+ izFb^RIHtHtHt^RIHt^RIHt^RI H t R.t !RR]]4t Rt R#) )sympifyAddImmutableMatrix) EvalfMixin) Printable) prec_to_dpsDyadicca]tRt^ toRtRtRt]R4tRt ] t Rt ] t Rt ] tRtR tR tR tR tR tRtRtRtRt]tRRltRRltRtRtRtRtRt Rt!Rt"Rt#Vt$R#)r a]A Dyadic object. See: https://en.wikipedia.org/wiki/Dyadic_tensor Kane, T., Levinson, D. Dynamics Theory and Applications. 1985 McGraw-Hill A more powerful way to represent a rigid body's inertia. While it is more complex, by choosing Dyadic components to be in body fixed basis vectors, the resulting matrix is equivalent to the inertia tensor. Fc .VnV^8Xd.p\V4^8wEd_^p\VP4Fwr4\V^,^,4\VPV,^,48XgKF\V^,^,4\VPV,^,48XgKVPV,^,V^,^,,V^,^,V^,^,3VPV&VP V^,4^pM V^8wgEK2VPP V^,4VP V^,4EKo^pV\VP48dVPV,^,^8HVPV,^,^8H,VPV,^,^8H,'d6VPP VPV,4V^,pV^, pKR#)z Just like Vector's init, you should not call this unless creating a zero dyadic. zd = Dyadic(0) Stores a Dyadic as a list of lists; the inner list has the measure number and the two unit vectors; the outerlist holds each unique unit vector pair. N)argslen enumeratestrremoveappend)selfinlistaddedivs&& Y/var/www/html/photoedit/myenv/lib/python3.14/site-packages/sympy/physics/vector/dyadic.py__init__Dyadic.__init__s Q;F&kQE!$)),1&#diil1o*>>VAYq\*c$))A,q/.BB$(IIaLOfQil$B$*1IaL&)A,$@DIIaLMM&),E-z   + fQi( #dii. 1aA%$))A,q/Q*>?YYq\!_)++   1.Q FA !c \#)zReturns the class Dyadic. )r rs&rfunc Dyadic.func@s  rc d\V4p\VPVP,4#)zThe add operator for Dyadic. ) _check_dyadicr r rothers&&r__add__Dyadic.__add__Es$e$dii%**,--rc \VP4p\V4p\\ V44F8pWV,^,,W#,^,W#,^,3W#&K: \ V4#)a%Multiplies the Dyadic by a sympifyable expression. Parameters ========== other : Sympafiable The scalar to multiply this Dyadic with Examples ======== >>> from sympy.physics.vector import ReferenceFrame, outer >>> N = ReferenceFrame('N') >>> d = outer(N.x, N.x) >>> 5 * d 5*(N.x|N.x) )listr rranger r )rr"newlistrs&& r__mul__Dyadic.__mul__Lsa&tyy/s7|$A!*Q-/A!*Q-)GJ%grc B^RIHpHp\V\4'd\ V4p\ ^4pVP F{pVP FhpWE^,V^,,V^,PV^,4,V^,PV^,4,, pKj K} V#V!V4pV!^4pVP F=pWE^,V^,,V^,PV4,, pK? V#)azThe inner product operator for a Dyadic and a Dyadic or Vector. Parameters ========== other : Dyadic or Vector The other Dyadic or Vector to take the inner product with Examples ======== >>> from sympy.physics.vector import ReferenceFrame, outer >>> N = ReferenceFrame('N') >>> D1 = outer(N.x, N.y) >>> D2 = outer(N.y, N.y) >>> D1.dot(D2) (N.x|N.y) >>> D1.dot(N.y) N.x )Vector _check_vector) sympy.physics.vector.vectorr,r- isinstancer r r dotouter)rr"r,r-olrv2s&& rr0 Dyadic.doths, F eV $ $!%(EBYY**BA$A,!A$((2a5/:adjjA>OPPB% "%(EBYYdQqTkQqTXXe_55 rc 2VP^V, 4#)z0Divides the Dyadic by a sympifyable expression. )r)r!s&&r __truediv__Dyadic.