface} \linebreak
\noindent\hyperlink{97_riemanns_theorem_and_horizontal_distribution}{9.7 Riemann's Theorem and Horizontal Distribution}\dotfill \pageref*{97_riemanns_theorem_and_horizontal_distribution} \linebreak
\noindent\hyperlink{10_geodesics}{10 Geodesics}\dotfill \pageref*{10_geodesics} \linebreak
\noindent\hyperlink{101_geodesics_and_jacobian_fields}{10.1 Geodesics and Jacobian Fields}\dotfill \pageref*{101_geodesics_and_jacobian_fields} \linebreak
\noindent\hyperlink{102_variational_principles_in_mechanics}{10.2 Variational Principles in Mechanics}\dotfill \pageref*{102_variational_principles_in_mechanics} \linebreak
\noindent\hyperlink{103_geodesics_spiders_and_the_universe}{10.3 Geodesics, Spiders, and the Universe}\dotfill \pageref*{103_geodesics_spiders_and_the_universe} \linebreak
\noindent\hyperlink{11_relativity_tensors_and_curvature}{11 Relativity, Tensors, and Curvature}\dotfill \pageref*{11_relativity_tensors_and_curvature} \linebreak
\noindent\hyperlink{111_heuristics_of_einsteins_theory}{11.1 Heuristic's of Einstein's Theory}\dotfill \pageref*{111_heuristics_of_einsteins_theory} \linebreak
\noindent\hyperlink{112_tensor_analysis}{11.2 Tensor analysis}\dotfill \pageref*{112_tensor_analysis} \linebreak
\noindent\hyperlink{113_hilberts_action_principle}{11.3 Hilbert's Action Principle}\dotfill \pageref*{113_hilberts_action_principle} \linebreak
\noindent\hyperlink{114_the_second_fundamental_form_in_the_riemannian_case}{11.4 The Second Fundamental Form in the Riemannian Case}\dotfill \pageref*{114_the_second_fundamental_form_in_the_riemannian_case} \linebreak
\noindent\hyperlink{115_the_geometry_of_einsteins_equations}{11.5 The Geometry of Einstein's Equations}\dotfill \pageref*{115_the_geometry_of_einsteins_equations} \linebreak
\noindent\hyperlink{12_curvature_and_topology_synges_theorem}{12 Curvature and Topology: Synge's Theorem}\dotfill \pageref*{12_curvature_and_topology_synges_theorem} \linebreak
\noindent\hyperlink{13_betti_numbers_and_de_rhams_theorem}{13 Betti Numbers and De Rham's Theorem}\dotfill \pageref*{13_betti_numbers_and_de_rhams_theorem} \linebreak
\noindent\hyperlink{131_singular_chains_and_their_boundaries}{13.1 Singular Chains and Their Boundaries}\dotfill \pageref*{131_singular_chains_and_their_boundaries} \linebreak
\noindent\hyperlink{132_the_singular_homology_groups}{13.2 The Singular Homology Groups}\dotfill \pageref*{132_the_singular_homology_groups} \linebreak
\noindent\hyperlink{133_homology_groups_of_familiar_manifolds}{13.3 Homology Groups of Familiar Manifolds}\dotfill \pageref*{133_homology_groups_of_familiar_manifolds} \linebreak
\noindent\hyperlink{134_de_rhams_theorem}{13.4 De Rham's Theorem}\dotfill \pageref*{134_de_rhams_theorem} \linebreak
\noindent\hyperlink{14_harmonic_forms}{14 Harmonic Forms}\dotfill \pageref*{14_harmonic_forms} \linebreak
\noindent\hyperlink{141_the_hodge_operators}{14.1 The Hodge Operators}\dotfill \pageref*{141_the_hodge_operators} \linebreak
\noindent\hyperlink{142_harmonic_forms}{14.2 Harmonic Forms}\dotfill \pageref*{142_harmonic_forms} \linebreak
\noindent\hyperlink{143_boundary_values_relative_homology_and_morse_theory}{14.3 Boundary Values, Relative Homology, and Morse Theory}\dotfill \pageref*{143_boundary_values_relative_homology_and_morse_theory} \linebreak
\noindent\hyperlink{ii_lie_groups_bundles_and_chern_forms}{II Lie Groups, Bundles and Chern Forms}\dotfill \pageref*{ii_lie_groups_bundles_and_chern_forms} \linebreak
\noindent\hyperlink{15_lie_groups}{15 Lie Groups}\dotfill \pageref*{15_lie_groups} \linebreak
\noindent\hyperlink{151_lie_groups_invariant_vector_fields_and_forms}{15.1 Lie Groups, Invariant Vector Fields and Forms}\dotfill \pageref*{151_lie_groups_invariant_vector_fields_and_forms} \linebreak
\noindent\hyperlink{152_one_parameter_subgroups}{15.2 One Parameter Subgroups}\dotfill \pageref*{152_one_parameter_subgroups} \linebreak
\noindent\hyperlink{153_the_lie_algebra_of_a_lie_group}{15.3 The Lie Algebra of a Lie Group}\dotfill \pageref*{153_the_lie_algebra_of_a_lie_group} \linebreak
\noindent\hyperlink{154_subgroups_and_subalgebras}{15.4 Subgroups and Subalgebras}\dotfill \pageref*{154_subgroups_and_subalgebras} \linebreak
\noindent\hyperlink{16_vector_bundles_in_geometry_and_physics}{16 Vector Bundles in Geometry and Physics}\dotfill \pageref*{16_vector_bundles_in_geometry_and_physics} \linebreak
\noindent\hyperlink{161_vector_bundles}{16.1 Vector Bundles}\dotfill \pageref*{161_vector_bundles} \linebreak
\noindent\hyperlink{162_poincars_theorem_and_the_euler_characteristic}{16.2 Poincar\'e{}`s Theorem and the Euler Characteristic}\dotfill \pageref*{162_poincars_theorem_and_the_euler_characteristic} \linebreak
\noindent\hyperlink{163_connections_in_a_vector_bundle}{16.3 Connections in a Vector Bundle}\dotfill \pageref*{163_connections_in_a_vector_bundle} \linebreak
\noindent\hyperlink{164_the_electromagnetic_connection}{16.4 The Electromagnetic Connection}\dotfill \pageref*{164_the_electromagnetic_connection} \linebreak
\noindent\hyperlink{fiber_bundles_gaussbonnet_and_topological_quantization}{Fiber Bundles, Gauss-Bonnet, and Topological Quantization}\dotfill \pageref*{fiber_bundles_gaussbonnet_and_topological_quantization} \linebreak
\noindent\hyperlink{171_fiber_bundles_and_principal_bundles}{17.1 Fiber Bundles and Principal Bundles}\dotfill \pageref*{171_fiber_bundles_and_principal_bundles} \linebreak
\noindent\hyperlink{172_coset_spaces}{17.2 Coset Spaces}\dotfill \pageref*{172_coset_spaces} \linebreak
\noindent\hyperlink{173_cherns_proof_of_the_gaussbonnetpoincar_theorem}{17.3 Chern's Proof of the Gauss-Bonnet-Poincar\'e{} Theorem}\dotfill \pageref*{173_cherns_proof_of_the_gaussbonnetpoincar_theorem} \linebreak
