The Poincaré lemma is a mathematical theorem and was named after the French mathematician Henri Poincaré .
Exact and closed differential forms
- A differential form of degree is called closed if applies. The outer derivation denotes .
![\omega](https://wikimedia.org/api/rest_v1/media/math/render/svg/48eff443f9de7a985bb94ca3bde20813ea737be8)
![k](https://wikimedia.org/api/rest_v1/media/math/render/svg/c3c9a2c7b599b37105512c5d570edc034056dd40)
![\ mathrm {d} \ omega = 0](https://wikimedia.org/api/rest_v1/media/math/render/svg/94beef0188c62c22491e78300a0b90442f9cd40f)
![\ mathrm {d}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a15022657616b297a2c995d1b314a3aa3442c0cb)
- A differential form of degree is called exact if there is a differential form such that it holds. The form is called a potential form of
![\omega](https://wikimedia.org/api/rest_v1/media/math/render/svg/48eff443f9de7a985bb94ca3bde20813ea737be8)
![k](https://wikimedia.org/api/rest_v1/media/math/render/svg/c3c9a2c7b599b37105512c5d570edc034056dd40)
![(k-1)](https://wikimedia.org/api/rest_v1/media/math/render/svg/e69f74fa2adbbab50f6969acb2af719045435461)
![\ nu](https://wikimedia.org/api/rest_v1/media/math/render/svg/c15bbbb971240cf328aba572178f091684585468)
![\ omega = \ mathrm {d} \ nu](https://wikimedia.org/api/rest_v1/media/math/render/svg/d617209cd9499fc02621a4fc1f4e42ca296e05fc)
![\ nu](https://wikimedia.org/api/rest_v1/media/math/render/svg/c15bbbb971240cf328aba572178f091684585468)
The potential form is not clearly defined, but only "except for re-calibration" (see below).
Because of this , every exact differential form is also closed. The Poincaré lemma specifies conditions under which the opposite statement also applies. In the proof, there is also a generalization of the lemma: An exact part can be split off from every differential form “by construction”.
![{\ mathrm {d}} \ circ {\ mathrm {d}} \ equiv 0](https://wikimedia.org/api/rest_v1/media/math/render/svg/40f5192f5ee11e39a37ec3487b2a1b8ff8140340)
statement
The Poincaré lemma says that every closed differential form defined on a star-shaped open set is exact.
![U \ subseteq \ mathbb {R} ^ {d}](https://wikimedia.org/api/rest_v1/media/math/render/svg/bb1a0be5553eae33cbd69b70e7ae9d849b26d058)
The statement can also be formulated more abstractly as follows: For a star-shaped open set , the -th De Rham cohomology vanishes for all :
![U \ subseteq \ mathbb {R} ^ {d}](https://wikimedia.org/api/rest_v1/media/math/render/svg/bb1a0be5553eae33cbd69b70e7ae9d849b26d058)
![k](https://wikimedia.org/api/rest_v1/media/math/render/svg/c3c9a2c7b599b37105512c5d570edc034056dd40)
![k> 0](https://wikimedia.org/api/rest_v1/media/math/render/svg/27b3af208b148139eefc03f0f80fa94c38c5af45)
![{\ mathrm {H}} _ {{{\ mathrm {dR}}}} ^ {k} (U) = 0](https://wikimedia.org/api/rest_v1/media/math/render/svg/39f35a0b23a7f538d436f49c7735be1fa39d23e7)
In the three-dimensional special case, the Poincaré lemma, translated into the language of vector analysis , says that an eddy-free vector field defined on a simply connected area as a gradient of a potential field ( ), a source-free vector field on a convex area through rotation of a vector potential ( ), and a scalar field density ("source density") can be represented as the divergence of a vector field ( ).
![\ Phi ({\ mathbf r})](https://wikimedia.org/api/rest_v1/media/math/render/svg/0aae204acbbf8529f6aebad36b0968d6eb4d8658)
![{\ vec A} ({\ mathbf r}, t)](https://wikimedia.org/api/rest_v1/media/math/render/svg/113498c1c9125152d2dfda8b05ce5ce8712f7f53)
![k = 2](https://wikimedia.org/api/rest_v1/media/math/render/svg/0bd301789e1f25a3da4be297ff637754ebee5f5d)
![k = 3](https://wikimedia.org/api/rest_v1/media/math/render/svg/662e06a2436f8a44fec791f5c794621f10dc8f30)
Proof (constructive)
Be the point around which is star-shaped. The Poincaré Lemma gives an explicit form, and although having the formula: any form unity can not necessary provided that a form assign, from which the desired potential shape results in unity: This associated shape can by itself define the following figure:
![x_ {0} \ in U](https://wikimedia.org/api/rest_v1/media/math/render/svg/2dee8010e46dd5891756fb4b02b43bc661f7c666)
![{\ displaystyle U \ subset \ mathbb {R} ^ {n}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c1caefb347c86337ea7cd0c354acd2294bd7d81d)
![(k-1)](https://wikimedia.org/api/rest_v1/media/math/render/svg/e69f74fa2adbbab50f6969acb2af719045435461)
![k](https://wikimedia.org/api/rest_v1/media/math/render/svg/c3c9a2c7b599b37105512c5d570edc034056dd40)
![{\ displaystyle \ textstyle \ omega ^ {k} = \ sum \ omega _ {I} {\ rm {d}} x_ {I}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0faa19d08a8a27ca25edd870a3bb152bc83444be)
![(k-1)](https://wikimedia.org/api/rest_v1/media/math/render/svg/e69f74fa2adbbab50f6969acb2af719045435461)
![{\ displaystyle P ^ {k-1} (\ omega ^ {k})}](https://wikimedia.org/api/rest_v1/media/math/render/svg/66cb2690af8a5104555bb905deba810be09ae9be)
-
.