__truediv__s||AI&&rc "V^8Xd \^4p\V4pVP.8XdVP.8XdR#VP.8XgVP.8XdR#\VP4\VP48H#)zKTests for equality. Is currently weak; needs stronger comparison testing TF)r r r setr!s&&r__eq__ Dyadic.__eq__sk A:1IEe$ IIO%**"2ii2o5::#3499~UZZ00rc W8X*#Nr!s&&r__ne__ Dyadic.__ne__s   rcVR,#r>rs&r__neg__Dyadic.__neg__s byrc2VPp\V4^8Xd \^4#.pVEFpV^,^8XdUVPRVP V^,4,R,VP V^,4,4KfV^,R 8XdUVPRVP V^,4,R,VP V^,4,4KV^,^8wgKVP V^,4p\ V^,\ 4'd RV,pVPR4'd VR,pRpMRpVPWe,VP V^,4,R,VP V^,4,4EK RPV4pVPR4'd VR,pV#VPR 4'd VR,pV#) r + z\otimes  - (%s)-rCNNNN rD) r r rr_printr/r startswithjoinrprinterarr2rarg_str str_startoutstrs&& r_latex Dyadic._latexs YY r7a<q6M Atqy %'..1"66D!..1./01 %!..1./%&"..1./0 1!..1.adC(($w.G%%c**%bkG %I %I )-qt0DD%&(/qt(<=>-0   U # #BZF   s # #BZF rc4aaVo!VV3RlR4pV!4#)c4<a]tRt^to^tVV3RltRtVtR#)Dyadic._pretty..Fakec ><S PpS p\V4^8Xd \^4#S P'dRMRp.pVEFpV^,^8XdDVP RVP V^,4VVP V^,4.4KUV^,R 8XdDVP RVP V^,4VVP V^,4.4KV^,^8wgK\ V^,\4'd/VPV^,4P4^,pMVP V^,4pVPR4'd VR,pRp MRp VP WRVP V^,4VVP V^,4.4EK RPV4p V PR4'd V R ,p V #V PR4'd V R,p V #) ru⊗|rHrIrKrLrPrMrNrD) r r r _use_unicodeextenddoprintr/rrQparensrRrS) rr kwargsrVmppbarr2rrWrXrYerUs &*, rrender#Dyadic._pretty..Fake.rendersVVr7a<q6M-4-A-A-A)sAtqy 5"%++ad"3"%"%++ad"3#56 1 5"%++ad"3"%"%++ad"3#561%adC00&)jj !!'&&,fhq'2G'*kk!A$&7G"--c22&-bkG(-I(-I 9s"%++ad"3"%"%++ad"3#569B$$U++#BZF &&s++#BZF rr>N)__name__ __module__ __qualname____firstlineno__baselineri__static_attributes____classdictcell__) __classdict__rhrUs@rFaker^sH- - rrsr>)rrUrsrhs&f @r_prettyDyadic._prettys 0 0 bv rc"RV,V,#rBr>r!s&&r__rsub__Dyadic.__rsub__sT U""rc dVPp\V4^8XdVP^4#.pVEFpV^,^8Xd\VPRVPV^,4,R,VPV^,4,R,4KmV^,R8Xd\VPRVPV^,4,R,VPV^,4,R,4KV^,^8wgKVPV^,4p\ V^,\ 4'd RV,pV^,R8Xd VR,pRpMR pVPWe,R ,VPV^,4,R,VPV^,4,R,4EK R P V4pVPR 4'd VR ,pV#VPR 4'd VR,pV#)zPrinting method. z + (r`)z - (rJrKrLrIrHz*(rMrNrPrD)r r rQrr/rrSrRrTs&& r _sympystrDyadic._sympystrs YY r7a<>>!$ $ Atqy &7>>!A$#77#=!..1./14561 &7>>!A$#77#=!..1./14561!..1.adC(($w.G1:$%bkG %I %I )-4!..1./ 'qt 457:;<).   U # #BZF   s # #BZF rc 2VPVR,4#)zThe subtraction operator. rD)r#r!s&&r__sub__Dyadic.__sub__*s||EBJ''rc ^RIHpV!V4p\^4pVPFEpW4^,V^,P V^,P V44,, pKG V#)a:Returns the dyadic resulting from the dyadic vector cross product: Dyadic x Vector. Parameters ========== other : Vector Vector to cross with. Examples ======== >>> from sympy.physics.vector import ReferenceFrame, outer, cross >>> N = ReferenceFrame('N') >>> d = outer(N.x, N.x) >>> cross(d, N.y) (N.x|N.z) )r-)r.r-r r r1cross)rr"r-r2rs&& rr Dyadic.cross.sX$ >e$ AYA A$!A$**adjj&79: :B rNc ^RIHpV!WV4#)agExpresses this Dyadic in alternate frame(s) The first frame is the list side