Vector calculus in Cartesian and spherical coordinates

Transcription

1 SageManifolds.0 Vecto calculus in Catesian and spheical coodinates This woksheet illustates some featues of SageManifolds (vesion.0, as included in SageMath 7.5) egading vecto calculus in the Euclidean 3-space by means of Catesian and spheical coodinates. Click hee to download the woksheet file (ipynb fomat). To un it, you must stat SageMath with the Jupyte notebook, via the command sage -n jupyte NB: a vesion of SageMath at least equal to 7.5 is equied to un this woksheet: In []: Out[]: vesion() 'SageMath vesion 7.5, Release Date: 07-0-' Euclidean 3-space and Catesian coodinates Fist we set up the notebook to display mathematical objects using LaTeX fomatting: In []: %display latex We then intoduce the Euclidean space as a 3-dimensional diffeentiable manifold: In [3]: M Manifold(3, 'M', stat_index) pint(m) 3-dimensional diffeentiable manifold M (x, y, z) We then intoduce the Catesian coodinates as the chat cat on : M In [4]: cat.<x,y,z> M.chat() pint(cat) cat Chat (M, (x, y, z)) Out[4]: (M, (x, y, z)) Spheical coodinates (, θ, ϕ) We intoduce spheical coodinates as the chat sphe on : M In [5]: sphe.<,th,ph> M.chat(':(0,+oo) th:(0,pi):\theta ph:(0,*pi):\phi ') pint(sphe) sphe Chat (M, (, th, ph)) Out[5]: (M, (, θ, ϕ))

2 SageManifolds.0 Spheical coodinates do not fom a egula coodinate system of the Euclidean space. So declaing that they span means that, stictly speaking, the manifold is not the whole Euclidean space, but the Euclidean space minus some half plane (the azimuthal oigin). Howeve, in this woksheet, this diffeence will not matte. The change of coodinates to chat cat: M (, θ, ϕ) (x, y, z) M is intoduced as a tansition map fom chat sphe In [6]: Out[6]: sphe_to_cat sphe.tansition_map(cat, [*sin(th)*cos(ph), *sin(th )*sin(ph), *cos(th)]) sphe_to_cat.display() x y z cos(ϕ) sin(θ) sin(ϕ) sin(θ) cos(θ) The invese is also set: In [7]: sphe_to_cat.set_invese(sqt(x^+y^+z^), atan(sqt(x^+y^),z), at an(y, x), vebosetue) Check of the invese coodinate tansfomation: th actan(*sin(th), *cos(th)) ph actan(*sin(ph)*sin(th), *cos(ph)*sin(th)) x x y y z z The check that the povided fomulas do coespond to the invese change of coodinates is passed, modulo some lack of simplification in some tigonometical fomulas involving the function actan. In [8]: Out[8]: cat_to_sphe sphe_to_cat.invese() cat_to_sphe.display() θ ϕ x actan( x, z) actan(y, x) The natual vecto fame of spheical coodinates is In [9]: Out[9]: sphe.fame() ( M, (,, θ ϕ )) We shall expand vecto and tenso fields on the othonomal fame spheical coodinates, which is elated to the natual fame means of the following field of automophisms: ( e, e, e 3 ) (/, /θ, /ϕ) associated with displayed above by

3 SageManifolds.0 In [0]: Out[0]: to_othonomal M.automophism_field() to_othonomal[sphe.fame(),,,sphe] to_othonomal[sphe.fame(),,,sphe] / to_othonomal[sphe.fame(),3,3,sphe] /(*sin(th)) to_othonomal.display(sphe.fame(), sphe) d + dθ + dϕ θ sin(θ) ϕ In othe wods, the change-of-basis matix is In []: Out[]: to_othonomal[sphe.fame(),:,sphe] sin(θ) 0 0 We constuct the othonomal fame fom the natual fame of spheical coodinates by this change of basis: In []: es sphe.fame().new_fame(to_othonomal, 'e') pint(es) es Vecto fame (M, (e_,e_,e_3)) Out[]: (M, ( e, e, e 3 )) In [3]: Out[3]: In [4]: Out[4]: In [5]: Out[5]: es[].display(sphe.fame(), sphe) e es[].display(sphe.fame(), sphe) e θ es[3].display(sphe.fame(), sphe) e 3 sin(θ) ϕ If we do not specify the fame and coodinates fo the display, we get it in tems of the default ones (i.e. Catesian fame and Catesian coodinates): In [6]: es[].display() Out[6]: e x y + ( x ) x ( x ) z + ( x ) z y 3

