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Vector Field Design

Albert Chern

Hairy ball theorem

Hairy ball theorem

Theorem (Hairy ball theorem)Every continuous vector field on a topological sphere musthave zeros (a.k.a. singularities).

(singularity in orientation) & (continuity) forces the vector field dipping to zero

Hairy ball theorem

Theorem (Hairy ball theorem)Every continuous vector field on a topological sphere musthave zeros (a.k.a. singularities).

Keenan Crane 2005

Poincaré–Hopf theorem

Theorem (General hairy ball theorem)Every continuous vector field on a closed surface musthave total index equal to the Euler characteristic.

Definition The index of a singularity is the turning number ofthe vector field around the singularity. That is, it is the number of full (counterclockwise) turns the vector makes when we walk counterclockwise around a small circle around the singularity.

Poincaré index in angle defectRecall:

(K dA)i = 2⇡�<latexit sha1_base64="crtdhnW4ONQ5qAZa6cErHdnzj6I=">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</latexit>

turning angle of a loop around the point i

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(K dA)i<latexit sha1_base64="tbu88BqPOn3S2PSSnF8f36qJM+4=">AAAU/HiczZhbb+PGFceV9Jaql2wav/WFW8NoGtiG6I1vwCZYr62FspEva603izXVxYgcSVzx5pmRVvKA/RR5TZ/6VvS136VAP0zPDDmHpCQGKJCHCBA58/ufc2YO58LLIAl8Llqt/3zw4c9+/otf/uqjXzd/89vf/f7jB5/84RWPp8ylN24cxOz1gHAa+BG9Eb4I6OuEURIOAvrtYHKq9G9nlHE/jl6KRUL7IRlF/tB3iQD05rNvnG3LO/nLW//tg83Wrr1/1Do+tlq7j+xHh0c2FA4eHRzvH1r2bkv/Nhv57+rtJ5/+1/FidxrSSLgB4fzWbiWiLwkTvhvQtOlMOU2IOyEjegvFiISU96XucmptAfGsYczgHwlL07KHJCHni3AAliERY76sKbhOu52K4VFf+lEyFTRys4aG08ASsaXytzyfUVcECygQl/nQV8sdE0ZcAVepEokPuUsC6n0p2JRuQ/FLN4TjtjIH475UjXuU+6Oommzm1pdjGsyoAO2MwgVitAf5xMEzyFcOoOCl8iY1pTCV0TrLkyAZkwEV0lGt5cbZqelE9L0bhyGJPOnEEU1LRtJOK52S8yHkuB35LlWFqkZcFwaRp83m1pbV67RPrq2nl6/bPd0CJzM6iOfSGcYxnJeanfExJUzx273+7e5+v2lZcgsOluVADnNdQqQxL0eTm3tpSRvziZ84771cLoTEG6qOVAEVkC/z59K2NmGyWnY5VBCQRKosy82VDSACo1zEzETV4lJ+459Ifi2V4Y+TH4xx++KsMs7A1M86u7x52m1bT69PTr9pv4TxD8mEEthUBEx4nJ1qXj4joR8s5OX561SeR9l8bady7WQ/j7qEjXLC0xWnfO73qPgBp2zmL7sOKstGOfbGJKHrm5D6OjzeOfjKslDbV/Dxwc5hGR5oeLhzVIaHGh7tHJfhkYbHO3brqwIeA3xst3bsvRK0W5ru7YB7ie41y1etPoNBbQY7T+HSrE1DK2tz0crahLSyNiutrE1NK+vzy6Q8SZhHatpGo4A6T6aRR4dw1/Iyztgqzy/KOWxqZ3CHC32YhdJEyHe7OKFRadZXfiuT6M/2wRfpelrXWt6vvDU3iDn9P5o7tNc1B9QsrViMYWXly08ty8tnK6uw2XTgkjhv3pj9Xb5Rm7ti19fIrg27uEB2YVivh6xn2NUVsivDOh1kHcNOT5GdGvb8ObLnaFfEO8WA1wW8LlopYAfhy3P50nl4bmoXqnZhal1V65raC1V7kdcGJA82lMTEgsCGDZANhwiHCEfIRsh8ZD6yd8jeIZsgmyCbIpsimyGbIXuP7D2yHjKObI5sjmyBbIHsHtk9MoaMIRMFdIQfeFRpRoxQiwoHZAJZgixBdofszjAf7LYdPxYkB3SeR2OhhDLaRW4wVU+oYP4uZ56vhmvby6tunAF1/KvjkdGIMuPtYUzfg5B5fwJ4BvZgaqj7BtdLD3YOzbDL4bQqh8WQwUNduOSsERqAcE9ZjEm3VMtbWpsRBvuHAHcowe0729ceOg9z30LXJ1+YPvN2Fo67TLZNS7xTQFyS/LyA5wh7BezhhYjMoA7CYlSFKKBAU3gwnOTrQRWLFaFq45IwLguiJBSxOiTEYYEy7kOLMEGuKpUmjE/WSOGl6hw9tcoz32Zl275MKIMtlX0uHdhxQx9umPk575XHpcfNDBPSc8IJZdGOvbsfTi1hhPmSMDfCQCmDospUlWF1oaoLrAoti0K/U/U7rCaqmpjqYqlRjHO/JNwb4W5JMJFdlSWsmOhdsY94FOZbxdyhuCw9WAhVDVZCLiVjf0kDgldwvBpWM/TmK94cvUcEnrGXZM2MAVHvPksGmhkDWEmr/cugMcm3gapJvg/ka0HlkO+F+qJgdmuSK3ITKjcH3kP5GL1VcoX7avKiyN4tu+uxqni7Kq8leVwK7q4Gd3XwtesBnmLgAeaapjWqD4/HX4d1Kof3W8lHUZ1OBFELDY57tSESEsHbTCp7UKjvBRml0q/tR0CjEbz0y252rusNoySVJ+pYYzGLg1S+gkONDq/tqezQWt1NUukmdarnc8hTH2ssYEeQalfIhrI9DSjDLVHXzCh/3XuGApQN7naRdrsGnvnMd8cBLW6wZ3ovrnwimFE2ICy9tfsymzpWNnWUoD4gyfKE2rTLtbTsUHlH/jyboFnUvAm5aa+07ntzbZPfpIbaZtkomOBjPSYCLHWiWD+R82WHESPeqoumZaeKD1/vxH/Yy52yYNVJ04oPo2Uvz5+tOilY2073bMlB3c+79fZjyvlqEx1F6y9bksAkdYT6FAQV9cmOqOTyEtwK07cPNs1HP6u+8Gpv1360u/fii80nZ/mXwY8af2z8qfFZw24cNp40Oo2rxk3DbUSN7xrfN/6+8beNf2z8c+NfmemHH+Q+nzYqv41//w95I4ma</latexit>

Poincaré index in angle defect

Index observation from a distance

Total turning angle of v= 2⇡X

singularitiesenclosed

index�Z

regionenclosed

K dA

<latexit sha1_base64="G+FBK1KpVg+lCZzUsfg4v1M4rks=">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</latexit>

Recall:

(K dA)i = 2⇡�<latexit sha1_base64="crtdhnW4ONQ5qAZa6cErHdnzj6I=">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</latexit>

turning angle of a loop around the point i

ä<latexit sha1_base64="1YCtNEF59DFpL6PXSMvvNH1IFRU=">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</latexit>

Ä<latexit sha1_base64="Eb78PIMkcDfdvx1fE7xCfH8c76U=">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</latexit>

Proof of Poincaré–Hopf theorem

Total turning angle of v= 2⇡X

singularitiesenclosed

index�Z

regionenclosed

K dA

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Theorem (General hairy ball theorem)Every continuous vector field on a closed surface musthave total index equal to the Euler characteristic.

Let the enclosed region be the entire surface. ApplyR

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Revising the notion of functionTheorem (General hairy ball theorem)Every continuous vector field on a closed surface musthave total index equal to the Euler characteristic.

• If vector field is like a function over a surface, what preventsus from assigning nonzero value everywhere?

• Unlike a traditional function f : S1! S2<latexit sha1_base64="4tWbB872UwEMIA8fZmn6xWjl3UE=">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</latexit>

a vector field takes value in different spacesu<latexit sha1_base64="9tVgvgrnxmWWXXTeNLM3qLRPmd8=">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</latexit>

u|x 2 TxM<latexit sha1_base64="BSsihvuKjUlrXcw/9JpCSKgeh6k=">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</latexit>

• In fact a more unifying picture to talk about “function” is to look at the function graph.

Revising the notion of functionf : S1! S2

<latexit sha1_base64="4tWbB872UwEMIA8fZmn6xWjl3UE=">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</latexit>

Plotting

Revising the notion of function

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Plotting

canvas for the plot

Revising the notion of function

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Plotting

canvas for the plot

Revising the notion of function

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Plotting a tangent vector field onu<latexit sha1_base64="9tVgvgrnxmWWXXTeNLM3qLRPmd8=">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</latexit>

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canvas for the plot

Revising the notion of function

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Plotting a tangent vector field onu<latexit sha1_base64="9tVgvgrnxmWWXXTeNLM3qLRPmd8=">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</latexit>

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canvas for the plot

Revising the notion of function

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The “canvas” as a (higher-dimensional) manifold is called a bundle. Here we have the tangent bundle .

canvas for the plot

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is called a section of the tangent bundle.

Twistedness of the canvas

Theorem (Hairy ball theorem)

The tangent bundle of a closed surface is twisted in such a way that every pair of sections must have the total signed intersection equals to the Euler characteristic.

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Optimal Sections of a Bundle

Optimal section

Goal Find the optimal section of a bundle.

canvas for the plot

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• Smoothest vector field

• Patterns best align with prescribed directions

• Automatically finds the optimal singularity locations.

Optimal sectionSmoothest tangent vector fields

[Knöppel, Crane, Pinkall & Schröder 2013]Globally optimal direction field

Optimal sectionImpainting n-direction field

[Brandt, Scandolo, Eisemann & Hildebrandt 2018]Modeling n-symmetric vector fields using higher-order energies

Optimal sectionNon-orthogonal cross fields

[Diamanti, Vaxman, Panozzo & Sorkine-Hornung 2014]Designing N-PolyVector fields with complex polynomials

Optimal sectionNon-orthogonal cross fields with physical constraints

[Sageman-Furnas, Chern, Ben-Chen & Vaxman 2019]Chebyshev nets from commuting polyvector field

Optimal sectionN-direction field with physical constraints

[Vekhter, Zhuo, Fandino, Huang & Vouga 2019]Weaving geodesic foliations

Optimal sectionComplex phase field

[Knöppel, Crane, Pinkall & Schröder 2015]Stripe patterns on surfaces

Optimal sectionComplex phase field for vortex detection

[Weißmann, Pinkall & Schröder 2014]Smoke rings from smoke

[Chern, Knöppel, Pinkall & Schröder 2017]Inside fluid: Clebsch maps for visualization and processing

Optimal section or phase field for quad meshing

[Fang, Bao, Tong, Desbrun & Huang 2018]Quadrangulation through Morse-parametrizationhybridization

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Optimal section phase field for nearly conformal deformations

[Chern, Pinkall & Schröder 2015]Close-to-conformal deformation of volumes

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Optimal section spinor field for nearly isometric reconstruction

[Chern, Knöppel, Pinkall & Schröder 2018]Shape from metric

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Optimal sectionFrame field for hexahedral meshing

[Corman & Crane 2019]Symmetric moving frame

Optimal sectionOctahedral variety phase field for hexahedral meshing

[Palmer, Bommes & Solomon 2020]Algebraic representations for volumetric frame fields

Common ingredient: gauge theory

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• Describe the connection(a.k.a. parallel transport)between adjacent value spaces

• Find an algebraic representation of the value space at each .

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• The derivatives (finite differences) of the field respect the connections.

• Model some energy to minimize.

Discrete tangent bundle

Discrete tangent bundleEach value spaceEi

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Discrete tangent bundleEach value spaceEi

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is viewed as a complex plane via an arbitrarybasis.

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gauge

i$ i = ri ei✓i 2 C

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vector field -valued function$ C<latexit sha1_base64="zzMjRZrnAzeoh8CbFM7ItmqAyjw=">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</latexit>

Discrete connection

For each adjacent value spaces

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✓i<latexit sha1_base64="W9gecnaumyu1OiN1zkrFEEqMiG4=">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</latexit>

i$ i = ri ei✓i 2 C

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vector field -valued function$ C<latexit sha1_base64="zzMjRZrnAzeoh8CbFM7ItmqAyjw=">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</latexit>

Discrete connectionFor each adjacent value spaces

i 2 Ei<latexit sha1_base64="yZXtFAd7yFdnYeMrF2BAUEP8pho=">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</latexit>

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ri<latexit sha1_base64="ZUgtIMBj4Zhp+X5dluuqYoakP80=">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</latexit>

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i$ i = ri ei✓i 2 C

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j$ j = rj ei✓ j 2 C

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Discrete connectionFor each adjacent value spaces

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i$ i = ri ei✓i 2 C

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j$ j = rj ei✓ j 2 C

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parallel transported basis

Discrete connectionFor each adjacent points

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i$ i = ri ei✓i 2 C

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1 j 2 Ej<latexit sha1_base64="NmCay2N8y3FGjyuKKsXtvW7LliE=">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</latexit>

j$ j = rj ei✓ j 2 C

<latexit sha1_base64="TYNG4J2oue/kPFoPA6z+RBetafo=">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</latexit>

↵i j<latexit sha1_base64="WJgEyWUoftDUeDGevhvOjhEmBbk=">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</latexit>

parallel transported basis

i, j<latexit sha1_base64="VKLi9gh4B7Y63+WA4D/+IdlcG7E=">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</latexit>

encode the parallel transportby an angle , with↵i j

<latexit sha1_base64="1H06LdcscL0wY7vO2iOB7QdwwyY=">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</latexit>

↵i j = �↵ ji mod 2⇡<latexit sha1_base64="plxzONyGiSWlQBZszxjuhvWkK3M=">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</latexit>

i.e. an angle-valued 1-form.It is called the connectionas an additional structure for the bundle.

Covariant derivativeThe derivative of the section is denoted by .