expression, the second frame is the right side; if Dyadic is in form A.x|B.y, you can express it in two different frames. If no second frame is given, the Dyadic is expressed in only one frame. Calls the global express function Parameters ========== frame1 : ReferenceFrame The frame to express the left side of the Dyadic in frame2 : ReferenceFrame If provided, the frame to express the right side of the Dyadic in Examples ======== >>> from sympy.physics.vector import ReferenceFrame, outer, dynamicsymbols >>> from sympy.physics.vector import init_vprinting >>> init_vprinting(pretty_print=False) >>> N = ReferenceFrame('N') >>> q = dynamicsymbols('q') >>> B = N.orientnew('B', 'Axis', [q, N.z]) >>> d = outer(N.x, N.x) >>> d.express(B, N) cos(q)*(B.x|N.x) - sin(q)*(B.y|N.x) )express)sympy.physics.vector.functionsr)rframe1frame2rs&&& rrDyadic.expressJs@ ;tV,,rc  VfTp\VUUu.uF+pVF"qCPV4PV4NK$ K- upp4P^^4#uuppi)aiReturns the matrix form of the dyadic with respect to one or two reference frames. Parameters ---------- reference_frame : ReferenceFrame The reference frame that the rows and columns of the matrix correspond to. If a second reference frame is provided, this only corresponds to the rows of the matrix. second_reference_frame : ReferenceFrame, optional, default=None The reference frame that the columns of the matrix correspond to. Returns ------- matrix : ImmutableMatrix, shape(3,3) The matrix that gives the 2D tensor form. Examples ======== >>> from sympy import symbols, trigsimp >>> from sympy.physics.vector import ReferenceFrame >>> from sympy.physics.mechanics import inertia >>> Ixx, Iyy, Izz, Ixy, Iyz, Ixz = symbols('Ixx, Iyy, Izz, Ixy, Iyz, Ixz') >>> N = ReferenceFrame('N') >>> inertia_dyadic = inertia(N, Ixx, Iyy, Izz, Ixy, Iyz, Ixz) >>> inertia_dyadic.to_matrix(N) Matrix([ [Ixx, Ixy, Ixz], [Ixy, Iyy, Iyz], [Ixz, Iyz, Izz]]) >>> beta = symbols('beta') >>> A = N.orientnew('A', 'Axis', (beta, N.x)) >>> trigsimp(inertia_dyadic.to_matrix(A)) Matrix([ [ Ixx, Ixy*cos(beta) + Ixz*sin(beta), -Ixy*sin(beta) + Ixz*cos(beta)], [ Ixy*cos(beta) + Ixz*sin(beta), Iyy*cos(2*beta)/2 + Iyy/2 + Iyz*sin(2*beta) - Izz*cos(2*beta)/2 + Izz/2, -Iyy*sin(2*beta)/2 + Iyz*cos(2*beta) + Izz*sin(2*beta)/2], [-Ixy*sin(beta) + Ixz*cos(beta), -Iyy*sin(2*beta)/2 + Iyz*cos(2*beta) + Izz*sin(2*beta)/2, -Iyy*cos(2*beta)/2 + Iyy/2 - Iyz*sin(2*beta) + Izz*cos(2*beta)/2 + Izz/2]]) )Matrixr0reshape)rreference_framesecond_reference_framerjs&&& r to_matrixDyadic.to_matrixmsdV " )%4 "?.?a,HIuuT{q),*?.//6wq!