4 SageManifolds.0 In [7]: es[].display() Out[7]: e xz yz + ( x x ) x ( x x ) x + ( x ) z y In [8]: es[3].display() Out[8]: e 3 y ( x ) x x + ( x ) y By constuction, the change of fame (/, /θ, /ϕ) ( e, e, e 3 ) change of fame (/x, /y, /z) ( e, e, e 3 ) by composition of (/x, /y, /z) (/, /θ, /ϕ) (/, /θ, /ϕ) ( e, e, e 3 ) is known. We fom the with : In [9]: Out[9]: M.set_change_of_fame(cat.fame(), es, M.change_of_fame(sphe.fame(), es) * M.change_o f_fame(cat.fame(), sphe.fame()), compute_invesefalse) M.change_of_fame(cat.fame(), es)[:, sphe] cos(ϕ) sin(θ) sin(ϕ) sin(θ) cos(θ) cos(ϕ) cos(θ) cos(θ) sin(ϕ) sin(θ) sin(ϕ) cos(ϕ) 0 Similaly, we fom the invese change of fame as: In [0]: Out[0]: M.set_change_of_fame(es, cat.fame(), M.change_of_fame(sphe.fame(), cat.fame()) * M.change_of_fame(es, sphe.fame()), compute_invesefalse) M.change_of_fame(es, cat.fame())[:, sphe] cos(ϕ) sin(θ) cos(ϕ) cos(θ) sin(ϕ) sin(ϕ) sin(θ) cos(θ) sin(ϕ) cos(ϕ) cos(θ) sin(θ) 0 At this stage, the manifold (use) atlas is In []: Out[]: M.atlas() [(M, (x, y, z)), (M, (, θ, ϕ))] The default chat is the fist one intoduced on the manifold (it can be changed by means of the function M.set_default_chat): In []: Out[]: M.default_chat() (M, (x, y, z)) The following vecto fames have been intoduced on the manifold: 4

5 SageManifolds.0 In [3]: Out[3]: M.fames() [( M, (,, )), ( M, (,, x y )), (M, ( e z θ, e ϕ, e 3 )) ] The default fame is the fist one intoduced on the manifold (it can be changed by means of the function M.set_default_fame): In [4]: Out[4]: M.default_fame() ( M, (, x, y z )) The following changes of fame have been defined: In [5]: Out[5]: M.changes_of_fame() {(( M, (,, )), (M, ( e x y, e z, e 3 )) ) : Field of tangent-space automophisms on the 3-dimensional diffeentiable manifold M ( (M, (,, )), ( M, (, e e e 3 x, y z ))) : Field of tangent-space automophisms on the 3-dimensional diffeentiable manifold M (( M, (,, )), ( M, x y (, z, θ ϕ ))) : Field of tangent-space automophisms on the 3-dimensional diffeentiable manifold M (( M, (,, )), ( M, θ (, ϕ x, y z ))) : Field of tangent-space automophisms on the 3-dimensional diffeentiable manifold M ( (M, (,, )), ( M, (, e e e 3, θ ϕ ))) : Field of tangent-space automophisms on the 3-dimensional diffeentiable manifold M (( M, (,, )), (M, ( e θ, e ϕ, e 3 )) ) : Field of tangent-space automophisms on the 3-dimensional diffeentiable manifold M } Vecto field defined in tems of its Catesian components We define the vecto field Catesian fame: U in tems of its components with espect to the default fame, i.e. the In [6]: U M.vecto_field(name'U') U[:] [function('u_x')(x,y,z), function('u_y')(x,y,z), function('u_z') (x,y,z)] U.display() Out[6]: U Ux (x, y, z) + U y (x, y, z) + U z (x, y, z) x y z 5