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i$ i = ri ei✓i 2 C

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j$ j = rj ei✓ j 2 C

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parallel transported basis

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<latexit sha1_base64="QmhZbEyZRQ6P+ojU7x5My4Mf9GE=">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</latexit>

In terms of the basis 1i<latexit sha1_base64="Hpv3MLcpg/JxpaF16VA9a4DwuIk=">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</latexit>

(dr )i j$ e�i↵i j j � i<latexit sha1_base64="6fa71kQVLBu90ojrXd8ooGbS+XY=">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</latexit>

(dr )i j$ j � ei↵i j i<latexit sha1_base64="bwdCnf3YMUWIEzQtXvLii515dTY=">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</latexit>

In terms of the basis 1 j<latexit sha1_base64="inLSXnpSMC+hAFk0OUomxLmrI74=">AAAVEniczZhbb9vIFce129tWvWW7fusLU8NosbAM0buJEyC7SGIr0KbyJVacDWJqgxE5Emnx5pmRInlAoB+hD31pP0rfir72C/SL9LlnhpxDUhIXKNCHChA18/ufc2bO3EhxnIYBF93uvz76+Ac//NGPf/LJT9s/+/kvfvmre5/++g1P5sylV24SJuztmHAaBjG9EoEI6duUURKNQ/rteHas9G8XlPEgiV+LVUpHEZnGwSRwiQD0xkli+v7m/b3d7kFXf6zNgl0UdlvF5+L9p5/92/ESdx7RWLgh4fza7qZiJAkTgRvSrO3MOU2JOyNTeg3FmESUj6TubmbtAfGsScLgGwtL06qHJBHnq2gMlhERPl/XFNymXc/F5NFIBnE6FzR284Ym89ASiaVyt7yAUVeEKygQlwXQV8v1CSOugBGqReIT7pKQel8JNqf7UPzKjeC6r8zBeCRV4x7lwTSuJ5u7jaRPwwUVoJ1QGCBGh5BPEr6AfOUYCl4mrzJTijIZb7N8FqY+GVMhHdVaYZz/tJ2YfnCTKCKxJ9UkZhUjaWe1TsnlBHLcjwOXqkJdI64Lk8jrWZAQUo1EkI7kcpW19/asYb/37NJ6fv62N9Rtc7Kg42QpnUmSwO9ahxbcp4Qpfn04uj54MGpbltyDi2U5kN1SlxBpzKvR5O5hVtF8PgtS54NXyKWQehPVkTqgAkaCBUtpW7u2Beu3GioMSSpVqtXmqgYQgVEuEmaianEtP///JL+uyvB/kx/Mce/spDbPwNTHOjm/ej7oWc8vnx3/ofca5j8iM0rgqBGwFXDdqhX7gkRBuJLnp28zeRrnK7mXya3b4DQeEDYtCM82nIpdMaTie5zyPbHuOq5tKOU49ElKtzch9Tg86Tz82rJQe6Dgk4edoyp8qOFR51EVHmn4qPO4Ch9p+Lhjd78u4WOAT+xuxz6sQLur6WEH3Cv0sF0dteYMxo0ZdJ7D0GxNQytbc9HK1oS0sjUrrWxNTSvb88ulIklYR2rZxtOQOk/nsUcncC/zcs7YJi8G5RSOuxO470UBrEJpIhTnYJLSuLLqa5+NRfQ7++GX2Xba1FrRr6I1N0w4/S+aO7K3NQfUbK1E+LCziu2ntuX5i41d2G47MCTOu3fm5Jfv1LGv2OUlskvDzs6QnRk2HCIbGnZxgezCsH4fWd+w42Nkx4a9fInsJdqV8Y4x4GUJL8tWSthH+PpUvnbun5ramaqdmdpA1Qam9krVXhW1MSmCTSQxsSCwYWNkkwnCCcIpsimyAFmA7AbZDbIZshmyObI5sgWyBbIPyD4gGyLjyJbIlshWyFbI7pDdIWPIGDJRQkcEoUeVZsQYtbh0QCaQpchSZLfIbg0LwG7fCRJBCkCXRTQWSSijXeyGc/XcCuY3BfMCNV37XlF1kxyo63eOR6ZTyoy3hzEDD0IW/QnhydiDpaHuG1xvPTg5NMMuR/O6HJVTBo970ZqzRmgAwh1lCSbdVS3vaW1BGJwfAtyhBLfv/Fy779wvfEtd/wTC9Jn38nDcZbJnWuL9EuKW5KclPEU4LOEQByI2kzqOylkVooQCTeGRcVbsB1Usd4Sq+RXBrwqiIpSx+iTCaYEynkOrKEWuKrUmjE/eSOml6hw9tcpz33bt2D5PKYMjlX0uHThxowBumMVv0SuPS4+bFSak50QzyuKOffAgmlvCCMs1YWmEsVLGZZWpKsPqSlVXWBVaFqV+q+q3WE1VNTXV1VqjGOduTbgzwu2aYCK7KkvYMfFNeY54FNZbzdyhuC092Ah1DXZCIaV+sKYBwRH0N8Nqht58w5uj95TAM/aarJkxIOpf0ZqBZsYAdtJm/3JoTIpjoG5SnAPFXlA5FGehHhTMbktyZW5C5ebAP1Tuo7dKrnTfTF6U2btVdz1XNW9X5bUm+5Xg7mZwVwffuh/gKQYeYC5p1qAG8Hj8TdSkcvjnK/k0btKJIGqjwfWwMURKYvg3k8khFJp7QaaZDBr7EdJ4KvxMDvLfpt4wSjL5TF0bLBZJmMk3cGnQ4Q99Jvu0UXfTTLppk+oFHPLU1wYLOBGkOhXyqezNQ8rwSNQ1M8vfDF+gAGWDBwOkg4GBJwELXD+k5Q32RJ/FtZcHC8rGhGXX9kjmS8fKl44S1GslWV1Qu3a1llUdav+RP88XaB61aELu2hutB95S2xQ3qYm2WTcKZ/hYj4kAy5w40U/kfN1hyoi36aJp1anmw7c78e/3cucs3HTStObDaNXLCxabTgo2tjM4WXNQ9/NBs71POd9soq9o87ClKSxSR6iXRFBRL/OISq4owa0we39v115/L7hZeHN4YH9xcPjqy92nr/6YvzP8pPWb1m9bv2/ZraPW01a/ddG6armtm9afWn9p/XXnzzt/2/n7zj9y048/Kt4zftaqfXb++R+3E5SO</latexit>

The magnitude is basis-choice independent.

|(dr )i j |<latexit sha1_base64="hh6Brjj1EuQKSpLZXHn8a2ko8X0=">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</latexit>

Covariant derivativeThe derivative of the section is denoted by .

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In terms of the basis 1i<latexit sha1_base64="Hpv3MLcpg/JxpaF16VA9a4DwuIk=">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</latexit>

(dr )i j$ e�i↵i j j � i<latexit sha1_base64="6fa71kQVLBu90ojrXd8ooGbS+XY=">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</latexit>

(dr )i j$ j � ei↵i j i<latexit sha1_base64="bwdCnf3YMUWIEzQtXvLii515dTY=">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</latexit>

In terms of the basis 1 j<latexit sha1_base64="inLSXnpSMC+hAFk0OUomxLmrI74=">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</latexit>

The magnitude is basis-choice independent.

|(dr )i j |<latexit sha1_base64="hh6Brjj1EuQKSpLZXHn8a2ko8X0=">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</latexit>

d =

2666664

...�1 1

...

3777775

<latexit sha1_base64="N/LWNRblvidR8K769Rc/hv8SSME=">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</latexit>

dr =

2666664

...�1 e�i↵i j

...

3777775

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j<latexit sha1_base64="crxY2PButeCEJQK1yJgnfOMBsSI=">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</latexit>

traditional derivative

connection derivative

Dirichlet energyThe magnitude is basis-choice independent.

|(dr )i j |<latexit sha1_base64="hh6Brjj1EuQKSpLZXHn8a2ko8X0=">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</latexit>

dr =

2666664

...�1 e�i↵i j

...

3777775

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i j2 halfedges

wi j |(dr )i j |2

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cotan weights

Dirichlet energyThe magnitude is basis-choice independent.

|(dr )i j |<latexit sha1_base64="hh6Brjj1EuQKSpLZXHn8a2ko8X0=">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</latexit>

dr =

2666664

...�1 e�i↵i j

...

3777775

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dr⇤¸ ⇥?1⇤ ⇥

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connection Laplacian| {z }

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Smoothest sectionE[ ] =

¸Lr

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The Dirichlet energy measures the smoothness of the section

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(the smaller, the smoother with respect to the parallel transport)

Ground state problem

min ¸[Lr]

s.t.X

i

(pointarea )i | i |2 = 1

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min

¸[Lr]

¸[?0]

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Lr = � ?0 <latexit sha1_base64="nCMt93atkUUJEIvXGUvgwo/PnMM=">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</latexit>

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smallest eigenvalue problem

Smoothest sectionE[ ] =

¸Lr

<latexit sha1_base64="9myDfU+AkS+6GE+x0V46OTxE4zQ=">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</latexit>

The Dirichlet energy measures the smoothness of the section

<latexit sha1_base64="tVgbUSXjGekpNNrNcla+TCBzLe0=">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</latexit>

(the smaller, the smoother with respect to the parallel transport)Ground state with pointwise norm=1 constraint

min ¸[Lr]

s.t. | i |2 = 1 8 i<latexit sha1_base64="7pdZNJGFMTw9HVMks9Ulmzc/ks4=">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</latexit>

min

¸Lr +

1"2

X

i

� pointweight

�i(| i |2 � 1)2

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Ginzburg–Landau model

Complex line bundle setup summary• The -valued function

describes the geometric object called a sectionC

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j$ j = rj ei✓ j 2 C

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↵i j<latexit sha1_base64="WJgEyWUoftDUeDGevhvOjhEmBbk=">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</latexit>

parallel transported basis

• We need to provide an angle-valued 1-form ↵

<latexit sha1_base64="hf+z9hs3qk7Skltw356CHJFNooo=">AAAVEniczZhbb9vIFce1l7Zb9bLZXb/1hVnDaLGwDNFZ34DsIomtQJvKl1hxNoipBiNyJNHmzTMjRfKAQD9CH/rSfpS+FX3tF+gX6XPPDDmHpCQuUGAfVoComd//nDNz5kaKwyTwuWi3//PBhx99/LOf/+KTXzZ/9evf/PbTB599/prHU+bSKzcOYvZmSDgN/IheCV8E9E3CKAmHAf1+eHus9O9nlHE/jl6JRUIHIRlH/sh3iQD02iFBMiHvHmy2d+y9w/bRkdXeeWQ/Oji0obD/aP9o78Cyd9r6s9nIPxfvPvviv44Xu9OQRsINCOfXdjsRA0mY8N2Apk1nymlC3FsyptdQjEhI+UDq7qbWFhDPGsUMvpGwNC17SBJyvgiHYBkSMeHLmoLrtOupGB0OpB8lU0EjN2toNA0sEVsqd8vzGXVFsIACcZkPfbXcCWHEFTBClUh8xF0SUO8bwaZ0G4rfuCFct5U5GA+katyj3B9H1WQzt4Gc0GBGBWgnFAaI0T7kEwfPIV85hIKXyqvUlMJURussn6q5GVIhHdVabpz9NJ2IvnfjMCSRJ504omnJSNpppVNyPoIctyPfpapQ1YjrwiTyahYkgFRD4ScDOV+kza0tq9/tPL20np2/6fR125zM6DCeS2cUx/C71KEZn1DCFL/eHVzv7A2aliW34GJZDmQ31yVEGvNyNLm5m5a0Cb/1E+e9l8uFkHgj1ZEqoAJGgvlzaVubsIwtuxwqCEgiVarl5soGEIFRLmJmompxKb/JTyS/tsrwx8kP5rhzdlKZZ2DqY52cXz3rdaxnl0+P/9h5BfMfkltK4KgRsBVw3aoV+5yEfrCQ56dvUnkaZSu5k8q12+A06hE2zglPV5zyXdGn4gecsj2x7DqsbCjl2J+QhK5vQupxeNza/9ayUNtT8PF+66AM9zU8aB2W4YGGh62jMjzU8Khlt78t4BHAx3a7Ze+WoN3WdLcF7iW62yyPWn0Gw9oMWs9gaNamoZW1uWhlbUJaWZuVVtamppX1+WVSniSsI7Vso3FAnSfTyKMjuJd5GWdsleeDcgrH3Qnc90IfVqE0EfJzME5oVFr1lc/KIvq9vf91up7WtZb3K2/NDWJO/4/mDux1zQE1WysWE9hZ+fZT2/L8+coubDYdGBLn7Vtz8su36thX7PIS2aVhZ2fIzgzr95H1Dbu4QHZhWLeLrGvY8TGyY8NevED2Au2KeMcY8LKAl0UrBewifHUqXzkPT03tTNXOTK2naj1Te6lqL/PakOTBRpKYWBDYsCGy0QjhCOEY2RiZj8xHdoPsBtktsltkU2RTZDNkM2Tvkb1H1kfGkc2RzZEtkC2Q3SO7R8aQMWSigI7wA48qzYgRalHhgEwgS5AlyO6Q3Rnmg92248eC5IDO82gslFBGu8gNpuq5Fcxvcub5arq2vbzqxhlQ1z85HhmPKTPeHsb0PQiZ9yeAJ2MPloa6b3C99eDk0Ay7HE6rclhMGTzuhUvOGqEBCPeUxZh0W7W8pbUZYXB+CHCHEty+s3PtofMw9y10/eML02feycJxl8mOaYl3C4hbkp8W8BRhv4B9HIjITOowLGZViAIKNIVHxtt8P6hisSNUbVISJmVBlIQiVpeEOC1QxnNoESbIVaXShPHJGim8VJ2jp1Z55tusHNvnCWVwpLKvpAMnbujDDTP/zXvlcelxs8KE9JzwlrKoZe/shVNLGGG+JMyNMFTKsKgyVWVYXajqAqtCy6LQ71T9DquJqiamulhqFOPcLwn3RrhbEkxkV2UJOya6Kc4Rj8J6q5g7FLelBxuhqsFOyKVk4i9pQHAEJ6thNUNvvuLN0XtM4Bl7SdbMGOh/rEsGmhkD2Emr/cugMcmPgapJfg7ke0HlkJ+FelAwuzXJFbkJlZsD/1D5BL1VcoX7avKiyN4tu+u5qni7Kq8leVIK7q4Gd3XwtfsBnmLgAeaSpjWqD4/H34V1Kod/vpKPozqdCKI2Glx3a0MkJIJ/M6nsQ6G+F2ScSr+2HwGNxmKSyl72W9cbRkkqn6prjcUsDlL5Gi41OvyhT2WX1upukko3qVM9n0Oe+lpjASeCVKdCNpWdaUAZHom6Zmb5u/5zFKBscK+HtNcz8MRnvjsJaHGDPdFnceXlwYyyIWHptT2Q2dKxsqWjBPVaSZYX1KZdrqVlh8p/5K+yBZpFzZuQm/ZK67431zb5TWqkbZaNglt8rMdEgKVOFOsncr7sMGbEW3XRtOxU8eHrnfgPe7lTFqw6aVrxYbTs5fmzVScFa9vpnSw5qPt5r95+QjlfbaKraP2wJQksUkeol0RQUS/ziEouL8GtMH33YNO8DrTqC693d+xHO7svv9588vLP2TvDTxq/a3zZ+EPDbhw0njS6jYvGVcNt3DT+0vhb4+8bf934x8Y/N/6VmX74Qf6e8YtG5bPx7/8BoaKU5A==</latexit>

• The Laplacian for is <latexit sha1_base64="092By7j01+qLKLw8leNwyWI8q6I=">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</latexit>

Lr = dr¸?1dr

<latexit sha1_base64="jrU+squYsndUn2uklAi2x75ZtrQ=">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</latexit>

where dr =

2666664

...�1 e�i↵i j

...

3777775

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Effect of connection

If the complex line bundle is the tangent bundle,

then there is a canonical ,↵<latexit sha1_base64="hf+z9hs3qk7Skltw356CHJFNooo=">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</latexit>

known as the Levi-Civita connection, that describes the metric-induced parallel transport.

