Seminar on Vector Field Analysis on Surfaces

Transcription

1 Seminar on Vector Field Analysis on Surfaces

2 Last time Intro Cool stuff VFs on 2D Euclidean Domains Arrows on the plane Div, curl and all that Helmholtz decomposition 2

3 today Arrows on surfaces Coordinate free? Grad/div/curl through stokes Line integrals Stokes Hairy ball theorem Harmonic vfs Hodge decomposition 3

4 Tangent Vector Fields on Surfaces 4

5 Vectors vs Scalars on surfaces Functions on a surface A number at every point Vectors on a surfaces An arrow at every point 5

6 Tangent Vector Fields the physics Wind 6

7 Tangent Vector Fields the math( ish) Tangent vf: a smooth map from the surface to arrow space : Means:, 0 How to represent the arrow? Length + angle Angle with respect to what? No 2D coordinates Can use 3D coordinates? How to define norm/grad/div/curl These are physical properties Define through physical relations 7 Some images from

8 Properties of vector fields length Can take vector in R3, and use standard definition for length OK as long as pointwise ( metric inherited from embedding 8

9 Properties of vector fields length Can take vector in R3, and use standard definition for length OK as long as pointwise ( metric inherited from embedding Also gives singularities Same singularity structures as in the plane Don t need to worry about change of frame 9

10 Properties of vector fields 2D divergence How much goes out how much comes in source sink 10

11 Properties of vector fields divergence on a surface How much goes out how much comes in Can also work on a surface! source 11

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13 Properties of vector fields divergence How to measure all the stuff that comes in stuff that goes out? Only the boundary matters! Only normal to the curve 13

14 Properties of vector fields divergence through boundary integrals V is a tangent vector field 14

15 Properties of vector fields divergence through boundary integrals is a tangent vector field Coordinate invariant? surface surface area of Boundary of normal to the boundary inner product in line integral 15

16 16 Line integrals a small aside

17 Divergence through boundary integrals 2D special case 17

18 Properties of vector fields 2D Vorticity/Curl How much turns left how much turns right ccw vortex cw vortex 18

19 Properties of vector fields 2D surface Vorticity/Curl How to measure How much turns left how much turns right? Only the boundary matters! Only tangent to the curve 19

20 Properties of vector fields curl through boundary integrals V is a tangent vector field 20

21 Properties of vector fields curl through boundary integrals is a tangent vector field surface surface area of Boundary of tangent to the boundary inner product in line integral 21

22 Boundary Integrals * rings a bell? 22 * 3d is more complicated

23 Boundary Integrals * rings a bell? div / curl in 2d basically the same definition Rotate the boundary tangent instead of rotating the vf 23 Rotation by /2

24 Div & curl = all you need Again 2 dofs per point Div says what happens normal to a curve Curl says what happens tangent to a curve This completely specifies the vector field in the frame of reference of the curve But weakly, since integrated So no need for coordinates 24

25 Stokes theorem (with forms) the boundary knows it all Huh? What s? What s? Exterior calculus perhaps later Special cases (in coordinates) Stokes theorem in R3 Divergence theorem fundamental theorem of calculus Green's Theorem 25

26 Singularities on a Surface Remember singularity = norm of vf is 0 They control the structure of the field What s the simplest vf you can have In the plane? Constant On a surface? Can there be a vf without singularities? 26

27 Hairy Ball Theorem A tangent vector field on a sphere will always have a singularity What s the minimal number of singularities? Intuition? 27

28 Hairy Torus Theorem? Can have a VF without singularities on the torus What s the difference? Unless the genus is 1, must have a singularity! 28

29 The Gradient directional derivative 29

30 30 Harmonic Vector Fields

31 Hodge Decomposition On a closed surface, any vector field V can be represented as: Curl free Div free Harmonic where are scalar functions, and H is a harmonic VF 31

32 Hodge Decomposition On a closed surface, any vector field V can be represented as: Curl free Div free Harmonic 32

33 Next time Discrete surfaces Discrete VFs PC on faces Non linear representations DEC, discrete 1 forms Derivations Discrete grad/div/curl 33

34 References Wind on earth: Line integral: Divergence, curl, stokes (including all the math ): S1 Exterior calculus versions of grad/div/curl: Hairy ball theorem video: math why you cant comb a hairy ball.html Topological constraints on singularities: 34