6 SageManifolds.0 We can ask fo its components in tems of the spheical othonomal fame: In [7]: Out[7]: U.display(es) x U x (x, y, z) + y U y (x, y, z) + z U z (x, y, z) U ( x ) x U z (x, y, z) + y U z (x, y, z) (x U x (x, y, z) + y U y (x, y, z))z + ( x x ) + ( y U x (x, y, z) x U y (x, y, z) x ) e 3 e e In [8]: Out[8]: U U.display_comp(es) U U 3 x U x (x,y,z)+y U y (x,y,z)+z U z (x,y,z) x + y + z x U z (x,y,z)+ y U z (x,y,z)(x U x (x,y,z)+y U y (x,y,z))z y U x(x,y,z)x U y (x,y,z) x + y x + y + z x + y The above components ae displayed in tems of the default chat (Catesian coodinates). If we want them in tems of spheical coodinates, we have to specify it, by setting the second agument of the function display to sphe: In [9]: Out[9]: U.display(es, sphe) U ( ( cos(ϕ) sin(θ), sin(ϕ) sin(θ), cos(θ)) cos(ϕ) sin(θ) U x + ( cos(ϕ) sin(θ), sin(ϕ) sin(θ), cos(θ)) sin(ϕ) sin(θ) U y + U z ( cos(ϕ) sin(θ), sin(ϕ) sin(θ), cos(θ)) cos(θ)) e + ( U x ( cos(ϕ) sin(θ), sin(ϕ) sin(θ), cos(θ)) cos(ϕ) cos(θ) + U y ( cos(ϕ) sin(θ), sin(ϕ) sin(θ), cos(θ)) cos(θ) sin(ϕ) U z ( cos(ϕ) sin(θ), sin(ϕ) sin(θ), cos(θ)) sin(θ)) e + ( U y ( cos(ϕ) sin(θ), sin(ϕ) sin(θ), cos(θ)) cos(ϕ) U x ( cos(ϕ) sin(θ), sin(ϕ) sin(θ), cos(θ)) sin(ϕ)) e 3 In [30]: Out[30]: U U.display_comp(es, sphe) U U 3 U x ( cos(ϕ) sin(θ), sin(ϕ) sin(θ), cos(θ)) cos(ϕ) sin(θ) + U y ( cos(ϕ) sin(θ), sin(ϕ) sin(θ), cos(θ)) sin(ϕ) sin(θ) + U z ( cos(ϕ) sin(θ), sin(ϕ) sin(θ), cos(θ)) cos(θ) U x ( cos(ϕ) sin(θ), sin(ϕ) sin(θ), cos(θ)) cos(ϕ) cos(θ) + U y ( cos(ϕ) sin(θ), sin(ϕ) sin(θ), cos(θ)) cos(θ) sin(ϕ) U z ( cos(ϕ) sin(θ), sin(ϕ) sin(θ), cos(θ)) sin(θ) ( cos(ϕ) sin(θ), sin(ϕ) sin(θ), cos(θ)) cos(ϕ) U y U x ( cos(ϕ) sin(θ), sin(ϕ) sin(θ), cos(θ)) sin(ϕ) 6

7 SageManifolds.0 We may also ask fo the components of U w..t. the natual fame of spheical coodinates: In [3]: Out[3]: U.display(sphe.fame()) x U x (x, y, z) + y U y (x, y, z) + z U z (x, y, z) U ( x ) ( x U z (x, y, z) + y U z (x, y, z) (x U x (x, y, z) + y U y (x, y, z))z) x + ( x 4 + x y + y 4 + ( x + y ) z ) θ y U x (x, y, z) x U y (x, y, z) + ( x + y ) ϕ In [3]: Out[3]: U.display(sphe.fame(), sphe) U ( U x ( cos(ϕ) sin(θ), sin(ϕ) sin(θ), cos(θ)) cos(ϕ) sin(θ) + U y ( cos(ϕ) sin(θ), sin(ϕ) sin(θ), cos(θ)) sin(ϕ) sin(θ) + U z ( cos(ϕ) sin(θ), sin(ϕ) sin(θ), cos(θ)) cos(θ)) U x ( cos(ϕ) sin(θ), sin(ϕ) sin(θ), cos(θ)) cos(ϕ) cos(θ) + U y ( cos(ϕ) sin(θ), sin(ϕ) sin(θ), cos(θ)) cos(θ) sin(ϕ) U + z ( cos(ϕ) sin(θ), sin(ϕ) sin(θ), cos(θ)) sin(θ) + U y U x ( cos(ϕ) sin(θ), sin(ϕ) sin(θ), cos(θ)) cos(ϕ) ( cos(ϕ) sin(θ), sin(ϕ) sin(θ), cos(θ)) sin(ϕ) sin(θ) ϕ θ Vecto field defined in tems of its spheical components Let us conside a vecto field ( e, e, e 3 ) fame es : V defined by its components with espect to the othonomal spheical In [33]: V M.vecto_field(name'V') V[es,:,sphe] [function('v_')(,th,ph), function('v_')(,th,ph), fu nction('v_3')(,th,ph)] V.display(es, sphe) Out[33]: V V (, θ, ϕ) e + V (, θ, ϕ) e + V 3 (, θ, ϕ) e 3 We may ask fo the components of this vecto field with espect to the Catesian fame (fist agument cat.fame()), each component being expessed in tems of spheical coodinates (second agument sphe): 7