Effect of connection

↵LeviCivita<latexit sha1_base64="MO0eYoKuZodBSRuqk9gJaja6beE=">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</latexit>

↵customized = ↵LeviCivita + v[<latexit sha1_base64="2hBlR2y6xWzvu0fKrxE95j1Y5JI=">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</latexit>

Levi-Civita connection

�1<latexit sha1_base64="hAaOmKNm2fmaXmAGaHYEztixSIk=">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</latexit>

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‣ In the picture �i j =P4

k=1 �0k

<latexit sha1_base64="+2ApzR/RXrmVygONXvDk73v2CCE=">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</latexit>

� 0k =�kP` �`

2⇡<latexit sha1_base64="sIgAXDBy7dCdMiS5IZKz44IBs3E=">AAAVKXiczZhbb9vIFce12213q96yXWMfdoGCqWFssbAN0alvQLJIYjvQpvIlVpxNY6rGiBxJjHjzzEiRPCDQ79HHPhbod+hr+9S3tuhbv0jPDDmHpCQuUKAPK0DUzO9/zpk5cyOpfhL4XLRa/3zv/e998P0ffPjRD5s/+vFPfvqzex///BWPJ8ylV24cxOx1n3Aa+BG9Er4I6OuEURL2A/pNf3yk9G+mlHE/jl6KeUJ7IRlG/sB3iQB0c2/X6VNBbsZfWI8sRwwYcWVOUunwSXgjHRoEacZ0ccdJ/Jt7661te/egdXhotbYf2A/2D2wo7D3YO9zdt+ztlv6sP9797Z//+Olnv7i4+fiTfzte7E5CGgk3IJxf261E9CRhwncDmjadCacJccdkSK+hGJGQ8p7U+aXWBhDPGsQMvpGwNC17SBJyPg/7YBkSMeKLmoKrtOuJGBz0pB8lE0EjN2toMAksEVtqsCzPZ9QVwRwKxGU+9NVyRwQGScCQViLxAXdJQL1Hgk3oJhQfuSFcN5U5GPekatyj3B9G1WQzt54c0WBKBWjHFAaI0S7kEwfPIF/Zh4KXyqvUlMJURqssnwTJiMBMSUe1lhtnP00nou/cOAxJ5EknjmhaMpJ2WumUnKmFsBn5LlWFqkZcFyaRp83mxobVbZ88ubSenr8+6eoWOJnSfjyTziCO4Xeh2SkfUcIUv97pXW/v9pqWJTfgYlkO5DDTJUQa83I0ub6TlrQRH/uJ887L5UJIvIHqSBVQAfkyfyZtax0Wq2WXQwUBSaTKstxc2QAiMMpFzExULS7kN/qO5NdSGf5/8oM5Pjk7rswzMPWxjs+vnnZOrKeXT45+c/IS5j8kY0rgBBKw4HF1qnX5jIR+MJfnp69TeRpl6/UklSsX+2nUIWyYE54uOeVrv0vFtzhlK3/RtV/ZNsqxOyIJXd2E1OPwcGvvK8tCbVfBh3tb+2W4p+H+1kEZ7mt4sHVYhgcaHm7Zra8KeAjwod3asndK0G5purMF7iW60yyPWn0G/doMtp7C0KxMQysrc9HKyoS0sjIrraxMTSur88ukPElYR2rZRsOAOo8nkUcHcIvzMs7YMs8H5RQOtWO4HYY+rEJpIuSnXZzQqLTqK5+lRfSFvffrdDWtay3vV96aG8Sc/g/N7durmgNqtlYsRrCz8u2ntuX5s6Vd2Gw6MCTOmzfmfJdv1OGu2OUlskvDzs6QnRnW7SLrGnZxgezCsHYbWduwoyNkR4Y9f47sOdoV8Y4w4GUBL4tWCthG+PJUvnTun5ramaqdmVpH1Tqm9kLVXuS1PsmDDSQxsSCwYX1kgwHCAcIhsiEyH5mP7C2yt8jGyMbIJsgmyKbIpsjeIXuHrIuMI5shmyGbI5sju0N2h4whY8hEAR3hBx5VmhEj1KLCAZlAliBLkN0iuzXMB7tNx48FyQGd5dFYKKGMdpEbTNTjLJi/zZnnq+na9PKqG2dAXX/neGQ4pMx4exjT9yBk3p8AHpg9WBrqvsH11oOTQzPscjipymExZfBQFy44a4QGINxRFmPSLdXyhtamhMH5IcAdSnD7zs61+8793LfQ9Y8vTJ/5SRaOu0yemJZ4u4C4JflpAU8RdgvYxYGIzKT2w2JWhSigQFN4MBzn+0EVix2haqOSMCoLoiQUsdokxGmBMp5D8zBBriqVJoxP1kjhpeocPbXKM99m5dg+TyiDI5V9KR04cUMfbpj5b94rj0uPmxUmpOeEY8qiLXt7N5xYwgizBWFmhL5S+kWVqSrD6lxV51gVWhaFfqvqt1hNVDUx1flCoxjnbkG4M8LtgmAiuypL2DHR2+Ic8eBdr2ruUNyWHmyEqgY7IZeSkb+gAcERHC2H1Qy9+ZI3R+8hgWfsBVkzY0DUu8+CgWbGAHbScv8yaEzyY6Bqkp8D+V5QOeRnoR4UzG5FckVuQuXmwHsoH6G3Sq5wX05eFNm7ZXc9VxVvV+W1II9Kwd3l4K4OvnI/wFMMPMBc0rRG9eHx+OuwTuXwfiv5MKrTiSBqo8F1pzZEQiJ4m0llFwr1vSDDVPq1/QhoNISXftnJfut6wyhJ5RN1rbGYxkEqX8GlRofX9lS2aa3uJql0kzrV8znkqa81FnAiSHUqZFN5MgkowyNR18wsf919hgKUDe50kHY6Bh77zHdHAS1usMf6LK78RTClrE9Yem33ZLZ0rGzpKEH92yTLC2rdLtfSskPlHfnLbIFmUfMm5Lq91LrvzbRNfpMaaJtFo2CMj/WYCLDUiWL9RM4XHYaMeMsumpadKj58tRP/di93woJlJ00rPoyWvTx/uuykYG07neMFB3U/79Tbjyjny020Fa0ftiSBRWr+HEzUX3ZEJZeX4FaY3txbN3/6WfWFVzvb9oPtnRf2+uMXv2/oz0eNzxu/bPyqYTf2G48b7cZF46rhNv7Q+Evjr42/rf1p7e9r/1j7V2b6/nvZb+OTRuWz9p//AjIkoDs=</latexit>

where

• The Levi-Civita connection is ↵i j = � ji ��i j �⇡ mod 2⇡<latexit sha1_base64="p8lvH1cQQoz5q6h6Zu3j50nVZBA=">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</latexit>

Gauge transformation

i 2 Ei<latexit sha1_base64="yZXtFAd7yFdnYeMrF2BAUEP8pho=">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</latexit>

1i 2 Ei<latexit sha1_base64="c+y1UlkmwGjOOhF/wNPaJ/Nvhfk=">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</latexit>

ri<latexit sha1_base64="ZUgtIMBj4Zhp+X5dluuqYoakP80=">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</latexit>

✓i<latexit sha1_base64="W9gecnaumyu1OiN1zkrFEEqMiG4=">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</latexit>

i$ i = ri ei✓i 2 C

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j$ j = rj ei✓ j 2 C

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↵i j<latexit sha1_base64="WJgEyWUoftDUeDGevhvOjhEmBbk=">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</latexit>

parallel transported basis

e i = ei'i i<latexit sha1_base64="8EDoNVfEsy3v6xI7dRJnFAqJ7U4=">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</latexit>

e1i = e�i'i 1i<latexit sha1_base64="vz7lMHBw44OvnimKioSWSr2g20Q=">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</latexit>

'i 2 R<latexit sha1_base64="lggegNElsc8Px8lYZaxh3/3lZ7c=">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</latexit>

(not angle-valued, but actual real-valued)

e↵i j = ↵i j + (d')i j

= ↵i j +' j �'i<latexit sha1_base64="Z0mlMbbrAzlUxZBcaUezGFK+bo0=">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</latexit>

n-direction field

n-direction field

n-direction field

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veci = ri e

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i veci

<latexit sha1_base64="R0zK0/8no63VngBCHyMFoEIUKYw=">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</latexit>

�i veci

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� veci

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crossi = ( vec

i )4

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z 7! z4<latexit sha1_base64="ky0qvemSp6d88YqQZ/pyvmPV1f4=">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</latexit>

n-direction field

crossi = ( vec

i )4

<latexit sha1_base64="mk1xhxN7Z5wyOr0oFha5WdjRwVg=">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</latexit>

parallel transport for veci

<latexit sha1_base64="yXrY5C9qIXUumzCZ/A/j/kG3HKQ=">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</latexit>

ei↵veci j vec

i<latexit sha1_base64="4LElRh2PCfsUS4/4LocDKMPhZz4=">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</latexit>

parallel transport for

ei↵crossi j cross

i = ei4↵veci j ( vec

i )4

<latexit sha1_base64="CgdW8fqLB+NLOxdne2LyKUm8ge0=">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</latexit>

crossi

<latexit sha1_base64="3Xs258DOM6TMBsSteYD2CC4k8bY=">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</latexit>

Therefore ↵crossi j = 4↵vec

i j<latexit sha1_base64="1q5YZQWv1kGvSXhSsIm/p8B0nMg=">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</latexit>

n-direction field

↵crossi j = 4↵vec

i j<latexit sha1_base64="1q5YZQWv1kGvSXhSsIm/p8B0nMg=">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</latexit>

Lr = dr¸?1dr

<latexit sha1_base64="jrU+squYsndUn2uklAi2x75ZtrQ=">AAAVk3iczZhbb9vIFcfl7bbdentJuvBL94WpYbRY2IborG9oskh8CbSpfIkVZ4OYSjAiRyJj3jwzUiQPCPRlP+R+hr73uWeGnENSErco0IcKEDXz+59zZs7cSHGQhgEX7fZPK5/94vNf/urXX/xm9cvf/u73f3jw8I9veDJmLr12kzBhbweE0zCI6bUIREjfpoySaBDSHwa3x0r/YUIZD5L4tZiltB+RURwMA5cIQB8e/Nh978RkEBLrqeUkYKkCSWdAR0EsIyJYMM28wsShsWdY9t4JYkGZS8K6scMFYR/squ1/jvbhwXp72949aB8eWu3tx/bj/QMbCnuP9w539y17u60/663ic/nh4Vf/crzEHUc0Fm5IOL+x26noS8JE4IY0W3XGnKbEvSUjegPFmESU96Uer8zaAOJZw4TBNxaWplUPSSLOZ9EALKGDPp/XFFym3YzF8KAvgzgdCxq7eUPDcWiJxFKDb3kBo64IZ1AgLgugr5brE0ZcGMl6JD7kMLTUeyrYmG5C8akbwXVTmYNxX42c71EejOJ6srlbX/o0nFAB2gmFAWK0B/kk4QvIVw6g4GXyOjOlKJPxMsvnYeqTARXSUa0VxvnPqhPTT24SRQRm0UlimlWMpJ3VOiWnQ8hxMw5cqgp1jbguTCKvZ0FCSDUSQdqX01m2urFh9Tqnz6+so4u3pz3dNicTOkim0hkmCfzOdWjCfUqY4jc7/Zvt3f6qZckNuFiWA9lNdQmRxrwaTa7vZBXN57dB6nzyCrkUUm+oOlIHVOTLWtrWOixjy66GCkOSSpVqtbmqAURglIuEmahanMvP/z/Jr60y/N/kB3N8en5Sm2dg6mOdXFwfdU+to6vnx38/fQ3zH5FbSuCsE7AVcN2qFfuCREE4kxdnbzN5Fucr+TSTS7fBWdwlbFQQni04FbuiR8XPOOV7Yt51UNtQyrHnk5Qub0LqcXiytfedZaG2q+CTva39KtzTcH/roAr3NTzYOqzCAw0Pt+z2dyU8BPjEbm/ZOxVotzXd2QL3Ct1ZrY5acwaDxgy2jmBolqahlaW5aGVpQlpZmpVWlqamleX55VKRJKwjtWzjUUidZ+PYo0O4B3o5Z2yRF4NyBsfdCdx4owBWoTQRinMwSWlcWfW1z8Ii+ou99222nDa1VvSraM0NE07/i+b27WXNATVbKxE+7Kxi+6ltefFiYReurjowJM67d+bkl+/Usa/Y1RWyK8POz5GdG9brIesZdnmJ7NKwTgdZx7DjY2THhr18iewl2pXxjjHgVQmvylZK2EH4+ky+dh6dmdq5qp2bWlfVuqb2StVeFbUBKYINJTGxILBhA2TDIcIhwhGyEbIAWYDsI7KPyG6R3SIbIxsjmyCbIPuE7BOyHjKObIpsimyGbIbsHtk9MoaMIRMldEQQelRpRoxRi0sHZAJZiixFdofszrAA7DadIBGkAHRaRGORhDLaxW44Vg/OYP6xYF6gpmvTK6pukgN1fe94ZDSizHh7GDPwIGTRnxAezT1YGuq+wfXWg5NDM+xyNK7LUTll8LgXzTlrhAYg3FOWYNJt1fKG1iaEwfkhwB1KcPvOz7VHzqPCt9T1TyBMn/lpHo67TJ6alninhLgl+VkJzxD2StjDgYjNpA6iclaFKKFAU3hkvC32gyqWO0LV/IrgVwVREcpYHRLhtEAZz6FZlCJXlVoTxidvpPRSdY6eWuW572rt2L5IKYMjlX0jHThxowBumMVv0SuPS4+bFSak50S3lMVb9vZuNLaEEaZzwtQIA6UMyipTVYbVmarOsCq0LEr9TtXvsJqqamqqs7lGMc79nHBvhLs5wUR2VZawY+KP5TniUVhvNXOH4rb0YCPUNdgJhZT6wZwGBEfQXwyrGXrzBW+O3iMCz9hzsmbGgKh/RXMGmhkD2EmL/cuhMSmOgbpJcQ4Ue0HlUJyFelAwuyXJlbkJlZsD/1C5j94qudJ9MXlRZu9W3fVc1bxdldec7FeCu4vBXR186X6Apxh4gLmiWYMawOPx91GTyuGfr+SjuEkngqiNBtedxhApieHfTCZ7UGjuBRllMmjsR0jjkfAz2c1/m3rDKMnkc3VtsJgkYSbfwKVBhz/0mezQRt1NM+mmTaoXcMhTXxss4ESQ6lTIp/J0HFKGR6KumVn+vvcCBSgb3O0i7XYNPAlY4PohLW+wJ/osrr08mFA2ICy7sfsyXzpWvnTK11GVBbVuV2tZ1aH2H/mbfIHmUYsm5Lq90HrgTbVNcZMaapt5o/AWH+sxEWCZEyf6iZzPO4wY8RZdNK061Xz4cif+817umIWLTprWfBitennBZNFJwcZ2uidzDup+3m229ynni010FG0etjSFReoI9ZIIKuplHlHJFSW4Fao3heZ1oNVceLOzbT/e3nn17fqzV//I3xl+0fq69efWX1t2a7/1rNVpXbauW27rnysPV/608vXa2trf1o7WTnLTz1aK94xftWqftbN/AxHJxf4=</latexit>

where dr =

2666664

...�1 e�i↵i j

...