\noindent\hyperlink{174_line_bundles_topological_quantization_and_berry_phase}{17.4 Line Bundles, Topological Quantization, and Berry Phase}\dotfill \pageref*{174_line_bundles_topological_quantization_and_berry_phase} \linebreak
\noindent\hyperlink{18_connections_and_associated_bundles}{18 Connections and Associated Bundles}\dotfill \pageref*{18_connections_and_associated_bundles} \linebreak
\noindent\hyperlink{181_forms_with_values_in_a_lie_algebra}{18.1 Forms with Values in a Lie Algebra}\dotfill \pageref*{181_forms_with_values_in_a_lie_algebra} \linebreak
\noindent\hyperlink{182_associated_bundles_and_connections}{18.2 Associated Bundles and Connections}\dotfill \pageref*{182_associated_bundles_and_connections} \linebreak
\noindent\hyperlink{183_form_sections_of_a_vector_bundle_curvature}{18.3 $r$-Form Sections of a Vector Bundle: Curvature}\dotfill \pageref*{183_form_sections_of_a_vector_bundle_curvature} \linebreak
\noindent\hyperlink{19_the_dirac_equation}{19 The Dirac Equation}\dotfill \pageref*{19_the_dirac_equation} \linebreak
\noindent\hyperlink{191_the_groups__and_}{19.1 The Groups $SO(3)$ and $SU(2)$}\dotfill \pageref*{191_the_groups__and_} \linebreak
\noindent\hyperlink{192_hamilton_clifford_and_dirac}{19.2 Hamilton, Clifford, and Dirac}\dotfill \pageref*{192_hamilton_clifford_and_dirac} \linebreak
\noindent\hyperlink{193_the_dirac_algebra}{19.3 The Dirac Algebra}\dotfill \pageref*{193_the_dirac_algebra} \linebreak
\noindent\hyperlink{194_the_dirac_operator_in_minkowski_space}{19.4 The Dirac Operator in Minkowski Space}\dotfill \pageref*{194_the_dirac_operator_in_minkowski_space} \linebreak
\noindent\hyperlink{195_the_dirac_operator_in_curved_spacetime}{19.5 The Dirac Operator in Curved Space-Time}\dotfill \pageref*{195_the_dirac_operator_in_curved_spacetime} \linebreak
\noindent\hyperlink{20_yangmills_fields}{20 Yang-Mills Fields}\dotfill \pageref*{20_yangmills_fields} \linebreak
\noindent\hyperlink{201_noethers_theorem_for_internal_symmetries}{20.1 Noether's Theorem for Internal Symmetries}\dotfill \pageref*{201_noethers_theorem_for_internal_symmetries} \linebreak
\noindent\hyperlink{202_weyls_gauge_invariance_revisited}{20.2 Weyl's Gauge Invariance Revisited}\dotfill \pageref*{202_weyls_gauge_invariance_revisited} \linebreak
\noindent\hyperlink{203_the_yangmills_nucleon}{20.3 The Yang-Mills Nucleon}\dotfill \pageref*{203_the_yangmills_nucleon} \linebreak
\noindent\hyperlink{204_compact_groups_and_yangmills_action}{20.4 Compact Groups and Yang-Mills Action}\dotfill \pageref*{204_compact_groups_and_yangmills_action} \linebreak
\noindent\hyperlink{205_the_yangmills_equation}{20.5 The Yang-Mills Equation}\dotfill \pageref*{205_the_yangmills_equation} \linebreak
\noindent\hyperlink{206_the_yangmills_instanton}{20.6 The Yang-Mills Instanton}\dotfill \pageref*{206_the_yangmills_instanton} \linebreak
\noindent\hyperlink{21_betti_numbers_and_covering_spaces}{21 Betti Numbers and Covering Spaces}\dotfill \pageref*{21_betti_numbers_and_covering_spaces} \linebreak
\noindent\hyperlink{211_biinvariant_forms_on_compact_groups}{21.1 Bi-invariant Forms on Compact Groups}\dotfill \pageref*{211_biinvariant_forms_on_compact_groups} \linebreak
\noindent\hyperlink{212_the_fundamental_group_and_covering_spaces}{21.2 The Fundamental Group and Covering Spaces}\dotfill \pageref*{212_the_fundamental_group_and_covering_spaces} \linebreak
\noindent\hyperlink{213_the_theorem_of_sb_myers_a_problem_set}{21.3 The Theorem of S.B. Myers: A Problem Set}\dotfill \pageref*{213_the_theorem_of_sb_myers_a_problem_set} \linebreak
\noindent\hyperlink{214_the_geometry_of_a_lie_group}{21.4 The Geometry of a Lie Group}\dotfill \pageref*{214_the_geometry_of_a_lie_group} \linebreak
\noindent\hyperlink{22_chern_forms_and_homotopy_groups}{22 Chern Forms and Homotopy Groups}\dotfill \pageref*{22_chern_forms_and_homotopy_groups} \linebreak
\noindent\hyperlink{221_chernforms_and_winding_numbers}{22.1 Chern-Forms and Winding Numbers}\dotfill \pageref*{221_chernforms_and_winding_numbers} \linebreak
\noindent\hyperlink{222_homotopies_and_extensions}{22.2 Homotopies and Extensions}\dotfill \pageref*{222_homotopies_and_extensions} \linebreak
\noindent\hyperlink{223_the_higher_homotopy_groups_}{22.3 The Higher Homotopy Groups $\pi_k(M)$}\dotfill \pageref*{223_the_higher_homotopy_groups_} \linebreak
\noindent\hyperlink{224_some_computations_of_homotopy_groups}{22.4 Some Computations of Homotopy Groups}\dotfill \pageref*{224_some_computations_of_homotopy_groups} \linebreak
\noindent\hyperlink{225_chern_forms_as_obstructions}{22.5 Chern Forms as Obstructions}\dotfill \pageref*{225_chern_forms_as_obstructions} \linebreak
\noindent\hyperlink{appendix_a_forms_in_continuum_mechanics}{Appendix A. Forms in Continuum Mechanics}\dotfill \pageref*{appendix_a_forms_in_continuum_mechanics} \linebreak
\noindent\hyperlink{appendix_b_harmonic_chains_and_kirchhoffs_circuit_law}{Appendix B. Harmonic Chains and Kirchhoff's Circuit Law}\dotfill \pageref*{appendix_b_harmonic_chains_and_kirchhoffs_circuit_law} \linebreak
\noindent\hyperlink{appendix_c_symmetries_quarks_and_meson_masses}{Appendix C. Symmetries, Quarks, and Meson Masses}\dotfill \pageref*{appendix_c_symmetries_quarks_and_meson_masses} \linebreak
\noindent\hyperlink{appendix_d_representations_and_hyperelastic_bodies}{Appendix D. Representations and Hyperelastic Bodies}\dotfill \pageref*{appendix_d_representations_and_hyperelastic_bodies} \linebreak
\noindent\hyperlink{appendix_e_orbits_and_morsebott_theory_in_compact_lie_groups}{Appendix E. Orbits and Morse-Bott Theory in Compact Lie Groups}\dotfill \pageref*{appendix_e_orbits_and_morsebott_theory_in_compact_lie_groups} \linebreak


