8/10/2019 Legendre transformation.pdf

1/9

http://en.wikipedia.org/w/index.php?title=Convex_functionhttp://en.wikipedia.org/w/index.php?title=Hamiltonian_mechanicshttp://en.wikipedia.org/w/index.php?title=Thermodynamicshttp://en.wikipedia.org/w/index.php?title=Legendre-Fenchel_transformationhttp://en.wikipedia.org/w/index.php?title=Duality_%28projective_geometry%29http://en.wikipedia.org/w/index.php?title=Integral_transformhttp://en.wikipedia.org/w/index.php?title=Fourier_transformhttp://en.wikipedia.org/w/index.php?title=Functional_equationhttp://en.wikipedia.org/w/index.php?title=Convex_functionhttp://en.wikipedia.org/w/index.php?title=Graph_of_a_functionhttp://en.wikipedia.org/w/index.php?title=Tangent_linehttp://en.wikipedia.org/w/index.php?title=Y-intercepthttp://en.wikipedia.org/w/index.php?title=Differentiablehttp://en.wikipedia.org/w/index.php?title=Supremumhttp://en.wikipedia.org/w/index.php?title=Convex_functionhttp://en.wikipedia.org/w/index.php?title=Convex_functionhttp://en.wikipedia.org/w/index.php?title=Real_variablehttp://en.wikipedia.org/w/index.php?title=Function_%28mathematics%29http://en.wikipedia.org/w/index.php?title=Real_numberhttp://en.wikipedia.org/w/index.php?title=List_of_transformshttp://en.wikipedia.org/w/index.php?title=Adrien-Marie_Legendrehttp://en.wikipedia.org/w/index.php?title=Adrien-Marie_Legendrehttp://en.wikipedia.org/w/index.php?title=Mathematicshttp://en.wikipedia.org/w/index.php?title=File%3ALegendreTransform1.png

8/10/2019 Legendre transformation.pdf

2/9

http://en.wikipedia.org/w/index.php?title=Integration_by_partshttp://en.wikipedia.org/w/index.php?title=Inverse_functionhttp://en.wikipedia.org/w/index.php?title=Derivative

8/10/2019 Legendre transformation.pdf

3/9

http://en.wikipedia.org/w/index.php?title=Symmetryhttp://en.wikipedia.org/w/index.php?title=Anglehttp://en.wikipedia.org/w/index.php?title=Cartesian_coordinate_systemhttp://en.wikipedia.org/w/index.php?title=Calculationhttp://en.wikipedia.org/w/index.php?title=Theoryhttp://en.wikipedia.org/w/index.php?title=Regular_Lagrangianhttp://en.wikipedia.org/w/index.php?title=Momentumhttp://en.wikipedia.org/w/index.php?title=Velocityhttp://en.wikipedia.org/w/index.php?title=Coordinate_systemhttp://en.wikipedia.org/w/index.php?title=Lagrangianhttp://en.wikipedia.org/w/index.php?title=Lagrangian_mechanicshttp://en.wikipedia.org/w/index.php?title=Hamiltonian_mechanicshttp://en.wikipedia.org/w/index.php?title=Classical_mechanicshttp://en.wikipedia.org/w/index.php?title=Thermodynamic_temperaturehttp://en.wikipedia.org/w/index.php?title=Gibbs_energyhttp://en.wikipedia.org/w/index.php?title=Helmholtz_energyhttp://en.wikipedia.org/w/index.php?title=Thermodynamic_free_energyhttp://en.wikipedia.org/w/index.php?title=Pressurehttp://en.wikipedia.org/w/index.php?title=Intensive_quantityhttp://en.wikipedia.org/w/index.php?title=Enthalpyhttp://en.wikipedia.org/w/index.php?title=Temperaturehttp://en.wikipedia.org/w/index.php?title=Entropyhttp://en.wikipedia.org/w/index.php?title=Helmholtz_free_energyhttp://en.wikipedia.org/w/index.php?title=Chemical_compositionhttp://en.wikipedia.org/w/index.php?title=Volumehttp://en.wikipedia.org/w/index.php?title=Entropyhttp://en.wikipedia.org/w/index.php?title=Extensive_quantityhttp://en.wikipedia.org/w/index.php?title=Internal_energy