} =.s1A c \VPUu.uF7p\V^,P!R/VBV^,V^,3.4NK9 up\^44#uupi)z(Calls .doit() on each term in the Dyadicr>)sumr r doit)rhintsrs&, rr Dyadic.doitsa!YY(&QqTYY//1qt<=>&()/4 4(=A"c ^RIHpV!W4#)a'Take the time derivative of this Dyadic in a frame. This function calls the global time_derivative method Parameters ========== frame : ReferenceFrame The frame to take the time derivative in Examples ======== >>> from sympy.physics.vector import ReferenceFrame, outer, dynamicsymbols >>> from sympy.physics.vector import init_vprinting >>> init_vprinting(pretty_print=False) >>> N = ReferenceFrame('N') >>> q = dynamicsymbols('q') >>> B = N.orientnew('B', 'Axis', [q, N.z]) >>> d = outer(N.x, N.x) >>> d.dt(B) - q'*(N.y|N.x) - q'*(N.x|N.y) )time_derivative)rr)rframers&& rdt Dyadic.dts2 Ct++rc \^4pVPF<pV\V^,P4V^,V^,3.4, pK> V#)zReturns a simplified Dyadic.)r r simplify)routrs& rrDyadic.simplifysIQiA 6AaDMMOQqT1Q489: :C rc  \VPUu.uF7p\V^,P!V/VBV^,V^,3.4NK9 up\^44#uupi)zSubstitution on the Dyadic. Examples ======== >>> from sympy.physics.vector import ReferenceFrame >>> from sympy import Symbol >>> N = ReferenceFrame('N') >>> s = Symbol('s') >>> a = s*(N.x|N.x) >>> a.subs({s: 2}) 2*(N.x|N.x) )rr r subs)rr rers&*, rr Dyadic.subsse !YY(&QqTYY771qtDEF&()/4 4(rc \V4'g \R4h\^4pVPF*wr4pW!!V4VP V4,, pK, V#)z/Apply a function to each component of a Dyadic.z`f` must be callable.)callable TypeErrorr r r1)rfrabcs&& r applyfuncDyadic.applyfuncsR{{34 4QiyyGA! 1Q41771:& &C! rcVP'gV#.p\V4pVPFDp\V4pV^,PVR7V^&VP \ V44KF \ V4#)r)n)r rr&evalfrtupler )rprecnew_argsdpsr new_inlists&& r _eval_evalfDyadic._eval_evalfslyyyK$iiFfJ"1IOOcO2JqM OOE*- . hrc .pVPFCp\V4pV^,PV4V^&VP\ V44KE \ V4#)a Replace occurrences of objects within the measure numbers of the Dyadic. Parameters ========== rule : dict-like Expresses a replacement rule. Returns ======= Dyadic Result of the replacement. Examples ======== >>> from sympy import symbols, pi >>> from sympy.physics.vector import ReferenceFrame, outer >>> N = ReferenceFrame('N') >>> D = outer(N.x, N.x) >>> x, y, z = symbols('x y z') >>> ((1 + x*y) * D).xreplace({x: pi}) (pi*y + 1)*(N.x|N.x) >>> ((1 + x*y) * D).xreplace({x: pi, y: 2}) (1 + 2*pi)*(N.x|N.x) Replacements occur only if an entire node in the expression tree is matched: >>> ((x*y + z) * D).xreplace({x*y: pi}) (z + pi)*(N.x|N.x) >>> ((x*y*z) * D).xreplace({x*y: pi}) x*y*z*(N.x|N.x) )r r&xreplacerrr )rrulerrrs&& rrDyadic.xreplacesWPiiFfJ&qM2248JqM OOE*- . hr)r r=)%rkrlrmrn__doc__ is_numberrpropertyrr#__radd__r)__rmul__r0__and__r6r:r?rErZrtrwr{r~r__xor__rrrrrrrrrrprq)rrs@rr r s I$L. H4H"JG'1 !"H4l#"H(4G!-F/=b4 ,84&  - - rcH\V\4'g \R4hV#)zA Dyadic must be supplied)r/r r)r"s&rr r s eV $ $344 LrN)sympyrrrrsympy.core.evalfrsympy.printing.defaultsrmpmath.libmp.libmpfr__all__r r r>rrrs399'-+ *P Y P fr