\hypertarget{i_manifolds_tensors_and_exterior_forms}{}\subsection*{{I Manifolds, Tensors and Exterior Forms}}\label{i_manifolds_tensors_and_exterior_forms}

\hypertarget{1_manifolds_and_vector_fields}{}\subsubsection*{{1 Manifolds and Vector Fields}}\label{1_manifolds_and_vector_fields}

\hypertarget{11_submanifolds_of_euclidean_space}{}\paragraph*{{1.1 Submanifolds of Euclidean space}}\label{11_submanifolds_of_euclidean_space}

\hypertarget{11a_submanifolds_of_}{}\paragraph*{{1.1a Submanifolds of $\mathbb{R}^N$}}\label{11a_submanifolds_of_}

\begin{itemize}%
\item [[Cartesian space]]


\item [[submanifold]]



\end{itemize}
\hypertarget{11b_the_geometry_of_jacobian_matrices_the_differential}{}\paragraph*{{1.1b The Geometry of Jacobian Matrices: The ``Differential''}}\label{11b_the_geometry_of_jacobian_matrices_the_differential}

\hypertarget{11c_the_main_theorem_on_submanifolds_of_}{}\paragraph*{{1.1c The main theorem on submanifolds of $\mathbb{R}^N$}}\label{11c_the_main_theorem_on_submanifolds_of_}

\begin{itemize}%
\item [[Whitney embedding theorem]]

\end{itemize}
\hypertarget{11d_a_nontrivial_example_the_configuration_space_of_a_rigid_body}{}\paragraph*{{1.1d A Nontrivial Example: The Configuration Space of a Rigid Body}}\label{11d_a_nontrivial_example_the_configuration_space_of_a_rigid_body}

\begin{itemize}%
\item [[rigid body]]

\end{itemize}
\hypertarget{12_manifolds}{}\paragraph*{{1.2 Manifolds}}\label{12_manifolds}

\begin{itemize}%
\item [[manifold]]