3777775

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Build

cross

Solve smallest eigenvalue problemLr cross = � ?0

cross<latexit sha1_base64="71VlaV6LMI8q1xClUPzdtsE6DRw=">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</latexit>

n-direction field

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veci = ri e

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i veci

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�i veci

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� veci

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4✓i<latexit sha1_base64="3u0OI/dfSSE1p1k3l2aeXIml0rY=">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</latexit>

crossi = ( vec

i )4

<latexit sha1_base64="mk1xhxN7Z5wyOr0oFha5WdjRwVg=">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</latexit>

z 7! z1/4<latexit sha1_base64="9rWpWtmMDJZq+YuscANTr1ASsKA=">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</latexit>

Turning angle, curvature,singularities

Poincaré index in angle defect

Index observation from a distance

Total turning angle of v= 2⇡X

singularitiesenclosed

index�Z

regionenclosed

K dA

<latexit sha1_base64="G+FBK1KpVg+lCZzUsfg4v1M4rks=">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</latexit>

Recall:

(K dA)i = 2⇡�<latexit sha1_base64="crtdhnW4ONQ5qAZa6cErHdnzj6I=">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</latexit>

turning angle of a loop around the point i

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Ä<latexit sha1_base64="Eb78PIMkcDfdvx1fE7xCfH8c76U=">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</latexit>

Poincaré index in angle defect

Total turning angle of v= 2⇡X

singularitiesenclosed

index�Z

regionenclosed

K dA

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For a general complex line bundle

• Turning angle from point i to point j arg

✓ j

ei↵i j i

<latexit sha1_base64="WFMbH40GopMRD7K+yofmVbaQ3iI=">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</latexit>

• Total turning angle after a round trip

arg

Ç i2

ei↵i1,i2 i1

å+ · · ·+ arg

Ç i1

ei↵in ,i1 in

å

<latexit sha1_base64="EL+5+r1LlRSZcmM9okGutvo7Ixw=">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</latexit>

Poincaré index in angle defect

Total turning angle of v= 2⇡X

singularitiesenclosed

index�Z

regionenclosed

K dA

<latexit sha1_base64="G+FBK1KpVg+lCZzUsfg4v1M4rks=">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</latexit>

For a general complex line bundle

• Turning angle from point i to point j arg

✓ j

ei↵i j i

<latexit sha1_base64="WFMbH40GopMRD7K+yofmVbaQ3iI=">AAAVT3iczZhbb+PGFceV9JLEvWSTbJ76wq1hNA1sQ/TGN2ATrNf2QtnKl7XWm8WaijEiR+LYvHlmpJU8INBvk0+Th7606Bdp34qeGZKHpCQGKNCHCDA98/ufc2bO3HgZJAETst3+53vv/+KXv/r1Bx9+tPKb3/7u9x8/+OTT1yIec5deunEQ8zcDImjAInopmQzom4RTEg4C+t3g9lDr300oFyyOXslZQvshGUVsyFwiAV0/OHAIHzkBHcovlJMIdn3jxGBv0e+VExLpDwZx4CmWOiRIfHKt2E2aGjtAnI18+efrB6vtTXt7r72/b7U3H9uPd/dsKOw83tnf3rXszbb5rbby3/n1J5/9y/FidxzSSLoBEeLKbieyrwiXzA1ouuKMBU2Ie0tG9AqKEQmp6CuTa2qtAfGsYczhL5KWoVUPRUIhZuEALHX/xbym4TLtaiyHe33FomQsaeRmDQ3HgSVjSw+c5TFOXRnMoEBczqCvlusTTlwJw1uLJIbCJQH1vpZ8TNeh+LUbwnVdm4NxX+nGPSrYKKonm7n1lU+DCZWgHVEYIE57kE8cPId8lZmPVF2mRSlMVbTM8kDP14DKyjQWPitORN+5cRiSyFNOHNG0Otd2WuuUmg4hx/WIuVQX6hpxXZhEUc+CBJBqKFnSV9NZurK2ZvU6xwcX1rOzN8c907YgEzqIp8oZxjH8n+vQRPiUcM2vtvpXm9v9FctSa3CxLAeym5oSIoNFNZpa3Uormi9uWeK883K5FBJvqDtSB1TCSHA2Vba1CsvYsquhgoAkSqdaba5qABE4FTLmRVQjzuXn/0zya+sM/z/5wRwfnx7V5hmY/llHZ5fPusfWs4uDw78cv4L5D8ktJXBOSdgKuG71in1OQhbM1NnJm1SdRNlKPk7V0m1wEnXh3MqJSBec8l3Ro/InnLI9Me86qG0o7djzSUKXN6HMODzZ2PnGslDb1vDJzsZuFe4YuLuxV4W7Bu5t7FfhnoH7G3b7mxLuA3xitzfsrQq024ZubYB7hW6tVEetOYNBYwYbz2BolqZhlKW5GGVpQkZZmpVRlqZmlOX5ZVKeJKwjvWyjUUCdp+PIo0O4EXoZ53yR54NyAsfdEdw0QwarUBUR8nMwTmhUWfW138Ii+pO981W6nDa1lvcrb80NYkH/h+Z27WXNAS22Vix92Fn59tPb8uz5wi5cWXFgSJy3b4uTX73Vx75mFxfILgp2eorstGC9HrJewc7PkZ0XrNNB1inY4SGyw4K9eIHsBdqV8Q4x4EUJL8pWSthB+OpEvXIenRS1U107LWpdXesWtZe69jKvDUgebKhIEQsCF2yAbDhEOEQ4QjZCxpAxZDfIbpDdIrtFNkY2RjZBNkH2Dtk7ZD1kAtkU2RTZDNkM2T2ye2QcGUcmS+hIFnhUa4UYoRaVDsgksgRZguwO2V3BGNitOyyWJAd0mkfjoYIy2kVuMNYPvWB+kzOP6ela9/KqG2dAX793PDIaUV54exiTeRAy708Aj9UeLA193xBm68HJYRh2ORzX5bCcMnjcC+ecDUIDEO4pjzHptm55zWgTwuH8kOAOJbh9Z+faI+dR7lvq5h+TRZ/FcRZOuFwdFy2JTglxS4qTEp4g7JWwhwMRFZM6CMtZlbKEEk3hkfE23w+6WO4IXfMrgl8VZEUoY3VIiNMCZTyHZmGCXFdqTRQ+WSOll64L9DSqyHxXasf2WUI5HKn8S6VflEIGN8z8f94rTyhPFCtMKs8JbymPNuzN7XBsyUKYzgnTQhhoZVBWua5yrM50dYZVaWRZ6ne6fofVRFeTojqbaxTj3M8J94VwNycUkV2dJeyY6KY8RzwK661m7lDclh5shLoGOyGXEp/NaUBwBP3FsIaht1jwFug9IvCMPScbVhiYt9g5A8MKA9hJi/3LYGGSHwN1k/wcyPeCziE/C82gYHZLkitzkzo3B95QhY/eOrnSfTF5WWbvVt3NXNW8XZ3XnOxXgruLwV0TfOl+gKcYeIC5oGmDyuDx+NuwSRXw5qvEKGrSiSR6o8F1qzFEQiJ4m0lVDwrNvSCjVLHGfgQ0Gkk/Vd3sf1NvOCWpOtDXBotJHKTqNVwadHihT1WHNupukio3aVI9JiBPc22wgBNB6VMhm8rjcUA5HommVszyt73nKEC5wN0u0m63gEeMM9cPaHmDPTJnce3jwYTyAeHpld1X2dKxsqWjBf1NSlUX1KpdraVVh9o78pfZAs2i5k2oVXuhdeZNjU1+kxoam3mj4BYf6zERYKkTxeaJXMw7jDjxFl0MrTrVfMRyJ/HTXu6YB4tOhtZ8OK16eWyy6KRhYzvdozkHfT/vNtv7VIjFJjqaNg9bksAidaT+SAQV/TGP6OTyEtwK0+sHq8XnQKu58Hpr0368ufXyq9WnL/+afTP8sPWH1h9bX7Ts1m7raavTOm9dttzWD60fW39v/ePh3x7+++F/Ps8/L77/Xl74rFX7ff7RfwHxva0t</latexit>

• Total turning angle after a round trip

arg

Ç i2

ei↵i1,i2 i1

å+ · · ·+ arg

Ç i1

ei↵in ,i1 in

å

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mod 2⇡= �i(↵i1,i2 + · · ·+↵in,i1)<latexit sha1_base64="boKXA0U6UhPVyAxnBFnxqeIvIpM=">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</latexit>

Poincaré index in angle defect

Total turning angle of v= 2⇡X

singularitiesenclosed

index�Z

regionenclosed

K dA

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arg

Ç i2

ei↵i1,i2 i1

å+ · · ·+ arg

Ç i1

ei↵in ,i1 in

å

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mod 2⇡= �i(↵i1,i2 + · · ·+↵in,i1)<latexit sha1_base64="boKXA0U6UhPVyAxnBFnxqeIvIpM=">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</latexit>

The curvature of the bundle enclosed by the loop is

� =X

around theloop

↵ik ,ik+1mod 2⇡

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ZK dA=

<latexit sha1_base64="di17I753+/h2bplqVt7wT5TKgdQ=">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</latexit>

Knowing which 2 pi branch of curvature for every face can give us Poincaré index.

Poincaré index in angle defectThe curvature of the bundle enclosed by the loop is

� =X

around theloop

↵ik ,ik+1mod 2⇡

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ZK dA=

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Knowing which 2 pi branch of curvature for every face can give us Poincaré index.

Curvature is gauge invariant

Complex line bundleA complex line bundle over a triangle mesh is equipped with

• angle-valued 1-form ↵<latexit sha1_base64="hf+z9hs3qk7Skltw356CHJFNooo=">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</latexit>

• real-valued 2-form �<latexit sha1_base64="e/6A8sFD77+v9yjXne9QjiudWLs=">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</latexit>

so that d↵= � mod 2⇡<latexit sha1_base64="y8xdRyNt7XBQR2B8aDLrsk0Qt7g=">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</latexit>

A complex-valued 0-form and the connection <latexit sha1_base64="092By7j01+qLKLw8leNwyWI8q6I=">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</latexit>

↵<latexit sha1_base64="hf+z9hs3qk7Skltw356CHJFNooo=">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</latexit>

are relative to an arbitrary reference basis. Under basis transformation,

7! ei' <latexit sha1_base64="cyqeNP7e0Ny6TEQLc4I9hzyM3bQ=">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</latexit>

↵ 7! ↵+ d'<latexit sha1_base64="duUAyC1rw2w9iNY+QwZYdGBLifM=">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</latexit>

The turning angle of along an edge is <latexit sha1_base64="092By7j01+qLKLw8leNwyWI8q6I=">AAAVEHiczZhbb9vIFce1u71s1Vt21299YWoYLRa2ITrrG5BdJLEVaFP5EivOBjHVYESORMa8eWakSB4Q6CfoQ1/aj9K3oq/9Bv0ife6ZIeeQlMQFCuxDBYia+f3POTNnbqQ4SsOAi07n3x99/MmPfvyTn376s/bPf/HLX/36wWefv+bJlLn02k3ChL0ZEU7DIKbXIhAhfZMySqJRSL8b3Z4o/bsZZTxI4ldikdJhRCZxMA5cIhRyUh68e7DZ2bX3jzrHx1Zn95H96PDIhsLBo4Pj/UPL3u3oz2ar+Fy+++yL/zhe4k4jGgs3JJzf2J1UDCVhInBDmrWdKacpcW/JhN5AMSYR5UOpO5tZW0A8a5ww+MbC0rTqIUnE+SIagWVEhM+XNQXXaTdTMT4ayiBOp4LGbt7QeBpaIrFU5pYXMOqKcAEF4rIA+mq5PmHEFTA+tUh8zF0SUu9rwaZ0G4pfuxFct5U5GA+latyjPJjE9WRzt6H0aTijArRTCgPE6ADyScLnkK8cQcHL5HVmSlEm43WWT8PUJyMqpKNaK4zzn7YT0w9uEkUk9qSTxDSrGEk7q3VKzseQ43YcuFQV6hpxXZhEXs+ChJBqJIJ0KOeLrL21ZQ163adX1rOLN92BbpuTGR0lc+mMkwR+lzo04z4lTPGbveHN7v6wbVlyCy6W5UB2c11CpDGvRpObe1lF8/ltkDofvEIuhdQbq47UARUwEiyYS9vahGVs2dVQYUhSqVKtNlc1gAiMcpEwE1WLS/n5/yf5dVSGP0x+MMfd89PaPANTH+v04vpZv2s9u3p68ofuK5j/iNxSAgeNgK2A61at2OckCsKFvDh7k8mzOF/J3Uyu3QZncZ+wSUF4tuJU7IoBFd/jlO+JZddRbUMpx4FPUrq+CanH4fHOwTeWhdq+go8Pdg6r8EDDw52jKjzU8GjnuAqPNDzesTvflPAY4GO7s2PvVaDd0XRvB9wrdK9dHbXmDEaNGew8g6FZm4ZW1uailbUJaWVtVlpZm5pW1ueXS0WSsI7Uso0nIXWeTGOPjuFO5uWcsVVeDMoZHHencNeLAliF0kQozsEkpXFl1dc+K4vod/bBV9l62tRa0a+iNTdMOP0fmju01zUH1GytRPiws4rtp7blxfOVXdhuOzAkztu35uSXb9Wxr9jVFbIrw87PkZ0bNhggGxh2eYns0rBeD1nPsJMTZCeGvXiB7AXalfFOMOBVCa/KVkrYQ/jqTL5yHp6Z2rmqnZtaX9X6pvZS1V4WtREpgo0lMbEgsGEjZOMxwjHCCbIJsgBZgOw9svfIbpHdIpsimyKbIZsh+4DsA7IBMo5sjmyObIFsgewe2T0yhowhEyV0RBB6VGlGjFGLSwdkAlmKLEV2h+zOsADstp0gEaQAdF5EY5GEMtrFbjhVT61g/r5gXqCma9srqm6SA3X9o+ORyYQy4+1hzMCDkEV/Qngu9mBpqPsG11sPTg7NsMvRtC5H5ZTB41605KwRGoBwT1mCSXdUy1tamxEG54cAdyjB7Ts/1x46DwvfUtc/gTB95t08HHeZ7JqWeK+EuCX5WQnPEA5KOMCBiM2kjqJyVoUooUBTeGS8LfaDKpY7QtX8iuBXBVERylg9EuG0QBnPoUWUIleVWhPGJ2+k9FJ1jp5a5blvu3ZsX6SUwZHKvpQOnLhRADfM4rfolcelx80KE9JzolvK4h17dz+aWsII8yVhboSRUkZllakqw+pCVRdYFVoWpX6n6ndYTVU1NdXFUqMY535JuDfC3ZJgIrsqS9gx8fvyHPEorLeauUNxW3qwEeoa7IRCSv1gSQOCI+ivhtUMvfmKN0fvCYFn7CVZM2NA1L+iJQPNjAHspNX+5dCYFMdA3aQ4B4q9oHIozkI9KJjdmuTK3ITKzYF/qNxHb5Vc6b6avCizd6vueq5q3q7Ka0n2K8Hd1eCuDr52P8BTDDzAXNGsQQ3g8fjbqEnl8M9X8kncpBNB1EaD615jiJTE8G8mkwMoNPeCTDIZNPYjpPFE+Jns579NvWGUZPKpujZYzJIwk6/h0qDDH/pM9mij7qaZdNMm1Qs45KmvDRZwIkh1KuRT2Z2GlOGRqGtmlr8dPEcBygb3+0j7fQNPAxa4fkjLG+ypPotrLw9mlI0Iy27socyXjpUvHSWol0qyuqA27WotqzrU/iN/mS/QPGrRhNy0V1oPvLm2KW5SY22zbBTe4mM9JgIsc+JEP5HzZYcJI96qi6ZVp5oPX+/Ev9/LnbJw1UnTmg+jVS8vmK06KdjYTv90yUHdz/vN9j7lfLWJnqLNw5amsEgdoV4SQUW9zCMquaIEt8Ls3YNN8zrQai683tu1H+3uvfxq88nLP+XvDD9t/ab129bvW3brsPWk1Wtdtq5bbstv/bn119bfNv6y8feNf2z8Mzf9+KPiPeMXrdpn41//BSpilBY=</latexit>

arg

✓ j

ei↵i j i

<latexit sha1_base64="WFMbH40GopMRD7K+yofmVbaQ3iI=">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</latexit>

The index formula: (Total turning angle)= 2⇡P

index�R�

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Complex line bundle

The index formula: (Total turning angle)= 2⇡P

index�R�

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Hairball theorem for complex line bundleOver the entire closed surface

2⇡X

index=Z

M�

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Definition. The degree of a complex line bundle is the integer1

2⇡

Z

M�

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Complex line bundleHairball theorem for complex line bundleOver the entire closed surface