8/10/2019 Legendre transformation.pdf

4/9

http://en.wikipedia.org/w/index.php?title=Capacitancehttp://en.wikipedia.org/w/index.php?title=Linearhttp://en.wikipedia.org/w/index.php?title=Battery_%28electricity%29http://en.wikipedia.org/w/index.php?title=Volthttp://en.wikipedia.org/w/index.php?title=Electrostaticshttp://en.wikipedia.org/w/index.php?title=Gradienthttp://en.wikipedia.org/w/index.php?title=Electric_chargehttp://en.wikipedia.org/w/index.php?title=Capacitancehttp://en.wikipedia.org/w/index.php?title=Capacitancehttp://en.wikipedia.org/w/index.php?title=Pistonhttp://en.wikipedia.org/w/index.php?title=Forcehttp://en.wikipedia.org/w/index.php?title=Cylinder_%28engine%29http://en.wikipedia.org/w/index.php?title=Gashttp://en.wikipedia.org/w/index.php?title=Forcehttp://en.wikipedia.org/w/index.php?title=Capacitorhttp://en.wikipedia.org/w/index.php?title=Physics

8/10/2019 Legendre transformation.pdf

5/9

http://en.wikipedia.org/w/index.php?title=Integration_by_partshttp://en.wikipedia.org/w/index.php?title=Substitution_rulehttp://en.wikipedia.org/w/index.php?title=Fundamental_theorem_of_calculushttp://en.wikipedia.org/w/index.php?title=Matrix_%28mathematics%29http://en.wikipedia.org/w/index.php?title=Invertible_matrixhttp://en.wikipedia.org/w/index.php?title=Symmetric_matrixhttp://en.wikipedia.org/w/index.php?title=Quadratic_formhttp://en.wikipedia.org/w/index.php?title=Domain_%28mathematics%29http://en.wikipedia.org/w/index.php?title=Exponential_functionhttp://en.wikipedia.org/w/index.php?title=File%3ALegendreExample.svghttp://en.wikipedia.org/w/index.php?title=Moment_generating_functionhttp://en.wikipedia.org/w/index.php?title=Large_deviations_theory

8/10/2019 Legendre transformation.pdf

6/9

http://en.wikipedia.org/w/index.php?title=Cotangent_bundlehttp://en.wikipedia.org/w/index.php?title=Cotangent_spacehttp://en.wikipedia.org/w/index.php?title=Dual_spacehttp://en.wikipedia.org/w/index.php?title=Real_vector_spacehttp://maze5.net/?page_id=733http://en.wikipedia.org/w/index.php?title=Supporting_hyperplanehttp://en.wikipedia.org/w/index.php?title=Epigraph_%28mathematics%29http://en.wikipedia.org/w/index.php?title=Convex_hullhttp://en.wikipedia.org/w/index.php?title=Scalar_producthttp://en.wikipedia.org/w/index.php?title=Gradienthttp://en.wikipedia.org/w/index.php?title=Open_sethttp://en.wikipedia.org/w/index.php?title=Envelope_%28mathematics%29http://en.wikipedia.org/w/index.php?title=Indexed_familyhttp://en.wikipedia.org/w/index.php?title=Y-intercepthttp://en.wikipedia.org/w/index.php?title=Slopehttp://en.wikipedia.org/w/index.php?title=Derivativehttp://en.wikipedia.org/w/index.php?title=Countable_sethttp://en.wikipedia.org/w/index.php?title=Tangenthttp://en.wikipedia.org/w/index.php?title=Graph_of_a_functionhttp://en.wikipedia.org/w/index.php?title=Strictly_convex_function

8/10/2019 Legendre transformation.pdf

7/9

http://en.wikipedia.org/w/index.php?title=Vladimir_Igorevich_Arnol%27dhttp://en.wikipedia.org/w/index.php?title=If_and_only_ifhttp://en.wikipedia.org/w/index.php?title=Orthogonal_matrixhttp://en.wikipedia.org/w/index.php?title=Adjoint_operatorhttp://en.wikipedia.org/w/index.php?title=Linear_transformationhttp://en.wikipedia.org/w/index.php?title=Homogeneous_function

8/10/2019 Legendre transformation.pdf

8/9

http://www.maths.qmul.ac.uk/~ht/archive/convex1.pdfhttp://www.maths.qmw.ac.uk/~ht/archive/lfth2.pdfhttp://maze5.net/?page_id=733http://maze5.net/?page_id=733http://www.maths.qmul.ac.uk/~ht/archive/convex1.pdfhttp://www.maths.qmw.ac.uk/~ht/archive/lfth2.pdfhttp://en.wikipedia.org/w/index.php?title=R._Tyrrell_Rockafellar

8/10/2019 Legendre transformation.pdf

9/9