\item [[smooth manifold]]



\end{itemize}
\hypertarget{13_tangent_vectors_and_mappings}{}\paragraph*{{1.3 Tangent Vectors and Mappings}}\label{13_tangent_vectors_and_mappings}

\begin{itemize}%
\item [[tangent bundle]]


\item [[coordinates]]



\end{itemize}
\hypertarget{14_vector_fields_and_flows}{}\paragraph*{{1.4 Vector Fields and Flows}}\label{14_vector_fields_and_flows}

\begin{itemize}%
\item [[vector field]]


\item [[flow]]



\end{itemize}
\hypertarget{2_tensors_and_exterior_forms}{}\subsubsection*{{2 Tensors and Exterior Forms}}\label{2_tensors_and_exterior_forms}

\hypertarget{21_covectors_and_riemannian_metrics}{}\paragraph*{{2.1 Covectors and Riemannian Metrics}}\label{21_covectors_and_riemannian_metrics}

\begin{itemize}%
\item [[differential form]]


\item [[Riemannian metric]]



\end{itemize}
\hypertarget{22_the_tangent_bundle}{}\paragraph*{{2.2 The Tangent Bundle}}\label{22_the_tangent_bundle}

\begin{itemize}%
\item [[tangent bundle]]

\end{itemize}
\hypertarget{23_the_cotangent_bundle_and_phase_space}{}\paragraph*{{2.3 The Cotangent Bundle and Phase Space}}\label{23_the_cotangent_bundle_and_phase_space}

\begin{itemize}%
\item [[cotangent bundle]]


\item [[phase space]]



\end{itemize}
\hypertarget{24_tensors}{}\paragraph*{{2.4 Tensors}}\label{24_tensors}

\begin{itemize}%
\item [[tensor]]

\end{itemize}
\hypertarget{25_the_grassmann_or_exterior_algebra}{}\paragraph*{{2.5 The Grassmann or Exterior Algebra}}\label{25_the_grassmann_or_exterior_algebra}

\begin{itemize}%
\item [[Grassmann algebra]]


\item [[exterior algebra]]



\end{itemize}
\hypertarget{26_exterior_differentiation}{}\paragraph*{{2.6 Exterior Differentiation}}\label{26_exterior_differentiation}

\begin{itemize}%
\item [[de Rham differential]]

\end{itemize}
\hypertarget{27_pullbacks}{}\paragraph*{{2.7 Pull-Backs}}\label{27_pullbacks}

\begin{itemize}%
\item [[pullback of a differential form]]

\end{itemize}
\hypertarget{28_orientation_and_pseudoforms}{}\paragraph*{{2.8 Orientation and Pseudoforms}}\label{28_orientation_and_pseudoforms}

\begin{itemize}%
\item [[orientation]]


\item [[pseudoform]]



\end{itemize}
\hypertarget{29_interior_products_and_vector_analysis}{}\paragraph*{{2.9 Interior Products and Vector Analysis}}\label{29_interior_products_and_vector_analysis}

\hypertarget{210_dictionary}{}\paragraph*{{2.10 Dictionary}}\label{210_dictionary}

\hypertarget{3_integration_of_differential_forms}{}\subsubsection*{{3 Integration of Differential Forms}}\label{3_integration_of_differential_forms}

\begin{itemize}%
\item [[integration]]


\item [[integration of differential forms]]



\end{itemize}
\hypertarget{31_integration_over_a_parameterized_subset}{}\paragraph*{{3.1 Integration over a Parameterized Subset}}\label{31_integration_over_a_parameterized_subset}

\hypertarget{32_integration_over_manifolds_with_boundary}{}\paragraph*{{3.2 Integration over Manifolds with Boundary}}\label{32_integration_over_manifolds_with_boundary}

\hypertarget{33_stokes_theorem}{}\paragraph*{{3.3 Stokes' Theorem}}\label{33_stokes_theorem}

\begin{itemize}%
\item [[Stokes' theorem]]

\end{itemize}
\hypertarget{34_integration_of_pseudoforms}{}\paragraph*{{3.4 Integration of Pseudoforms}}\label{34_integration_of_pseudoforms}

\hypertarget{35_maxwells_equations}{}\paragraph*{{3.5 Maxwell's Equations}}\label{35_maxwells_equations}

\begin{itemize}%
\item [[Maxwell's equations]]

\end{itemize}
\hypertarget{4_the_lie_derivative}{}\subsubsection*{{4 The Lie Derivative}}\label{4_the_lie_derivative}

\begin{itemize}%
\item [[Lie derivative]]

\end{itemize}
\hypertarget{41_the_lie_derivative_of_a_vector_field}{}\paragraph*{{4.1 The Lie Derivative of a Vector Field}}\label{41_the_lie_derivative_of_a_vector_field}

\hypertarget{42_the_lie_derivative_of_a_form}{}\paragraph*{{4.2 The Lie Derivative of a Form}}\label{42_the_lie_derivative_of_a_form}

\hypertarget{43_differentiation_of_integrals}{}\paragraph*{{4.3 Differentiation of Integrals}}\label{43_differentiation_of_integrals}

\hypertarget{44_a_problem_set_on_hamiltonian_mechanics}{}\paragraph*{{4.4 A Problem Set on Hamiltonian Mechanics}}\label{44_a_problem_set_on_hamiltonian_mechanics}

\begin{itemize}%
\item [[Hamiltonian mechanics]]