2⇡X

index=Z

M�

<latexit sha1_base64="WQ+XBYPg9rTSF5mgk3oAu/ETVg8=">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</latexit>

Paint some 2-form total integral being 2pi integer, can you build the rest of the ingredients? (i.e 1-form )↵<latexit sha1_base64="vrE2lD5p5ODTbqZcLXK+OqiZgSY=">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</latexit>

Singularities in optimal n-direction field

n=1

degree = 2

↵= ↵LC<latexit sha1_base64="pq3FGNBBfuTIkfq+lzSVAKT46qk=">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</latexit>

� = K dA<latexit sha1_base64="Ge/Fz2e9LzK9nA/o6t7MFY0Iv18=">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</latexit>

Singularities in optimal n-direction field

n=2

degree = 4

↵= 2↵LC<latexit sha1_base64="oYvMwglLa6C5NPE2PZEBQD8AkfA=">AAAVDXiczZhbbxvHFceZtGlTNm2dRi9FX9YRhBaBKHDZ2LIBO7At0WBc6mLRcgxrGWO4OyTX3JtmhjSpwaL9BvkUeU2e8lb0tZ+hQD9Mz8zunN0luQEC9KECtJz5/c85M2duexklgc9Fu/3v997/2c8/+MUvP/xV89cf/ea3v7v18e9f8njOXHrpxkHMXo0Ip4Ef0Uvhi4C+Shgl4SigX41mR0r/akEZ9+PohVgldBiSSeSPfZcIQG9u/cEhQTIl1kOrk5W+lg4Lrf5R+ubWbvugrf+szYKdF3Yb+d/5m48/+Y/jxe48pJFwA8L5ld1OxFASJnw3oGnTmXOaEHdGJvQKihEJKR9KnUNq7QHxrHHM4D8SlqZlD0lCzlfhCCxDIqZ8XVNwm3Y1F+N7Q+lHyVzQyM0aGs8DS8SWGhDL8xl1RbCCAnGZD3213ClhxBUwbJVIfMxdElDvoWBzug/Fh24I131lDsZDqRr3KPcnUTXZzG0opzRYUAHaMYUBYnQA+cTBU8hXjqDgpfIyNaUwldE2y8dqkkZUSEe1lhtnP00nou/cOAxJ5EknjmhaMpJ2WumUXI4hx/3Id6kqVDXiujCJPG029/asQa/7+MJ6cvaqO9AtcLKgo3gpnXEcw+9asws+pYQpftUZXh3cGTYtS+7BxbIcyGGpS4g05uVocreTlrQpn/mJ887L5UJIvLHqSBVQAfkyfylta9e2YJWWQwUBSaTKstxc2QAiMMpFzExULa7lN/0/ya+tMvzf5Adz3D09rswzMPVnHZ9dPul3rScXj4/+2n0B8x+SGSVwyghY8Lg61bp8SkI/WMmzk1epPImy9dpN5dbFfhL1CZvkhKcbTvnaH1DxI07Zyl93HVW2jXIcTElCtzch9Tg8aN39wrJQu6Pgg7utwzK8q+Fh614ZHmp4r3W/DO9peL9lt78o4H2AD+x2y+6UoN3WtNMC9xLtNMujVp/BqDaD1hMYmq1paGVrLlrZmpBWtmalla2paWV7fpmUJwnrSC3baBJQ59E88ugYbmNexhnb5PmgnMChdgy3vNCHVShNhPy0ixMalVZ95W9jEf3Jvvt5up3WtZb3K2/NDWJOf0Jzh/a25oCarRWLKeysfPupbXn2dGMXNpsODInz+rU53+VrdbgrdnGB7MKw01Nkp4YNBsgGhp2fIzs3rNdD1jPs6AjZkWHPniF7hnZFvCMMeFHAi6KVAvYQvjiRL5zbJ6Z2qmqnptZXtb6pPVe153ltRPJgY0lMLAhs2AjZeIxwjHCCbILMR+Yje4vsLbIZshmyObI5sgWyBbJ3yN4hGyDjyJbIlshWyFbIbpDdIGPIGDJRQEf4gUeVZsQItahwQCaQJcgSZNfIrg3zwW7f8WNBckCXeTQWSiijXeQGc/XICuZvc+b5arr2vbzqxhlQ168dj0wmlBlvD2P6HoTM+xPAQ7EHS0PdN7jeenByaIZdDudVOSymDB7qwjVnjdAAhBvKYky6rVre09qCMDg/BLhDCW7f2bl227md+xa6/vGF6TPvZuG4y2TXtMR7BcQtyU8KeIJwUMABDkRkJnUUFrMqRAEFmsKD4SzfD6pY7AhVm5aEaVkQJaGI1SMhTguU8RxahQlyVak0YXyyRgovVefoqVWe+TYrx/ZZQhkcqewz6cCJG/pww8x/8155XHrcrDAhPSecURa17IM74dwSRliuCUsjjJQyKqpMVRlWV6q6wqrQsij0a1W/xmqiqomprtYaxTg3a8KNEa7XBBPZVVnCjoneFueIR2G9VcwditvSg41Q1WAn5FIy9dc0IDiC082wmqE33/Dm6D0h8Iy9JmtmDPQL6pqBZsYAdtJm/zJoTPJjoGqSnwP5XlA55GehHhTMbktyRW5C5ebAeyifordKrnDfTF4U2btldz1XFW9X5bUmT0vB3c3grg6+dT/AUww8wFzQtEb14fH4y7BO5fB+K/kkqtOJIGqjwbVTGyIhEbzNpHIAhfpekEkq/dp+BDSawEu/7Ge/db1hlKTysbrWWCziIJUv4VKjw2t7Knu0VneTVLpJner5HPLU1xoLOBGkOhWyqezOA8rwSNQ1M8tfDp6iAGWD+32k/b6Bxz7z3WlAixvssT6LK58IFpSNCEuv7KHMlo6VLR0lqC9Ksrygdu1yLS07VN6RP8sWaBY1b0Lu2hut+95S2+Q3qbG2WTcKZvhYj4kAS50o1k/kfN1hwoi36aJp2aniw7c78R/3cucs2HTStOLDaNnL8xebTgrWttM/XnNQ9/N+vf2Ucr7ZRE/R+mFLElikjlCfgqCiPtkRlVxegluh+h5or3/92yy87BzYfznoPP9899Hzv2dfBj9s/LHxaePPDbtx2HjU6DXOG5cNt/G3xreN7xrf73yz88POP3b+mZm+/17+NfGTRuVv51//BVGFkJU=</latexit>

� = 2K dA<latexit sha1_base64="2Cd+TJqH2Na20GxVre0mx7J9DA0=">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</latexit>

Singularities in optimal n-direction field

n=3

degree = 6

↵= 3↵LC<latexit sha1_base64="gPE5PFB8iCTmWV5YvHk9gW3fmTg=">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</latexit>

� = 3K dA<latexit sha1_base64="VsVgISNharPnvgT1VIb5Ic2bAJM=">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</latexit>

Singularities in optimal n-direction field

n=10

degree = 20

↵= 10↵LC<latexit sha1_base64="qarlq6AAx/XNw4EKX3OcmiAs5Ls=">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</latexit>

� = 10K dA<latexit sha1_base64="6FKuVhjEzAVhxwrDcTqiQ2uPizs=">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</latexit>

Singularities in optimal n-direction field

n=20

degree = 40

↵= 20↵LC<latexit sha1_base64="3/JSjKda4E9DlPjahwW14Jc17UE=">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</latexit>

� = 20K dA<latexit sha1_base64="Y06nUolGaH8VfpzUQzfWjDZvHYs=">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</latexit>

Singularities in optimal n-direction fieldsingularity of optimal field

independent random samples

Gauge theory in physics

Continuous theory

(dr )i j = e�i↵i j j � i<latexit sha1_base64="p2/RiaUSvd6T2m0SLfMvwcl2zE4=">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</latexit>

• In the discrete setting

• Towards continuous j ⇡ i + d ( ~i j) + · · ·

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(dr )( ~i j)⇡�1� i↵( ~i j)� � i + d ( ~i j)�� i

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= d ( ~i j)� i↵( ~i j) i +O(| ~i j|2)<latexit sha1_base64="Oq1qw6xu1fjEnT6tsTPm7QEJ7XI=">AAAVWHiczZhbj9vGFceV9JJEvTlJ/dQXuotFnXS1EOXuDbAD27trKK724pXXMbxUFiNyJM6Kt50ZydJOCPRrtJ+in6ZA+tJ+ij73zJAzJCUxQIA+VIComd//nDNz5kZSwyQgjLfb33/w4U9++rOff/TxJ81f/PJXv/7NvU8/e8PiKXXxpRsHMX07RAwHJMKXnPAAv00oRuEwwN8MJ4dS/2aGKSNx9JovEjwI0TgiI+IiDuj63kvrieU5CSMPnRl2BblJv7BalhMi7g+HceAJkjooSHxU6NL6mlh/tM4efqfhd992vri+t9Hetnf22wcHVnv7kf1ob9+Gwu6j3YOdPcvebqvPxlP7+7/C52/n159+/h/Hi91piCPuBoixK7ud8IFAlBM3wGnTmTKcIHeCxvgKihEKMRsIlXRqbQLxrFFM4RtxS9Gyh0AhY4twCJYyG7asSbhOu5ry0f5AkCiZchy5WUOjaWDx2JIjaHmEYpcHCygglxLoq+X6iCKXwzhXIrERc1GAvSecTvEWFJ+4IVy3pDkYD4Rs3MOMjKNqspnbQPg4mGEO2hGGAaK4D/nEwQvIV6jZScVlqkthKqJ1ls/k7A0xF8Wkap+mE+H3bhyGKPKEE0c4LRkJO610SsxHkONWRFwsC1UNuS5MIqtmgQJINeQkGYj5Im1ublr97vGzC+v52dvjvmqboRkexnPhjOIYfpc6NGM+RlTyq87gantn0LQssQkXCxYoYFUySGFWjiY2OmlJ89mEJM57L5cLIfFGsiNVgDmMBCVzYVsbsIwtuxwqCFAiZKrl5soGEIFixmOqoypxKT///yS/tszwf5MfzPHx6VFlnoHJj3V0dvm8d2w9v3h2+Ofj1zD/IZpgBAcWh61g1q1csS9QSIKFODt5m4qTKFvJx6lYuw1Ooh6i45ywdMUp3xV9zH/AKdsTy67DyoaSjn0fJXh9E0KNw+PW7leWZbQdCR/vtvbKcFfBvdZ+Ge4puN86KMN9BQ9advurAh4AfGy3W3anBO22op0WuJdop1ketfoMhrUZtJ7D0KxNQylrc1HK2oSUsjYrpaxNTSnr88ukPElYR3LZRuMAO0+nkYdHcEf0Mk7pKs8H5QSOuyO4e4YEVqHQEfJzME5wVFr1lc/KIvqDvfundD2tay3vV96aG8QM/4jm9ux1zQHVWyvmPuysfPvJbXn2YmUXNpsODInz7p0++cU7eexLdnFh2IVmp6eGnWrW7xvW1+z83LBzzbpdw7qaHR4adqjZy5eGvTR2RbxDE/CigBdFKwXsGvj6RLx2Hpzo2qmsnepaT9Z6uvZK1l7ltSHKg40E0rEgsGZDw0YjA0cGjg0bG0YMI4bdGHZj2MSwiWFTw6aGzQybGfbesPeG9Q1jhs0Nmxu2MGxh2J1hd4ZRw6hhvIAOJ4GHpabFyGhR4WAYNywxLDHs1rBbzQjYbTkk5igHeJ5Ho6GAsrGL3GAqn37B/CZnHpHTteXlVTfOgLx+63hoPMZUe3smJvEgZN6fAJ6vPVga8r7B1NaDk0Mx0+VwWpXDYsrgcS9cclbIGIBwh2lskm7LljeVNkMUzg8O7lCC23d2rj1wHuS+ha5+CNd9ZsdZOOZScaxbYt0Cmi3JTgp4YmC/gH0zEJGe1GFYzCrnBeTGFB4ZJ/l+kMViR8iaXxL8ssBLQhGri0IzLVA259AiTAyXlUoT2idrpPCSdWY8lcoy32bl2D5LMIUjlX4pHDhxQwI3zPw375XHhMf0CuPCc8IJplHL3t4JpxbXwnxJmGthKJVhUaWySk11IasLU+VK5oV+K+u3pprIaqKri6VGTZy7JeFOC7dLgo7syixhx0Q3xTniYVhvFXMHm23pwUaoarATcinxyZIGxIygvxpWMePNVryZ8R4jeMZekhXTBuqddslAMW0AO2m1fxnUJvkxUDXJz4F8L8gc8rNQDYrJbk1yRW5c5ubAGyrzjbdMrnBfTZ4X2btldzVXFW9X5rUk+6Xg7mpwVwVfux/gKQYeYC5wWqMSeDz+OqxTGbz5CjaO6nTEkdxocO3UhkhQBG8zqehDob4XaJwKUtuPAEdj7qeil/3W9YZilIpn8lpjMYuDVLyBS40OL/Sp6OJa3U1S4SZ1qkcY5KmuNRZwIgh5KmRTeTwNMDVHoqrpWf66/8IIUNa41zO019PwiFDi+gEubrBH6iyu/Hkww3SIaHplD0S2dKxs6UhB/jklygtqwy7X0rJD5R35y2yBZlHzJsSGvdI68ebKJr9JjZTNslEwMY/1JhFgqRPF6omcLTuMKfJWXRQtO1V82Hon9sNe7pQGq06KVnwoLnt5ZLbqJGFtO72jJQd5P+/V2/uYsdUmupLWD1uSwCJ1uPyTCCryzzwkk8tLcCtMr+9t6L8DrfrCm862/Wi788reePrqLw31+bjxu8bvGw8bdmOv8bTRbZw3Lhtu4++NfzT+1fj3b/95v3H/o/ufZKYffpD9Nj5vVD73P/svm7GwOg==</latexit>

dr = d � i↵ <latexit sha1_base64="rizX3M4Ncv5VDRw6ok0Eqebvfho=">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</latexit>

Continuous theory

• Continuous covariant derivative

dr = d � i↵ <latexit sha1_base64="rizX3M4Ncv5VDRw6ok0Eqebvfho=">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</latexit>

dr = d � i↵^<latexit sha1_base64="5tyQ01ypZuLAdCbAXFvd81T4dZI=">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</latexit>

: ⌦k(M ;C)! ⌦k+1(M ;C)<latexit sha1_base64="AlZr3TSXBbKGakFI164h+++6il0=">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</latexit>

• Double covariant exterior derivativedrdr

<latexit sha1_base64="AjVya/sdgjiUVkzKnv8IWZ03lPY=">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</latexit>

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curvature of the connection

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q1<latexit sha1_base64="yy4qUVyhIJeIyXK/z1alHJ/r0N0=">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</latexit>

q2<latexit sha1_base64="+dOhe2cdBHO03TaIQvitpJqzVJo=">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</latexit>