8 SageManifolds.0 In [34]: Out[34]: In [35]: Out[35]: V.display(cat.fame(), sphe) V ( V (, θ, ϕ) cos(ϕ) cos(θ) + V (, θ, ϕ) cos(ϕ) sin(θ) V 3 (, θ, ϕ) sin(ϕ)) x + ( V (, θ, ϕ) cos(θ) sin(ϕ) + V (, θ, ϕ) sin(ϕ) sin(θ) + V 3 (, θ, ϕ) cos(ϕ)) y + ( V (, θ, ϕ) cos(θ) V (, θ, ϕ) sin(θ)) z V.display_comp(cat.fame(), sphe) V x V y V z V (, θ, ϕ) cos(ϕ) cos(θ) + V (, θ, ϕ) cos(ϕ) sin(θ) V 3 (, θ, ϕ) sin(ϕ) V (, θ, ϕ) cos(θ) sin(ϕ) + V (, θ, ϕ) sin(ϕ) sin(θ) + V 3 (, θ, ϕ) cos(ϕ) V (, θ, ϕ) cos(θ) V (, θ, ϕ) sin(θ) Euclidean metic The standad Euclidean metic is intoduced as a Riemannian metic on diag(,, ) espect to the Catesian fame ae : M, whose components with In [36]: Out[36]: g M.iemannian_metic('g') g[,], g[,], g[3,3],, g.display() g dx dx + dy dy + dz dz The components of g with espect to the spheical coodinates ae then: In [37]: g.display(sphe.fame(), sphe) Out[37]: g d d + dθ dθ + sin (θ) dϕ dϕ ( e, e, e 3 ) diag(,, ) : Since es is an othonomal fame, the components of with espect to it ae g In [38]: g.display(es, sphe) Out[38]: g e e + e e + e 3 e 3 The covaiant deivative opeato is intoduced as the (Levi-Civita) connection associated with : g In [39]: nabla g.connection() pint(nabla) nabla Levi-Civita connection nabla_g associated with the Riemannian metic g on the 3-dimensional diffeentiable manifold M Out[39]: g The connection coefficient with espect to the natual fame of spheical coodinates (Chistoffel symbols) ae: 8

9 SageManifolds.0 In [40]: Out[40]: nabla.display(sphe.fame(), sphe) Γ θ θ Γ ϕ ϕ Γ θ θ Γ θ θ Γ θ ϕ ϕ Γ ϕ ϕ Γ ϕ θ ϕ Γ ϕ ϕ Γ ϕ ϕ θ sin (θ) cos(θ) sin(θ) cos(θ) sin(θ) cos(θ) sin(θ) while those with espect to the othonomal spheical fame ae: In [4]: nabla.display(es, sphe) Out[4]: Γ Γ 3 3 Γ Γ 3 3 Γ 3 3 Γ 3 3 cos(θ) sin(θ) cos(θ) sin(θ) The covaiant deivative of U is In [4]: nabu nabla(u) pint(nabu) Tenso field nabla_g(u) of type (,) on the 3-dimensional diffeentiab le manifold M In [43]: Out[43]: nabu.display() U x U x U x U y g U dx + dy + dz + dx x x y x z x x y U y U y U z U z U z + dy + dz + dx + dy + y y z y x z y z z z dz while the covaiant deivative of V is In [44]: nabv nabla(v) pint(nabv) Tenso field nabla_g(v) of type (,) on the 3-dimensional diffeentiab le manifold M 9

10 SageManifolds.0 In [45]: Out[45]: nabv.display(es, sphe) g V V θ V V + e (, θ, ϕ) e ( e ) e V V (, θ, ϕ) sin(θ) + 3 ϕ + sin(θ) e e 3 V e e V V (, θ, ϕ) + V V (, θ, ϕ) cos(θ) θ + e e + 3 ϕ ( ) sin(θ) e e 3 V 3 V e 3 e θ e 3 e V V (, θ, ϕ) cos(θ) + (, θ, ϕ) sin(θ) + + V 3 ϕ sin(θ) e 3 e 3 In [ ]: 0