\end{itemize}
\hypertarget{5_the_poincar_lemma_and_potentials}{}\subsubsection*{{5 The Poincar\'e{} Lemma and Potentials}}\label{5_the_poincar_lemma_and_potentials}

\begin{itemize}%
\item [[electromagnetic potential]]

\end{itemize}
\hypertarget{6_holonomic_and_nonholonomic_constraints}{}\subsubsection*{{6 Holonomic and Nonholonomic Constraints}}\label{6_holonomic_and_nonholonomic_constraints}

\hypertarget{ii_geometry_and_topology}{}\subsection*{{II Geometry and Topology}}\label{ii_geometry_and_topology}

\hypertarget{7__and_minkowski_space}{}\subsubsection*{{7 $\mathbb{R}^3$ and Minkowski Space}}\label{7__and_minkowski_space}

\hypertarget{71_curvature_and_special_relativity}{}\paragraph*{{7.1 Curvature and Special Relativity}}\label{71_curvature_and_special_relativity}

\begin{itemize}%
\item [[curvature]]


\item [[special relativity]]



\end{itemize}
\hypertarget{72_electromagnetism_in_minkowski_space}{}\paragraph*{{7.2 Electromagnetism in Minkowski Space}}\label{72_electromagnetism_in_minkowski_space}

\begin{itemize}%
\item [[electromagnetism]]

\end{itemize}
\hypertarget{8_the_geometry_of_surfaces_in_}{}\subsubsection*{{8 The Geometry of Surfaces in $\mathbb{R}^3$}}\label{8_the_geometry_of_surfaces_in_}

\hypertarget{81_the_first_and_second_fundamental_form}{}\paragraph*{{8.1 The First and Second Fundamental Form}}\label{81_the_first_and_second_fundamental_form}

\hypertarget{82_gaussian_and_mean_curvature}{}\paragraph*{{8.2 Gaussian and Mean Curvature}}\label{82_gaussian_and_mean_curvature}

\hypertarget{83_the_brouwer_degree_of_a_map_a_problem_set}{}\paragraph*{{8.3 The Brouwer Degree of a Map: A Problem Set}}\label{83_the_brouwer_degree_of_a_map_a_problem_set}

\hypertarget{84_area_mean_curvature_and_soap_bubbles}{}\paragraph*{{8.4 Area, Mean Curvature, and Soap Bubbles}}\label{84_area_mean_curvature_and_soap_bubbles}

\begin{itemize}%
\item [[area]]


\item [[mean curvature]]



\end{itemize}
\hypertarget{85_gauss_theorema_egregium}{}\paragraph*{{8.5 Gauss' \emph{Theorema Egregium}}}\label{85_gauss_theorema_egregium}

\begin{itemize}%
\item [[theorema egregium]]

\end{itemize}
\hypertarget{86_geodesics}{}\paragraph*{{8.6 Geodesics}}\label{86_geodesics}

\begin{itemize}%
\item [[geodesic]]

\end{itemize}
\hypertarget{87_the_parallel_displacement_of_levicivita}{}\paragraph*{{8.7 The Parallel Displacement of Levi-Civita}}\label{87_the_parallel_displacement_of_levicivita}

\begin{itemize}%
\item [[parallel transport]]


\item [[Levi-Civita connection]]



\end{itemize}
\hypertarget{9_covariant_differentiation_and_curvature}{}\subsubsection*{{9 Covariant Differentiation and Curvature}}\label{9_covariant_differentiation_and_curvature}

\hypertarget{91_covariant_differentiation}{}\paragraph*{{9.1 Covariant Differentiation}}\label{91_covariant_differentiation}

\begin{itemize}%
\item [[covariant derivative]]

\end{itemize}
\hypertarget{92_the_riemannian_connection}{}\paragraph*{{9.2 The Riemannian Connection}}\label{92_the_riemannian_connection}

\begin{itemize}%
\item [[Levi-Civita connection]]

\end{itemize}
\hypertarget{93_cartans_exterior_covariant_differential}{}\paragraph*{{9.3 Cartan's Exterior Covariant Differential}}\label{93_cartans_exterior_covariant_differential}

\hypertarget{94_change_of_basis_anf_gauge_transformations}{}\paragraph*{{9.4 Change of Basis anf Gauge Transformations}}\label{94_change_of_basis_anf_gauge_transformations}

\begin{itemize}%
\item [[basis]]


\item [[gauge transformation]]



\end{itemize}
\hypertarget{95_the_curvature_forms_in_a_riemannian_manifold}{}\paragraph*{{9.5 The Curvature Forms in a Riemannian Manifold}}\label{95_the_curvature_forms_in_a_riemannian_manifold}

\hypertarget{96_parallel_displacement_and_curvature_on_a_surface}{}\paragraph*{{9.6 Parallel Displacement and Curvature on a Surface}}\label{96_parallel_displacement_and_curvature_on_a_surface}

\hypertarget{97_riemanns_theorem_and_horizontal_distribution}{}\paragraph*{{9.7 Riemann's Theorem and Horizontal Distribution}}\label{97_riemanns_theorem_and_horizontal_distribution}

\hypertarget{10_geodesics}{}\subsubsection*{{10 Geodesics}}\label{10_geodesics}

\hypertarget{101_geodesics_and_jacobian_fields}{}\paragraph*{{10.1 Geodesics and Jacobian Fields}}\label{101_geodesics_and_jacobian_fields}

\begin{itemize}%
\item [[geodesic]]