Electrodynamics

charged particle

q1<latexit sha1_base64="yy4qUVyhIJeIyXK/z1alHJ/r0N0=">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</latexit>

q2<latexit sha1_base64="+dOhe2cdBHO03TaIQvitpJqzVJo=">AAAVD3iczZjLjtvIFYY1k9tEuXkynVU2dBqdBJNWQ5TTN8AzsN3dhsZRX9xyeww3FaNEliRavHVVSZa6QCAvkEU2yaNkF2SbR8g7ZJ11ThVZh6QkDhAgiwgQWfX955yqUzdSGiaBz0W7/c+PPv7Wt7/z3e998v3mD374ox//5MGnP33N4xlz6Y0bBzF7MyScBn5Eb4QvAvomYZSEw4B+PZyeKP3rOWXcj6NXYpnQQUjGkT/yXSIA9e/edd492G7v2ftH7eNjq733yH50eGRD4eDRwfH+oWXvtfVn+8lv/vWz6JcX3at3n372b8eL3VlII+EGhPNbu52IgSRM+G5A06Yz4zQh7pSM6S0UIxJSPpC6r6m1A8SzRjGDbyQsTcsekoScL8MhWIZETPiqpuAm7XYmRkcD6UfJTNDIzRoazQJLxJZK3PJ8Rl0RLKFAXOZDXy13QhhxBQxPJRIfcZcE1PtCsBndheIXbgjXXWUOxgOpGvco98dRNdnMbSAnNJhTAdophQFitA/5xMFzyFcOoeCl8iY1pTCV0SbLp0EyIUMqpKNay42zW9OJ6Ac3DkMSedKJI5qWjKSdVjolFyPIcTfyXaoKVY24Lkwir2ZBAkg1FH4ykItl2tzZsfrds6fX1rPLN2d93TYnczqMF9IZxTHcVzo05xNKmOK3ncHt3v6gaVlyBy6W5UB2C11CpDEvR5PbnbSkTfjUT5wPXi4XQuKNVEeqgAoYCeYvpG1twzK27HKoICCJVKmWmysbQARGuYiZiarFlfwm/yf5tVWG/5v8YI7PLk4r8wxMfazTy5tnvTPr2fXTk9+dvYL5D8mUEjhnBGwFXLdqxT4noR8s5eX5m1SeR9lKPkvlxm1wHvUIG+eEp2tO+a7oU/ENTtmeWHUdVjaUcuxPSEI3NyH1ODxuHXxpWajtK/j4oHVYhgcaHraOyvBQw6PWcRkeaXjcsttfFvAY4GO73bI7JWi3Ne20wL1EO83yqNVnMKzNoPUMhmZjGlrZmItWNiaklY1ZaWVjalrZnF8m5UnCOlLLNhoH1Hkyizw6ggeZl3HG1nk+KOdw3J3CQy/0YRVKEyE/B+OERqVVX/msLaJf2Qe/TTfTutbyfuWtuUHM6X/R3KG9qTmgZmvFYgI7K99+altePl/bhc2mA0PivH1rTn75Vh37il1fI7s27OIC2YVh/T6yvmFXV8iuDOt2kXUNOzlBdmLYixfIXqBdEe8EA14X8LpopYBdhK/O5Svn4bmpXajahan1VK1nai9V7WVeG5I82EgSEwsCGzZENhohHCEcIxsj85H5yN4je49simyKbIZshmyObI7sA7IPyPrIOLIFsgWyJbIlsntk98gYMoZMFNARfuBRpRkxQi0qHJAJZAmyBNkdsjvDfLDbdfxYkBzQRR6NhRLKaBe5wUy9tIL5+5x5vpquXS+vunEG1PX3jkfGY8qMt4cxfQ9C5v0J4LXYg6Whnhtcbz04OTTDLoezqhwWUwave+GKs0ZoAMI9ZTEm3VYt72htThicHwLcoQSP7+xce+g8zH0LXd98YfrMz7Jw3GXyzLTEuwXELcnPC3iOsF/APg5EZCZ1GBazKkQBBZrCK+M03w+qWOwIVZuUhElZECWhiNUlIU4LlPEcWoYJclWpNGF8skYKL1Xn6KlVnvk2K8f2ZUIZHKnsc+nAiRv68MDM73mvPC49blaYkJ4TTimLWvbefjizhBEWK8LCCEOlDIsqU1WG1aWqLrEqtCwK/U7V77CaqGpiqsuVRjHO/Ypwb4S7FcFEdlWWsGOi98U54lFYbxVzh+K29GAjVDXYCbmUTPwVDQiO4GQ9rGbozde8OXqPCbxjr8iaGQOifhWtGGhmDGAnrfcvg8YkPwaqJvk5kO8FlUN+FupBwew2JFfkJlRuDvxC5RP0VskV7uvJiyJ7t+yu56ri7aq8VuRJKbi7HtzVwTfuB3iLgReYa5rWqD68Hn8V1qkcfvlKPo7qdCKI2mhw7dSGSEgEv2ZS2YdCfS/IOJV+bT8CGo3FJJW97F7XG0ZJKp+qa43FPA5S+RouNTr8oE9ll9bqbpJKN6lTPZ9DnvpaYwEnglSnQjaVZ7OAMjwSdc3M8lf95yhA2eBeD2mvZ+Cpz3x3EtDiAXuqz+LKnwdzyoaEpbf2QGZLx8qWjhLUf0qyvKC27XItLTtUfiN/ni3QLGrehNy211r3vYW2yR9SI22zahRM8bUeEwGWOlGs38j5qsOYEW/dRdOyU8WHb3bi3+zlzliw7qRpxYfRspfnz9edFKxtp3e64qCe5716+wnlfL2JrqL1w5YksEgdof4kgor6M4+o5PISPArTdw+2zd+BVn3hdWfPfrTXeWlvP3n5h4b+fNL4eeMXjV837MZh40mj27hq3DTcxrjxx8afG3/Z+tPWX7f+tvX3zPTjj7J747NG5bP1j/8AigSV+Q==</latexit>

Electrodynamics

electric fieldE 2 ⌦1(M ;R)

<latexit sha1_base64="8HuwDDV6tEAVjwFZieRV32BMRrk=">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</latexit>

d ? E = ⇢<latexit sha1_base64="viw5upORk5/+8afDNMJpgt35mKQ=">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</latexit>

charge densityn-form

q1<latexit sha1_base64="yy4qUVyhIJeIyXK/z1alHJ/r0N0=">AAAVD3iczZjLjtvIFYY1k9tEuXkynVU2dBqdBJNWQ2ynb4BnYLtbhsZRX9xyeww3FaNEliRavHVVSZa6QCAvkEU2yaNkF2SbR8g7ZJ11ThVZh6QkDhAgiwgQWfX955yqUzdSGiaBz0W7/c+PPv7Wt7/z3e998v3mD374ox//5MGnP33N4xlz6Y0bBzF7MyScBn5Eb4QvAvomYZSEw4B+PZyeKv3rOWXcj6NXYpnQQUjGkT/yXSIA9e/e2e8ebLf37IPj9smJ1d57ZD86OrahcPjo8OTgyLL32vqz/eQ3//pZ9MuL7tW7Tz/7t+PF7iykkXADwvmt3U7EQBImfDegadOZcZoQd0rG9BaKEQkpH0jd19TaAeJZo5jBNxKWpmUPSULOl+EQLEMiJnxVU3CTdjsTo+OB9KNkJmjkZg2NZoElYkslbnk+o64IllAgLvOhr5Y7IYy4AoanEomPuEsC6n0h2IzuQvELN4TrrjIH44FUjXuU++OommzmNpATGsypAO2MwgAx2od84uA55CuHUPBSeZOaUpjKaJPl0yCZkCEV0lGt5cbZrelE9IMbhyGJPOnEEU1LRtJOK52SixHkuBv5LlWFqkZcFyaRV7MgAaQaCj8ZyMUybe7sWP1u5+m19ezyTaev2+ZkTofxQjqjOIb7SofmfEIJU/x2f3C7dzBoWpbcgYtlOZDdQpcQaczL0eT2flrSJnzqJ84HL5cLIfFGqiNVQAWMBPMX0ra2YRlbdjlUEJBEqlTLzZUNIAKjXMTMRNXiSn6T/5P82irD/01+MMedi7PKPANTH+vs8uZZr2M9u356+rvOK5j/kEwpgXNGwFbAdatW7HMS+sFSXp6/SeV5lK3kTio3boPzqEfYOCc8XXPKd0Wfim9wyvbEquuwsqGUY39CErq5CanH4XHr8EvLQu1AwceHraMyPNTwqHVchkcaHrdOyvBYw5OW3f6ygCcAH9vtlr1fgnZb0/0WuJfofrM8avUZDGszaD2DodmYhlY25qKVjQlpZWNWWtmYmlY255dJeZKwjtSyjcYBdZ7MIo+O4EHmZZyxdZ4Pyjkcd2fw0At9WIXSRMjPwTihUWnVVz5ri+hX9uFv0820rrW8X3lrbhBz+l80d2Rvag6o2VqxmMDOyref2paXz9d2YbPpwJA4b9+ak1++Vce+YtfXyK4Nu7hAdmFYv4+sb9jVFbIrw7pdZF3DTk+RnRr24gWyF2hXxDvFgNcFvC5aKWAX4atz+cp5eG5qF6p2YWo9VeuZ2ktVe5nXhiQPNpLExILAhg2RjUYIRwjHyMbIfGQ+svfI3iObIpsimyGbIZsjmyP7gOwDsj4yjmyBbIFsiWyJ7B7ZPTKGjCETBXSEH3hUaUaMUIsKB2QCWYIsQXaH7M4wH+x2HT8WJAd0kUdjoYQy2kVuMFMvrWD+Pmeer6Zr18urbpwBdf2945HxmDLj7WFM34OQeX8CeC32YGmo5wbXWw9ODs2wy+GsKofFlMHrXrjirBEagHBPWYxJt1XLO1qbEwbnhwB3KMHjOzvXHjoPc99C1zdfmD7zThaOu0x2TEu8W0Dckvy8gOcI+wXs40BEZlKHYTGrQhRQoCm8Mk7z/aCKxY5QtUlJmJQFURKKWF0S4rRAGc+hZZggV5VKE8Yna6TwUnWOnlrlmW+zcmxfJpTBkco+lw6cuKEPD8z8nvfK49LjZoUJ6TnhlLKoZe8dhDNLGGGxIiyMMFTKsKgyVWVYXarqEqtCy6LQ71T9DquJqiamulxpFOPcrwj3RrhbEUxkV2UJOyZ6X5wjHoX1VjF3KG5LDzZCVYOdkEvJxF/RgOAITtbDaobefM2bo/eYwDv2iqyZMSDqV9GKgWbGAHbSev8yaEzyY6Bqkp8D+V5QOeRnoR4UzG5DckVuQuXmwC9UPkFvlVzhvp68KLJ3y+56rirersprRZ6UgrvrwV0dfON+gLcYeIG5pmmN6sPr8Vdhncrhl6/k46hOJ4KojQbX/doQCYng10wq+1Co7wUZp9Kv7UdAo7GYpLKX3et6wyhJ5VN1rbGYx0EqX8OlRocf9Kns0lrdTVLpJnWq53PIU19rLOBEkOpUyKayMwsowyNR18wsf9V/jgKUDe71kPZ6Bp75zHcnAS0esGf6LK78eTCnbEhYemsPZLZ0rGzpKEH9pyTLC2rbLtfSskPlN/Ln2QLNouZNyG17rXXfW2ib/CE10jarRsEUX+sxEWCpE8X6jZyvOowZ8dZdNC07VXz4Zif+zV7ujAXrTppWfBgte3n+fN1Jwdp2emcrDup53qu3n1DO15voKlo/bEkCi9QR6k8iqKg/84hKLi/BozB992Db/B1o1Rde7+/Zj/b2X9rbT17+oaE/nzR+3vhF49cNu3HUeNLoNq4aNw23MW78sfHnxl+2/rT1162/bf09M/34o+ze+KxR+Wz94z91W5X4</latexit>

q2<latexit sha1_base64="+dOhe2cdBHO03TaIQvitpJqzVJo=">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</latexit>

Electrodynamics

electric fieldE 2 ⌦1(M ;R)

<latexit sha1_base64="8HuwDDV6tEAVjwFZieRV32BMRrk=">AAAVIniczZhbj9vGFceVNG1T9RKn2be+0Fksmgarhbju3lAnsL0rQ3G1F0tex/BSMUbkSKTF286MZGkHBPo12qc+9KX9KH0r+hSgn6PPPTMkD0lJDBAgDyUgcub3P+fMnOHMkNQo9j0u2u1v33v/Rx/8+Cc//fBnzZ//4pe/+ujex79+yaMZs+m1HfkRezUinPpeSK+FJ3z6KmaUBCOffj2anir96zll3IvCF2IZ02FAJqE39mwiAL25t9WxvNCwLgM6Id+Yn53/wer3f/fm3nZ7zzw4bp+cGO29B+aDo2MTCocPDk8Ojgxzr62P7Ufmt3+G4y9Xbz7+5L+WE9mzgIbC9gnnN2Y7FkNJmPBsnyZNa8ZpTOwpmdAbKIYkoHwodfcTYweIY4wjBr9QGJqWPSQJOF8GI7AMiHD5qqbgJu1mJsbHQ+mF8UzQ0E4bGs98Q0SGGgvD8Ri1hb+EArGZB301bJcwYgsYsUokPuY28anzhWAzugvFL+wAzrvKHIyHUjXuUO5NwmqyqdtQutSfUwHaGYUBYnQA+UT+U8hXjqDgJPI6yUtBIsNNlo/92CUjKqSlWsuM00vTCuk7OwoCEjrSikKalIykmVQ6JRdjyHE39GyqClWN2DbcRF7NgviQaiC8eCgXy6S5s2MMup3HfePJ5avOQLfNyZyOooW0xlEE15UOzblLCVP8Zn94s3cwbBqG3IGTYViQ3UKXEGnMy9Hk9n5S0lw+9WLrnZPJhRA7Y9WRKqACRoJ5C2ka2zCNDbMcyvdJLFWq5ebKBhCBUS4ilkfV4kp+7v9Jfm2V4Q+TH9zjzsVZ5T4DU4dxdnn9pNcxnvQfn/6x8wLuf0CmlMDWI2Ap4LxVM/YpCTx/KS/PXyXyPExncieRG5fBedgjbJIRnqw5ZatiQMV3OKVrYtV1VFlQynHgkphubkLqcXjYOvzSMFA7UPDhYeuoDA81PGodl+GRhsetkzI81vCkZba/LOAJwIdmu2Xul6DZ1nS/Be4lut8sj1p9BqPaDFpPYGg2pqGVjbloZWNCWtmYlVY2pqaVzfmlUpYkzCM1bcOJT61Hs9ChY3i2OSlnbJ1ng3IO290ZPAcDD2ahzCNk+2AU07A06yvH2iT6rXn4+2QzrWst61fWmu1HnH6P5o7MTc0BzZdWJFxYWdnyU8vy8unaKmw2LRgS6/XrfOeXr9W2r1i/j6yfs4sLZBc5GwyQDXJ2dYXsKmfdLrJuzk5PkZ3m7NkzZM/Qroh3igH7BewXrRSwi/DFuXxh3T/PaxeqdpHXeqrWy2vPVe15VhuRLNhYkjwWBM7ZCNl4jHCMcIJsgsxD5iF7i+wtsimyKbIZshmyObI5snfI3iEbIOPIFsgWyJbIlsjukN0hY8gYMlFAS3i+Q5WWiyFqYeGATCCLkcXIbpHd5swDu13LiwTJAF1k0VggoYx2oe3P1HssmL/NmOOp27XrZFU7SoE6f2M5ZDKhLPd2MKbnQMisPz68KTswNdRzg+ulBzuHZtjlYFaVg+KWwetesOKsERqAcEdZhEm3Vcs7WpsTBvuHAHcoweM73dfuW/cz30LXF0/kfeadNBy3mezkLfFuAXFJ8vMCniMcFHCAAxHmN3UUFHdViAIKNIVXxmm2HlSxWBGq5pYEtyyIklDE6pIAbwuUcR9aBjFyVak0kfukjRReqs7RU6s89W1Wtu3LmDLYUtnn0oIdN/DggZlds145XDo8n2FCOlYwpSxsmXsHwcwQubBYERa5MFLKqKgyVWVYXarqEqtCy6LQb1X9FquxqsZ5dbnSKMa5WxHucuF2Rcgj2ypLWDHh22IfcSjMt4q5RXFZOrAQqhqshEyKXW9FA4Ij6K6H1Qy9+Zo3R+8JgXfsFVmz3ICor6IVA81yA1hJ6/1LYW6SbQNVk2wfyNaCyiHbC/WgYHYbkityEyo3C75QuYveKrnCfT15UWRvl931vap42yqvFdktBbfXg9s6+Mb1AG8x8ALTp0mN6sHr8VdBncrhy1fySVinE0HUQoPzfm2ImITwNZPIARTqe0EmifRq++HTcCLcRPbSa11vGCWJfKzONRbzyE/kSzjV6PBBn8gurdXtOJF2XKc6Hoc89bnGAnYEqXaF9FZ2Zj5luCXqWn6Xvxo8RQHKOe71kPZ6OTzzmGe7Pi0esGd6L678eTCnbERYcmMOZTp1jHTqKEH9zSTLE2rbLNeSskPlG/nzdIKmUbMm5La51rrnLLRN9pAaa5tVI3+Kr/WYCLDECiP9Rs5XHSaMOOsumpadKj58sxP/bi97xvx1J00rPoyWvRxvvu6kYG07vbMVB/U879Xbu5Tz9Sa6itYPWxzDJLWE+pMIKurPPKKSy0rwKEze3NvO/w406gsv9/fMB3v7z83tR8//1NDHh43fND5tfNYwG0eNR41u46px3bAbd42/Nv7e+MfW37b+ufWvrX+npu+/l14bnzQqx9Z//gfDxp3J</latexit>

d ? E = ⇢<latexit sha1_base64="viw5upORk5/+8afDNMJpgt35mKQ=">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</latexit>

charge densityn-form

F = q2E<latexit sha1_base64="3vLsmChFQSyIzPzC6IBlgYO6JPc=">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</latexit>