\end{itemize}
\hypertarget{102_variational_principles_in_mechanics}{}\paragraph*{{10.2 Variational Principles in Mechanics}}\label{102_variational_principles_in_mechanics}

\begin{itemize}%
\item [[variational calculus]]


\item [[Euler-Lagrange equations]]



\end{itemize}
\hypertarget{103_geodesics_spiders_and_the_universe}{}\paragraph*{{10.3 Geodesics, Spiders, and the Universe}}\label{103_geodesics_spiders_and_the_universe}

\begin{itemize}%
\item [[observable universe]]

\end{itemize}
\hypertarget{11_relativity_tensors_and_curvature}{}\subsubsection*{{11 Relativity, Tensors, and Curvature}}\label{11_relativity_tensors_and_curvature}

\hypertarget{111_heuristics_of_einsteins_theory}{}\paragraph*{{11.1 Heuristic's of Einstein's Theory}}\label{111_heuristics_of_einsteins_theory}

\begin{itemize}%
\item [[general covariance]]

\end{itemize}
\hypertarget{112_tensor_analysis}{}\paragraph*{{11.2 Tensor analysis}}\label{112_tensor_analysis}

\hypertarget{113_hilberts_action_principle}{}\paragraph*{{11.3 Hilbert's Action Principle}}\label{113_hilberts_action_principle}

\begin{itemize}%
\item [[Einstein-Hilbert action]]

\end{itemize}
\hypertarget{114_the_second_fundamental_form_in_the_riemannian_case}{}\paragraph*{{11.4 The Second Fundamental Form in the Riemannian Case}}\label{114_the_second_fundamental_form_in_the_riemannian_case}

\hypertarget{115_the_geometry_of_einsteins_equations}{}\paragraph*{{11.5 The Geometry of Einstein's Equations}}\label{115_the_geometry_of_einsteins_equations}

\begin{itemize}%
\item [[Einstein's equations]]

\end{itemize}
\hypertarget{12_curvature_and_topology_synges_theorem}{}\subsubsection*{{12 Curvature and Topology: Synge's Theorem}}\label{12_curvature_and_topology_synges_theorem}

\hypertarget{13_betti_numbers_and_de_rhams_theorem}{}\subsubsection*{{13 Betti Numbers and De Rham's Theorem}}\label{13_betti_numbers_and_de_rhams_theorem}

\hypertarget{131_singular_chains_and_their_boundaries}{}\paragraph*{{13.1 Singular Chains and Their Boundaries}}\label{131_singular_chains_and_their_boundaries}

\begin{itemize}%
\item [[singular chain complex]]

\end{itemize}
\hypertarget{132_the_singular_homology_groups}{}\paragraph*{{13.2 The Singular Homology Groups}}\label{132_the_singular_homology_groups}

\begin{itemize}%
\item [[singular homology]]

\end{itemize}
\hypertarget{133_homology_groups_of_familiar_manifolds}{}\paragraph*{{13.3 Homology Groups of Familiar Manifolds}}\label{133_homology_groups_of_familiar_manifolds}

\begin{itemize}%
\item [[Betti number]]

\end{itemize}
\hypertarget{134_de_rhams_theorem}{}\paragraph*{{13.4 De Rham's Theorem}}\label{134_de_rhams_theorem}

\begin{itemize}%
\item [[de Rham theorem]]

\end{itemize}
\hypertarget{14_harmonic_forms}{}\subsubsection*{{14 Harmonic Forms}}\label{14_harmonic_forms}

\hypertarget{141_the_hodge_operators}{}\paragraph*{{14.1 The Hodge Operators}}\label{141_the_hodge_operators}

\begin{itemize}%
\item [[Hodge star operator]]

\end{itemize}
\hypertarget{142_harmonic_forms}{}\paragraph*{{14.2 Harmonic Forms}}\label{142_harmonic_forms}

\begin{itemize}%
\item [[harmonic differential form]]

\end{itemize}
\hypertarget{143_boundary_values_relative_homology_and_morse_theory}{}\paragraph*{{14.3 Boundary Values, Relative Homology, and Morse Theory}}\label{143_boundary_values_relative_homology_and_morse_theory}

\begin{itemize}%
\item [[relative homology]]


\item [[Morse theory]]



\end{itemize}
\hypertarget{ii_lie_groups_bundles_and_chern_forms}{}\subsection*{{II Lie Groups, Bundles and Chern Forms}}\label{ii_lie_groups_bundles_and_chern_forms}

\hypertarget{15_lie_groups}{}\subsubsection*{{15 Lie Groups}}\label{15_lie_groups}

\hypertarget{151_lie_groups_invariant_vector_fields_and_forms}{}\paragraph*{{15.1 Lie Groups, Invariant Vector Fields and Forms}}\label{151_lie_groups_invariant_vector_fields_and_forms}

\begin{itemize}%
\item [[Lie group]]


\item [[Maurer-Cartan form]]



\end{itemize}
\hypertarget{152_one_parameter_subgroups}{}\paragraph*{{15.2 One Parameter Subgroups}}\label{152_one_parameter_subgroups}

\hypertarget{153_the_lie_algebra_of_a_lie_group}{}\paragraph*{{15.3 The Lie Algebra of a Lie Group}}\label{153_the_lie_algebra_of_a_lie_group}

\begin{itemize}%
\item [[Lie algebra]]


\item [[Lie theory]]



\end{itemize}
\hypertarget{154_subgroups_and_subalgebras}{}\paragraph*{{15.4 Subgroups and Subalgebras}}\label{154_subgroups_and_subalgebras}

\begin{itemize}%
\item [[subgroup]]

\end{itemize}
\hypertarget{16_vector_bundles_in_geometry_and_physics}{}\subsubsection*{{16 Vector Bundles in Geometry and Physics}}\label{16_vector_bundles_in_geometry_and_physics}

\hypertarget{161_vector_bundles}{}\paragraph*{{16.1 Vector Bundles}}\label{161_vector_bundles}

\begin{itemize}%
\item [[vector bundle]]

\end{itemize}
\hypertarget{162_poincars_theorem_and_the_euler_characteristic}{}\paragraph*{{16.2 Poincar\'e{}`s Theorem and the Euler Characteristic}}\label{162_poincars_theorem_and_the_euler_characteristic}

\begin{itemize}%
\item [[Euler characteristic]]

\end{itemize}
\hypertarget{163_connections_in_a_vector_bundle}{}\paragraph*{{16.3 Connections in a Vector Bundle}}\label{163_connections_in_a_vector_bundle}

\begin{itemize}%
\item [[connection on a vector bundle]]

\end{itemize}
\hypertarget{164_the_electromagnetic_connection}{}\paragraph*{{16.4 The Electromagnetic Connection}}\label{164_the_electromagnetic_connection}

\begin{itemize}%
\item [[line bundle with connection]]

\end{itemize}
\hypertarget{fiber_bundles_gaussbonnet_and_topological_quantization}{}\subsubsection*{{Fiber Bundles, Gauss-Bonnet, and Topological Quantization}}\label{fiber_bundles_gaussbonnet_and_topological_quantization}

\hypertarget{171_fiber_bundles_and_principal_bundles}{}\paragraph*{{17.1 Fiber Bundles and Principal Bundles}}\label{171_fiber_bundles_and_principal_bundles}

\begin{itemize}%
\item [[fiber bundle]]


\item [[principal bundle]]



\end{itemize}
\hypertarget{172_coset_spaces}{}\paragraph*{{17.2 Coset Spaces}}\label{172_coset_spaces}

\begin{itemize}%
\item [[coset]]

\end{itemize}
\hypertarget{173_cherns_proof_of_the_gaussbonnetpoincar_theorem}{}\paragraph*{{17.3 Chern's Proof of the Gauss-Bonnet-Poincar\'e{} Theorem}}\label{173_cherns_proof_of_the_gaussbonnetpoincar_theorem}

\begin{itemize}%
\item [[Gauss-Bonnet theorem]]

\end{itemize}
\hypertarget{174_line_bundles_topological_quantization_and_berry_phase}{}\paragraph*{{17.4 Line Bundles, Topological Quantization, and Berry Phase}}\label{174_line_bundles_topological_quantization_and_berry_phase}

\begin{itemize}%
\item [[line bundle]]


\item [[Berry phase]]



\end{itemize}
\hypertarget{18_connections_and_associated_bundles}{}\subsubsection*{{18 Connections and Associated Bundles}}\label{18_connections_and_associated_bundles}

\hypertarget{181_forms_with_values_in_a_lie_algebra}{}\paragraph*{{18.1 Forms with Values in a Lie Algebra}}\label{181_forms_with_values_in_a_lie_algebra}

\begin{itemize}%
\item [[Lie algebra valued 1-form]]

\end{itemize}
\hypertarget{182_associated_bundles_and_connections}{}\paragraph*{{18.2 Associated Bundles and Connections}}\label{182_associated_bundles_and_connections}

\begin{itemize}%
\item [[associated bundle]]


\item [[connection on a bundle]]



\end{itemize}
\hypertarget{183_form_sections_of_a_vector_bundle_curvature}{}\paragraph*{{18.3 $r$-Form Sections of a Vector Bundle: Curvature}}\label{183_form_sections_of_a_vector_bundle_curvature}

\begin{itemize}%
\item [[section]]


\item [[curvature]]



\end{itemize}
\hypertarget{19_the_dirac_equation}{}\subsubsection*{{19 The Dirac Equation}}\label{19_the_dirac_equation}

\hypertarget{191_the_groups__and_}{}\paragraph*{{19.1 The Groups $SO(3)$ and $SU(2)$}}\label{191_the_groups__and_}

\begin{itemize}%
\item [[special orthogonal group]]


\item [[special unitary group]]



\end{itemize}
\hypertarget{192_hamilton_clifford_and_dirac}{}\paragraph*{{19.2 Hamilton, Clifford, and Dirac}}\label{192_hamilton_clifford_and_dirac}

\begin{itemize}%
\item [[Clifford algebra]]


\item [[spinor]]



\end{itemize}
\hypertarget{193_the_dirac_algebra}{}\paragraph*{{19.3 The Dirac Algebra}}\label{193_the_dirac_algebra}

\begin{itemize}%
\item [[Lorentz group]]

\end{itemize}
\hypertarget{194_the_dirac_operator_in_minkowski_space}{}\paragraph*{{19.4 The Dirac Operator in Minkowski Space}}\label{194_the_dirac_operator_in_minkowski_space}

\begin{itemize}%
\item [[Dirac operator]]

\end{itemize}
\hypertarget{195_the_dirac_operator_in_curved_spacetime}{}\paragraph*{{19.5 The Dirac Operator in Curved Space-Time}}\label{195_the_dirac_operator_in_curved_spacetime}

\hypertarget{20_yangmills_fields}{}\subsubsection*{{20 Yang-Mills Fields}}\label{20_yangmills_fields}