Coulomb electric force

q1<latexit sha1_base64="yy4qUVyhIJeIyXK/z1alHJ/r0N0=">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</latexit>

q2<latexit sha1_base64="+dOhe2cdBHO03TaIQvitpJqzVJo=">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</latexit>

Electrodynamics

magnetic field

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v1<latexit sha1_base64="jbLQ8qkGUIs8+I5vd33/erLwbBQ=">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</latexit>

� 2 ⌦2(M ;R)<latexit sha1_base64="FipJXRJaGI22CAB4XrQpXDXwlic=">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</latexit>

d� = 0<latexit sha1_base64="H7GEE4ngpb1hXNvhlsAT6zL+XeE=">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</latexit>

d ? � = J<latexit sha1_base64="VP/qq1hgvfOKNGbAiSpPPFI2XkM=">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</latexit>

electriccurrentflux form

q1<latexit sha1_base64="yy4qUVyhIJeIyXK/z1alHJ/r0N0=">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</latexit>

q2<latexit sha1_base64="+dOhe2cdBHO03TaIQvitpJqzVJo=">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</latexit>

Electrodynamics

magnetic field

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v1<latexit sha1_base64="jbLQ8qkGUIs8+I5vd33/erLwbBQ=">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</latexit>

� 2 ⌦2(M ;R)<latexit sha1_base64="FipJXRJaGI22CAB4XrQpXDXwlic=">AAAVJXiczZhbj9vGFceV9JaqN6dZoA95obtYNA1WC1Hu3hAnsL0rQ3Gl3bXkdQwvFWNEjiRavO3MSJZ2QKBfo33tQ1/aj9K3okCe+i363DND8pCUxAAF8hABomZ+/3POzJkbSY0iz+Wi2fzmvfd/8MMf/fgnH/y0/rOf/+KXv7r34a9f8nDObHpth17IXo0Ip54b0GvhCo++ihgl/sijX41mZ0r/akEZd8PghVhFdOiTSeCOXZsIQG/u/cYaUUEsN7AufTohX7c+6X1m9fu/f3Nvt3lgHp40T0+N5sED88HxiQmFowdHp4fHhnnQ1J/dR+Y3f4bPX67efPjRfy0ntOc+DYTtEc5vzGYkhpIw4doejevWnNOI2DMyoTdQDIhP+VDqDGJjD4hjjEMG30AYmhY9JPE5X/kjsPSJmPJ1TcFt2s1cjE+G0g2iuaCBnTQ0nnuGCA01HIbjMmoLbwUFYjMX+mrYU8KILWDQSpH4mNvEo87ngs3pPhQ/t3247itzMB5K1bhDuTsJyskmbkM5pd6CCtDOKQwQowPIJ/SeQr5yBAUnltdxVvJjGWyzfOxFUwIzJi3VWmqc/NStgL6zQ98ngSOtMKBxwUiacalTcjmGHPcD16aqUNaIbcMk8nIWxINUfeFGQ7lcxfW9PWPQaT/uG08uX7UHum1OFnQULqU1DkP4XevQgk8pYYrftIY3B4fDumHIPbgYhgXZLXUJkca8GE3utuKCNuUzN7LeOamcC5EzVh0pAypgJJi7lKaxC8vYMIuhPI9EUqVabK5oABEY5SJkWVQtruU3/Z7k11QZfjf5wRy3L85L8wxMfYzzy+sn3bbxpP/47I/tFzD/PplRAqePgK2A61at2KfEd72VvOy9imUvSFZyO5Zbt0Ev6BI2SQmPN5zSXTGg4luckj2x7joqbSjlOJiSiG5vQupxeNg4+sIwUDtU8OFR47gIjzQ8bpwU4bGGJ43TIjzR8LRhNr/I4SnAh2azYbYK0Gxq2mqAe4G26sVRq85gVJlB4wkMzdY0tLI1F61sTUgrW7PSytbUtLI9v0RKk4R1pJZtMPGo9WgeOHQMtzcn4Yxt8nRQenDcncOt0HdhFcosQnoOhhENCqu+9NlYRL8zj/4Qb6dVraX9SluzvZDT/6O5Y3Nbc0CzrRWKKeysdPupbXn5dGMX1usWDIn1+nV28svX6thXrN9H1s/YxQWyi4wNBsgGGbu6QnaVsU4HWSdjZ2fIzjL27BmyZ2iXxzvDgP0c9vNWcthB+KInX1j3e1ntQtUuslpX1bpZ7bmqPU9rI5IGG0uSxYLAGRshG48RjhFOkE2QuchcZG+RvUU2QzZDNkc2R7ZAtkD2Dtk7ZANkHNkS2RLZCtkK2R2yO2QMGUMmcmgJ13Oo0jIxQC3IHZAJZBGyCNktstuMuWC3b7mhICmgyzQa8yWU0S6wvbl6lAXztylzXDVd+05atcMEqOvXlkMmE8oybwdjug6ETPvjwcOyA0tD3Te43npwcmiGXfbnZdnPpwwe9/w1Z43QAIQ7ykJMuqla3tPagjA4PwS4Qwlu38m5dt+6n/rmuv5xRdZn3k7CcZvJdtYS7+QQtyTv5bCHcJDDAQ5EkE3qyM9nVYgcCjSFR8ZZuh9UMd8RqjYtCNOiIApCHqtDfJwWKOM5tPIj5KpSaiLzSRrJvVSdo6dWeeJbLx3blxFlcKSyT6UFJ67vwg0z/U175XDp8GyFCelY/oyyoGEeHPpzQ2TCck1YZsJIKaO8ylSVYXWlqiusCi2LXL9V9VusRqoaZdXVWqMY525NuMuE2zUhi2yrLGHHBG/zc8SBt8GyuUVxWzqwEcoa7IRUiqbumgYER3C6GVYz9OYb3hy9JwSesddkzTIDot6K1gw0ywxgJ232L4GZSXoMlE3ScyDdCyqH9CzUg4LZbUkuz02o3Cx4Q+VT9FbJ5e6byYs8e7vorueq5G2rvNbkaSG4vRnc1sG37gd4ioEHmD6NK1QXHo+/9KtUDm++kk+CKp0IojYaXFuVISISwNtMLAdQqO4FmcTSreyHR4OJmMaym/xW9YZREsvH6lphsQi9WL6ES4UOL/Sx7NBK3Y5iaUdVquNyyFNfKyzgRJDqVEimsj33KMMjUdeyWf5y8BQFKGe420Xa7Wbw3GWuPfVofoM912dx6c+DBWUjwuIbcyiTpWMkS0cJ6p8mWVxQu2axFhcdSu/InyYLNImaNiF3zY3WXWepbdKb1FjbrBt5M3ysx0SAxVYQ6idyvu4wYcTZdNG06FTy4dud+Ld72XPmbTppWvJhtOjluItNJwUr2+merzmo+3m32n5KOd9soqNo9bBFESxSS6g/iaCi/swjKrm0BLfC+M293ezvQKO68LJ1YD44aD03dx89/1NNfz6ofVz7be2Tmlk7rj2qdWpXteuaXYtrf639vfaPnb/t/HPnXzv/Tkzffy/5rX1UK312/vM/KIGfew==</latexit>

d� = 0<latexit sha1_base64="H7GEE4ngpb1hXNvhlsAT6zL+XeE=">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</latexit>

d ? � = J<latexit sha1_base64="VP/qq1hgvfOKNGbAiSpPPFI2XkM=">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</latexit>

electriccurrentflux form

v2<latexit sha1_base64="O91oORp+jcwmemheqGgHPlbgYZk=">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</latexit>

F = �q2iv2�

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F= q2v2 ⇥ B<latexit sha1_base64="U3nPvb88I3xu3AkU1lw7ENCOeVU=">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</latexit>

Lorentz magnetic force

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Electrodynamics

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Coulomb electric force

Coulomb electric force

Lorentz magnetic force

Lorentz magnetic force

This figure seems to violate the conservation of momentum

Vector potential• The magnetic field satisfies� 2 ⌦2(M)

<latexit sha1_base64="Q0YIa+tETIXZQJY7RzV3mlKyoLI=">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</latexit>

d� = 0<latexit sha1_base64="7coRYzLsz+P9JPZGifzRFkT2qsQ=">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</latexit>

• Kelvin (1851) and Helmholtz (1858): by Hodge decomposition� = d↵

<latexit sha1_base64="iGDuDAvtqpI3h9VUK9dIh3WaqKk=">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</latexit>

• The potential is not unique↵ 2 ⌦1(M)<latexit sha1_base64="23p8JbHRhKaFLIRpGNg7T9I5hcA=">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</latexit>

↵, ↵+ d'<latexit sha1_base64="aGyy38ddFNc7TTRSVY67Kee++bU=">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</latexit>

are equality valid representation.

• A change of choice ↵ 7! e↵= ↵+ d'<latexit sha1_base64="+gbf6uS01fZt5k9fkrsNmbsgY8o=">AAAVJHiczZhbb9vIFce129tWvWV3/dYXpobRYmsborO+Ackiia1Am8qXWHE2iKkNRuRIYsSbZ0ay5AGBfop+hH6KvrZPfSv6UKDoZ+mZIeeQlMQFFuhDBYic+f3POcPDuXDIQRL4XLRa//ro4x/88Ec//sknP23+7Oe/+OWvHnz62RseT5lLr904iNnbAeE08CN6LXwR0LcJoyQcBPSbweRE6d/MKON+HL0Wi4T2QzKK/KHvEgHo/YOWQ4JkTJyQJFzEzp3vUeEHHs3wEys7W7+3PGdGWDL23z/YbO3a+0et42OrtfvIfnR4ZEPh4NHB8f6hZe+29G+zkf8u33/6+b8dL3anIY2EGxDOb+xWIvqSMOG7AU2bzpTThLgTMqI3UIxISHlf6tRSawuIZw1jBv9IWJqWPSQJOV+EA7AMiRjzZU3BddrNVAyP+tKPkqmgkZs1NJwGlogtdZ8sz2fUFcECCsRlPlyr5Y4JI66Au1mJxIfcJQH1ngg2pdtQfOKGcNxW5mDcl6pxj3J/FFWTzdz6ckyDGRWgnVK4QYz2IJ84eAH5ygEUvFRep6YUpjJaZ/lMddOACumo1nLj7NR0InrnxmFIIk86cUTTkpG008pFyfkQctyOfJeqQlUjrgudyNNmc2vL6nXaz66s5xdv2z3dAiczOojn0hnGMZyXmp3xMSVM8Zu9/s3ufr9pWXILDpblQA5zXUKkMS9Hk5t7aUkb84mfOHdeLhdC4g3VhVQBFZAv8+fStjZhsFp2OVQQkESqLMvNlQ0gAqMwOZiJqsWl/Mb/J/m1VIb/m/ygj9vnp5V+BqZ+1unF9fNu23p+9ezkD+3X0P8hmVACi4+AAY+jU43LFyT0g4W8OHubyrMoG6/tVK4d7GdRl7BRTni64pSP/R4V3+GUjfxl10Fl2ijH3pgkdH0TUt+HxzsHX1kWavsKPj7YOSzDAw0Pd47K8FDDo53jMjzS8HjHbn1VwGOAj+3Wjr1XgnZL070dcC/RvWb5rtVnMKjNYOc53Jq1aWhlbS5aWZuQVtZmpZW1qWllfX6ZlCcJ40gN22gUUOfpNPLoEJ5uXsYZW+X5TTmDRe0UnoShD6NQmgj5ahcnNCqN+spvZRD91j74Ml1P61rLrytvzQ1iTr9Hc4f2uuaAmqkVizHMrHz6qWl58WJlFjabDtwS5907s77Ld2pxV+zqCtmVYefnyM4N6/WQ9Qy7vER2aVing6xj2MkJshPDXr5E9hLtingnGPCqgFdFKwXsIHx9Jl87D89M7VzVzk2tq2pdU3ulaq/y2oDkwYaSmFgQ2LABsuEQ4RDhCNkImY/MR/YB2QdkE2QTZFNkU2QzZDNkd8jukPWQcWRzZHNkC2QLZPfI7pExZAyZKKCjN4RKM2KEWlQ4IBPIEmQJsltkt4b5YLft+LEgOaDzPBoLJZTRLnKDqdrJgvmHnHm+6q5tL6+6cQbU8VvHI6MRZcbbw5i+ByHz6wlgr+zB0FDPDa6nHqwcmuElh9OqHBZdBpu6cMlZIzQA4Z6yGJNuqZa3tAb7aVg/BLhDCR7f2br20HmY+xa6PvnCXDNvZ+G4y2TbtMQ7BcQpyc8KeIawV8Ae3ojIdOogLHpViAIKNIWN4SSfD6pYzAhVG5eEcVkQJaGI1SEhdguUcR1ahAlyVak0YXyyRgovVefoqVWe+TYry/ZFQhksqewL6cCKG/rwwMzP+VV5XHrcjDAhPSecUBbt2Lv74dQSRpgvCXMjDJQyKKpMVRlWF6q6wKrQsij0W1W/xWqiqompLpYaxTj3S8K9EW6XBBPZVVnCjIk+FOsIvP2RqrlDcVp6MBGqGsyEXII3wyUNCN7B8WpYzdCbr3hz9B4R2GMvyZoZA/2KumSgmTHI3lyXLDJoTPJloGqSrwP5XFA55GuhvimY3ZrkityEys2B91A+Rm+VXOG+mrwosnfL7rqvKt6uymtJHpeCu6vBXR187XyAXQxsYK5oWqP6sD3+OqxTObzfSj6K6nQiiJpocNyrDZGQCN5mUtmDQv1VkFEq/drrCGg0gpd+2c3OdVfDKEnlM3WssZjFQSrfwKFGh9f2VHZore4mqXSTOtXzOeSpjzUWsCJItSpkXdmeBpThkqhrppe/7r1AAcoGd7tIu10DT33mu+OAFg/YU70WVz4RzCgbEJbe2H2ZDR0rGzpKUB+aZHlAbdrlWlp2qLwjf5EN0Cxq3oTctFda9725tskfUkNts2wUTHBbj4kAS50o1jtyvuwwYsRbddG07FTx4eud+Hd7uVMWrDppWvFhtOzl+bNVJwVr2+meLjmo53m33n5MOV9toqNo/W1LEhikjlCfgqCiPtkRlVxegkdh+v7BpvnoZ9UX3uzt2o929159ufn01R+zL4OfNH7d+E3jdw27cdh42ug0LhvXDbfxp8ZfGn9t/G3jzxt/3/jHxj8z048/yr8mft6o/Db+819ONJti</latexit>

is called agauge transformation

• All physically observable quantities should be gauge-invariant.

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Electrodynamics

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Coulomb electric force

Coulomb electric force

Lorentz magnetic force

Lorentz magnetic force This figure seems to violate the conservation of momentumProblem solved if we revise the momentum as

(but now the momentum is not gauge invariant)p[ = mv[ + q↵

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Quantum mechanics

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Quantum mechanics

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v2<latexit sha1_base64="kondWRy6dVmxwX7kfbyyZ2pCm88=">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</latexit>

1 = r1ei✓1<latexit sha1_base64="e71IeAYCKkqeRgRb90R9DgWus1I=">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</latexit>

2 = r2ei✓2<latexit sha1_base64="hLIWhwBSzPX8wtWLt9IYY2067xA=">AAAVGniczZhbb9vIFce1u73sqrds1299YWq4LRa2ISrrG5AsktgKtKl8iRVng5heY0SOJEa8eWakSB4Q6AfoR+in6Gv71Leir30p0A/TM0POISmJCxToQwWYnvn9zzkzZ268DJLA56LV+tdHH3/ygx/+6Mefftb8yU9/9vNfPPj8l294PGUuvXLjIGZvB4TTwI/olfBFQN8mjJJwENBvB5NjpX87o4z7cfRaLBJ6E5JR5A99lw 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</latexit>

p[ = r2d✓<latexit sha1_base64="RATiq9a7hXI4wrxs/dTFUuI9/Vs=">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</latexit>

Quantum mechanics

Quantum mechanics

p[ = r2d✓<latexit sha1_base64="RATiq9a7hXI4wrxs/dTFUuI9/Vs=">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</latexit>

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Z1

2m|p[|2 = 1

2m

Z d ? d

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Total kinetic energy is our Dirichlet energy

Quantum electrodynamics

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Z1

2m|p[|2 = 1

2m

Z d ? d

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Total kinetic energy is our Dirichlet energy

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= (�idr) <latexit sha1_base64="X7SBI+Gba1FxZOiNEptYyE5QvQM=">AAAVGniczZhbbxvHFceZ9JYyvTiN3vqyjqA2DUSBK0c3wA5sSzIYl7pYtBzDWsYY7g7JNfemmSFNarBAP0A/Qj5FXtunvhV97UuBfpiemd05u0tyAxToQwloNfP7n3Nmztz2MkgCn4t2+18ffPijH//kpz/76OfNj3/xy1/9+t4nv3nF4ylz6bUbBzF7PSCcBn5Er4UvAvo6YZSEg4B+M5gcK/2bGWXcj6OXYpHQfkhGkT/0XSIAvb33u0eOG0fvnIT7n7eckIjxYBAHnvRT71snIoOA/EFpb+9ttnfsvcP20ZHV3nlgPzg4tKGw/2D/aO/Asnfa+rfZyH+Xbz/59N+OF7vTkEbCDQjnN3Y7EX1JmPDdgKZNZ8ppQtwJGdEbKEYkpLwvdUKptQXEs4Yxg79IWJqWPSQJOV+EA7BUXebLmoLrtJupGB72pR8lU0EjN2toOA0sEVtqdCzPZ9QVwQIKxGU+9NVyx4QRV8AYViLxIXdJQL1Hgk3pNhQfuSFct5U5GPelatyj3B9F1WQzt74c02BGBWgnFAaI0R7kEwfPIF+ppyCV16kphamM1lk+CZIxGVAhi5kzPk0nou/dOAxJ5EknjmhaMpJ2WumUnA8hx+3Id6kqVDXiujCJPG02t7asXuf0yZX19OL1aU+3wMmMDuK5dIZxDP+Xmp3xMSVM8Zvd/s3OXr9pWXILLpblQA5zXUKkMS9Hk5u7aUkb84mfOO+9XC6ExBuqjlQBFZAv8+fStjZhsVp2OVQQkESqLMvNlQ0gAqNcxMxE1eJSfuP/k/zaKsP/TX4wx6fnJ5V5BqZ+1snF9dPuqfX06snxH09fwvyHZEIJHDkCFjyuTrUun5HQDxby4ux1Ks+ibL2epnLtYj+LuoSNcsLTFad87feo+AGnbOUvuw4q20Y59sYkoeubkHocHrb2v7Is1PYUfLjfOijDfQ0PWodleKDhYeuoDA81PGrZ7a8KeATwod1u2bslaLc13W2Be4nuNsujVp/BoDaD1lMYmrVpaGVtLlpZm5BW1mallbWpaWV9fpmUJwnrSC3baBRQ5/E08ugQ7mlexhlb5fmgnMGhdgL3v9CHVShNhPy0ixMalVZ95beyiH5v73+Zrqd1reX9yltzg5jT/6K5A3tdc0DN1orFGHZWvv3Utrx4trILm00HhsR588ac7/KNOtwVu7pCdmXY+Tmyc8N6PWQ9wy4vkV0a1ukg6xh2fIzs2LDnz5E9R7si3jEGvCrgVdFKATsIX57Jl879M1M7V7VzU+uqWtfUXqjai7w2IHmwoSQmFgQ2bIBsOEQ4RDhCNkLmI/ORvUP2DtkE2QTZFNkU2QzZDNl7ZO+R9ZBxZHNkc2QLZAtkd8jukDFkDJkooCP8wKNKM2KEWlQ4IBPIEmQJsltkt4b5YLft+LEgOaDzPBoLJZTRLnKDqXp+BfN3OfN8NV3bXl514wyo67eOR0Yjyoy3hzF9D0Lm/QngCdmDpaHuG1xvPTg5NMMuh9OqHBZTBg914ZKzRmgAwh1lMSbdVi1vaW1GGJwfAtyhBLfv7Fy779zPfQtd//OF6TM/zcJxl8lT0xLvFBC3JD8r4BnCXgF7OBCRmdRBWMyqEAUUaAoPhpN8P6hisSNUbVwSxmVBlIQiVoeEOC1QxnNoESbIVaXShPHJGim8VJ2jp1Z55tusHNsXCWVwpLIvpAMnbujDDTP/n/fK49LjZoUJ6TnhhLKoZe/shVNLGGG+JMyNMFDKoKgyVWVYXajqAqtCy6LQb1X9FquJqiamulhqFOPcLQl3RrhdEkxkV2Wp3/mKc8SjsN4q5g7FbenBRqhqsBNyKRn7SxoQHMHxaljN0JuveHP0HhF4xl6SNTMGRL37LBloZgxgJ632L4PGJD8Gqib5OZDvBZVDfhbqQcHs1iRX5CZUbg68h/IxeqvkCvfV5EWRvVt213NV8XZVXkvyuBTcXQ3u6uBr9wM8xcADzBVNa1QfHo+/DutUDu+3ko+iOp0IojYaXHdrQyQkgreZVPagUN8LMkqlX9uPgEYjeOmX3ex/XW8YJal8oq41FrM4SOUruNTo8Nqeyg6t1d0klW5Sp3o+hzz1tcYCTgSpToVsKk+nAWV4JOqameWve89QgLLB3S7SbtfAE5/57jigxQ32RJ/FlU8EM8oGhKU3dl9mS8fKlo4S1OclWV5Qm3a5lpYdKu/IX2QLNIuaNyE37ZXWfW+ubfKb1FDbLBsFE3ysx0SApU4U6ydyvuwwYsRbddG07FTx4eud+A97uVMWrDppWvFhtOzl+bNVJwVr2+meLDmo+3m33n5MOV9toqNo/bAlCSxSR6hPQVBRn+yISi4vwa0wfXtv03z0s+oLr3Z37Ac7uy++3Hz84k/Zl8GPGr9tfNb4vGE3DhqPG53GZeO64Tb+3Pi+8ZfGXze+2/jbxt83/pGZfvhB/jXx00blt/HP/wA6rpdW</latexit>

Quantum electrodynamics

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Total kinetic energy is our Dirichlet energy

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Quantum electrodynamicsGauge transform in quantum electrodynamics

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Curvature of the bundleis gauge invariant

Interference pattern is gauge invariant

Singularity location is gauge invariant

Aharonov–Bohm effect

What determines the complex line bundle?By Hodge decomposition Z

globalloop

<latexit sha1_base64="4PLrlymtxZV+VDVo5eSn1CpDAiw=">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</latexit>

� = d↵<latexit sha1_base64="iGDuDAvtqpI3h9VUK9dIh3WaqKk=">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</latexit>

and allows us to reconstruct ↵<latexit sha1_base64="vrE2lD5p5ODTbqZcLXK+OqiZgSY=">AAAU+HiczZhbb9vIFce129tWvWW7fusLU8NosbAM0VnfgOwiia1Am8qXWHE2iKkGI3Ik0eLNMyNF8oBAP0Jf26e+FX3ttynQD9MzQ84hKYkLFOhDBYic+f3POTOHc+FlmAQ+F+32vz759Ac//NGPf/LZT5s/+/kvfvmrR5//+i2PZ8ylN24cxOzdkHAa+BG9Eb4I6LuEURIOA/rdcHqq9O/mlHE/jt6IZUIHIRlH/sh3iQD01iFBMiEfHm239+yD4/bJidXee2I/OTq2oXD45PDk4Miy99r6t93If1cfPv/i344Xu7OQRsINCOe3djsRA0mY8N2Apk1nxmlC3CkZ01soRiSkfCB1d1NrB4hnjWIG/0hYmpY9JAk5X4ZDsAyJmPBVTcFN2u1MjI4H0o+SmaCRmzU0mgWWiC2Vu+X5jLoiWEKBuMyHvlruhDDiCrhClUh8xF0SUO9rwWZ0F4pfuyEcd5U5GA+katyj3B9H1WQzt4Gc0GBOBWhnFC4Qo33IJw5eQr5yCAUvlTepKYWpjDZZPldjM6RCOqq13Dg7NZ2IfnTjMCSRJ504omnJSNpppVNyMYIcdyPfpapQ1YjrwiDytNnc2bH63c7za+vF5btOX7fAyZwO44V0RnEM55Vm53xCCVP8dn9wu3cwaFqW3IGDZTmQw0KXEGnMy9Hk9n5a0iZ86ifORy+XCyHxRqojVUAF5Mv8hbStbZisll0OFQQkkSrLcnNlA4jAKBcxM1G1uJLf5P8kv7bK8H+TH4xx5+KsMs7A1M86u7x50etYL66fn/6h8wbGPyRTSmBDETDhcXaqefmShH6wlJfn71J5HmXztZPKjZP9POoRNs4JT9ec8rnfp+J7nLKZv+o6rCwb5difkIRubkLq6/C0dfiNZaF2oODTw9ZRGR5qeNQ6LsMjDY9bJ2V4rOFJy25/U8ATgE/tdsveL0G7rel+C9xLdL9Zvmr1GQxrM2i9gEuzMQ2tbMxFKxsT0srGrLSyMTWtbM4vk/IkYR6paRuNA+o8m0UeHcEdy8s4Y+s8vyjnsKmdwd0t9GEWShMh3+3ihEalWV/5rU2i39mHX6WbaV1reb/y1twg5vS/aO7I3tQcULO0YjGBlZUvP7UsL1+urcJm04FL4rx/b/Z3+V5t7opdXyO7NuziAtmFYf0+sr5hV1fIrgzrdpF1DTs9RXZq2KtXyF6hXRHvFANeF/C6aKWAXYRvzuUb5/G5qV2o2oWp9VStZ2qvVe11XhuSPNhIEhMLAhs2RDYaIRwhHCMbI/OR+cjukN0hmyKbIpshmyGbI5sj+4jsI7I+Mo5sgWyBbIlsiewB2QMyhowhEwV0hB94VGlGjFCLCgdkAlmCLEF2j+zeMB/sdh0/FiQHdJFHY6GEMtpFbjBTT6dgfpczz1fDtevlVTfOgDr+0fHIeEyZ8fYwpu9ByLw/ATz/ejA11H2D66UHO4dm2OVwVpXDYsjgoS5ccdYIDUB4oCzGpNuq5R2tzQmD/UOAO5Tg9p3ta4+dx7lvoeuTL0yfeScLx10mO6Yl3i0gLkl+XsBzhP0C9vFCRGZQh2ExqkIUUKApPBhO8/WgisWKULVJSZiUBVESilhdEuKwQBn3oWWYIFeVShPGJ2uk8FJ1jp5a5Zlvs7JtXyaUwZbKvpQO7LihDzfM/Jz3yuPS42aGCek54ZSyqGXvHYQzSxhhsSIsjDBUyrCoMlVlWF2q6hKrQsui0O9V/R6riaomprpcaRTjPKwID0a4XxFMZFdlCSsmuiv2EY/CfKuYOxSXpQcLoarBSsilZOKvaEDwCk7Ww2qG3nzNm6P3mMAz9oqsmTHQ76UrBpoZA1hJ6/3LoDHJt4GqSb4P5GtB5ZDvhfqiYHYbkityEyo3B95D+QS9VXKF+3ryosjeLbvrsap4uyqvFXlSCu6uB3d18I3rAZ5i4AHmmqY1qg+Px9+GdSqH91vJx1GdTgRRCw2O+7UhEhLB20wq+1Co7wUZp9Kv7UdAozG89Mtedq7rDaMklc/VscZiHgepfAuHGh1e21PZpbW6m6TSTepUz+eQpz7WWMCOINWukA1lZxZQhluirplR/rb/EgUoG9zrIe31DDzzme9OAlrcYM/0Xlz5RDCnbEhYemsPZDZ1rGzqKEF9PJLlCbVtl2tp2aHyjvxlNkGzqHkTcttea933Ftomv0mNtM2qUTDFx3pMBFjqRLF+IuerDmNGvHUXTctOFR++2Yl/v5c7Y8G6k6YVH0bLXp4/X3dSsLad3tmKg7qf9+rtJ5Tz9Sa6itZftiSBSeoI9SkIKuqTHVHJ5SW4FaYfHm2bj35WfeHt/p79ZG//9Vfbz17/Kfsy+FnjN43fNn7fsBtHjWeNbuOqcdNwG3eNPzf+0vjr1sPW37b+vvWPzPTTT/KviV80Kr+tf/4HZX6JsA==</latexit>

up to a factor of d'<latexit sha1_base64="DG72CCC69LkO7lgfyIBKgyElC9I=">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</latexit>

Optimal singularity placement

• Prescribe magnetic field (e.g. as Gaussian curvature)and global period

• Construct and thus

Rgloballoop↵

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Lr = dr¸?1 dr

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• Find the singularities ofthe ground state.

Optimal singularity placement

[Knöppel, Crane, Pinkall & Schröder 2015]Stripe patterns on surfaces

Optimal singularity placementComplex phase field for vortex detection

[Weißmann, Pinkall & Schröder 2014]Smoke rings from smoke

[Chern, Knöppel, Pinkall & Schröder 2017]Inside fluid: Clebsch maps for visualization and processing


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