\hypertarget{201_noethers_theorem_for_internal_symmetries}{}\paragraph*{{20.1 Noether's Theorem for Internal Symmetries}}\label{201_noethers_theorem_for_internal_symmetries}

\begin{itemize}%
\item [[Noether's theorem]]

\end{itemize}
\hypertarget{202_weyls_gauge_invariance_revisited}{}\paragraph*{{20.2 Weyl's Gauge Invariance Revisited}}\label{202_weyls_gauge_invariance_revisited}

\begin{itemize}%
\item [[gauge invariance]]

\end{itemize}
\hypertarget{203_the_yangmills_nucleon}{}\paragraph*{{20.3 The Yang-Mills Nucleon}}\label{203_the_yangmills_nucleon}

\begin{itemize}%
\item [[nucleon]]

\end{itemize}
\hypertarget{204_compact_groups_and_yangmills_action}{}\paragraph*{{20.4 Compact Groups and Yang-Mills Action}}\label{204_compact_groups_and_yangmills_action}

\begin{itemize}%
\item [[compact Lie group]]


\item [[Yang-Mills theory]]



\end{itemize}
\hypertarget{205_the_yangmills_equation}{}\paragraph*{{20.5 The Yang-Mills Equation}}\label{205_the_yangmills_equation}

\begin{itemize}%
\item [[Yang-Mills equation]]

\end{itemize}
\hypertarget{206_the_yangmills_instanton}{}\paragraph*{{20.6 The Yang-Mills Instanton}}\label{206_the_yangmills_instanton}

\begin{itemize}%
\item [[Yang-Mills instanton]]

\end{itemize}
\hypertarget{21_betti_numbers_and_covering_spaces}{}\subsubsection*{{21 Betti Numbers and Covering Spaces}}\label{21_betti_numbers_and_covering_spaces}

\hypertarget{211_biinvariant_forms_on_compact_groups}{}\paragraph*{{21.1 Bi-invariant Forms on Compact Groups}}\label{211_biinvariant_forms_on_compact_groups}

\hypertarget{212_the_fundamental_group_and_covering_spaces}{}\paragraph*{{21.2 The Fundamental Group and Covering Spaces}}\label{212_the_fundamental_group_and_covering_spaces}

\begin{itemize}%
\item [[fundamental group]]


\item [[covering space]]


\item [[universal covering space]]



\end{itemize}
\hypertarget{213_the_theorem_of_sb_myers_a_problem_set}{}\paragraph*{{21.3 The Theorem of S.B. Myers: A Problem Set}}\label{213_the_theorem_of_sb_myers_a_problem_set}

\hypertarget{214_the_geometry_of_a_lie_group}{}\paragraph*{{21.4 The Geometry of a Lie Group}}\label{214_the_geometry_of_a_lie_group}

\begin{itemize}%
\item [[Lie group]]

\end{itemize}
\hypertarget{22_chern_forms_and_homotopy_groups}{}\subsubsection*{{22 Chern Forms and Homotopy Groups}}\label{22_chern_forms_and_homotopy_groups}

\hypertarget{221_chernforms_and_winding_numbers}{}\paragraph*{{22.1 Chern-Forms and Winding Numbers}}\label{221_chernforms_and_winding_numbers}

\begin{itemize}%
\item [[invariant polynomial]]


\item [[Chern form]]



\end{itemize}
\hypertarget{222_homotopies_and_extensions}{}\paragraph*{{22.2 Homotopies and Extensions}}\label{222_homotopies_and_extensions}

\begin{itemize}%
\item [[homotopy]]


\item [[extension]]



\end{itemize}
\hypertarget{223_the_higher_homotopy_groups_}{}\paragraph*{{22.3 The Higher Homotopy Groups $\pi_k(M)$}}\label{223_the_higher_homotopy_groups_}

\begin{itemize}%
\item [[homotopy group]]

\end{itemize}
\hypertarget{224_some_computations_of_homotopy_groups}{}\paragraph*{{22.4 Some Computations of Homotopy Groups}}\label{224_some_computations_of_homotopy_groups}

\hypertarget{225_chern_forms_as_obstructions}{}\paragraph*{{22.5 Chern Forms as Obstructions}}\label{225_chern_forms_as_obstructions}

\begin{itemize}%
\item [[obstruction]]

\end{itemize}
\hypertarget{appendix_a_forms_in_continuum_mechanics}{}\subsubsection*{{Appendix A. Forms in Continuum Mechanics}}\label{appendix_a_forms_in_continuum_mechanics}

\hypertarget{appendix_b_harmonic_chains_and_kirchhoffs_circuit_law}{}\subsubsection*{{Appendix B. Harmonic Chains and Kirchhoff's Circuit Law}}\label{appendix_b_harmonic_chains_and_kirchhoffs_circuit_law}

\hypertarget{appendix_c_symmetries_quarks_and_meson_masses}{}\subsubsection*{{Appendix C. Symmetries, Quarks, and Meson Masses}}\label{appendix_c_symmetries_quarks_and_meson_masses}

\hypertarget{appendix_d_representations_and_hyperelastic_bodies}{}\subsubsection*{{Appendix D. Representations and Hyperelastic Bodies}}\label{appendix_d_representations_and_hyperelastic_bodies}

\hypertarget{appendix_e_orbits_and_morsebott_theory_in_compact_lie_groups}{}\subsubsection*{{Appendix E. Orbits and Morse-Bott Theory in Compact Lie Groups}}\label{appendix_e_orbits_and_morsebott_theory_in_compact_lie_groups}

category: reference



\end{document}