(The roof symbol in the -th column on the right-hand side means that the corresponding differential is omitted.)
![{\ displaystyle i _ {\ alpha}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f19d3a4ad88aa52f941355851bd15f38b85df2ef)
Now one shows directly that the following identity applies: what formally corresponds to the product rule of differentiation and splits the properties represented by into two parts, of which the second has the property sought.
![\ omega ^ {k} \ equiv {\ mathrm P} ^ {{{k}}} ({{\ rm {d}}} \ omega ^ {k}) + {{\ rm {d}}} {{ \ mathrm P} ^ {{k-1}} (} \ omega ^ {k} {\ mathrm)} \ ,,](https://wikimedia.org/api/rest_v1/media/math/render/svg/a2def2dbb8b43bac21bb4ce1bc46740523a6a169)
![\ omega ^ {k}](https://wikimedia.org/api/rest_v1/media/math/render/svg/454e0a9b97a57f3e7ad93736d0bab6f6bf6043f4)
Because of the prerequisite and because of this, this initially applies without restricting the generality even without the foremost of the right-hand side, namely because the requirement only considers the form at the zero point, so that, as with the total differential, a function from up to so-called Calibration transformations (see below) can also be deduced.
![{{\ rm {d}}} \ omega ^ {k} \ equiv 0](https://wikimedia.org/api/rest_v1/media/math/render/svg/476f6022bd694aeefb0a9353e53bf21253ae548c)
![{\ mathrm {d}} \ circ {\ mathrm {d}} = 0](https://wikimedia.org/api/rest_v1/media/math/render/svg/36bc7196c77736d637b3f319ec83db0aa5fdbdbb)
![0 \ equiv {{\ mathrm d}} P ^ {{k}} ({{\ mathrm {d}} {\ omega} ^ {k}} \ to 0).](https://wikimedia.org/api/rest_v1/media/math/render/svg/75f26145b47827e6d9f665903d0b18c372088bc7)
![\ mathrm {d}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a15022657616b297a2c995d1b314a3aa3442c0cb)
![{\ mathrm d} \ omega ^ {k} \ to 0](https://wikimedia.org/api/rest_v1/media/math/render/svg/23e2b2166567b5cde33e9665994d9af38d64d4dc)
![{\ mathrm d} P ^ {k}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d12a4f129f1a7dc16b6e5352cd9a3db97fbd370f)
![{\ mathrm d} P ^ {k} = 0](https://wikimedia.org/api/rest_v1/media/math/render/svg/200edae83b1e20a6f878a1f81ba6695951a44d18)
![{\ mathrm P} ^ {k} = 0](https://wikimedia.org/api/rest_v1/media/math/render/svg/f76c0989deeb2421d81c117151667c7b1a28e6d8)
This leaves only the last term of the above identity, and the required statement follows: with![\ omega ^ {k} \ equiv {\ mathrm {d}} \ eta ^ {{k-1}},](https://wikimedia.org/api/rest_v1/media/math/render/svg/09662fdec18a1e99831d965a99652d5570b75331)
The given identity also generalizes Poincaré's lemma by breaking down any differential form into an inexact (“anholonomic”) and an exact (“holonomic”) part (the bracketed names correspond to the so-called constraining forces in analytical mechanics ). At the same time, it corresponds to the decomposition of any vector field into a vortex part and a source part.
![\omega](https://wikimedia.org/api/rest_v1/media/math/render/svg/48eff443f9de7a985bb94ca3bde20813ea737be8)
In the language of homological algebra , a contracting homotopy , e.g. B. contracted to the central point of the star-shaped area considered here .
![P](https://wikimedia.org/api/rest_v1/media/math/render/svg/b4dc73bf40314945ff376bd363916a738548d40a)
Re-calibration
What is so defined is not the only form whose external differential is. All others differ from one another by the differential of a -form: If and are two such -forms, then there exists a -form such that it holds.
![\ eta ^ {{k-1}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c8c572f3c7114dfd9cb5e20422ea23e7a45c488f)
![(k-1)](https://wikimedia.org/api/rest_v1/media/math/render/svg/e69f74fa2adbbab50f6969acb2af719045435461)
![{\ omega} ^ {k}](https://wikimedia.org/api/rest_v1/media/math/render/svg/325965d6bb6a7599017b2960b29c0ae9abab87c8)
![(k-2)](https://wikimedia.org/api/rest_v1/media/math/render/svg/130fc191919e1fc707bfb047c27f6fe708852835)
![\ eta _ {2} ^ {{k-1}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/30d3ba242c342613ae922fa7e5543c6255fbbfb4)
![\ eta _ {1} ^ {{k-1}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/055b680970d3c6ce198250e0e836fa81aff691d4)
![(k-1)](https://wikimedia.org/api/rest_v1/media/math/render/svg/e69f74fa2adbbab50f6969acb2af719045435461)
![(k-2)](https://wikimedia.org/api/rest_v1/media/math/render/svg/130fc191919e1fc707bfb047c27f6fe708852835)
![\ xi ^ {{k-2}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2603415f31f0802c7c984c373196f445e1c5fc99)
![\ eta _ {2} ^ {{k-1}} = \ eta _ {1} ^ {{k-1}} + {\ mathrm d} \ xi ^ {{k-2}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e6bcfb61c5511d34ac5e9eb84775122b2afb2c60)
The addition is also known as calibration transformation or re-calibration of .
![+ \, {\ mathrm d} \ xi ^ {{k-2}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/af2011fc37de3d35253212c5fd198333745b01b4)
![\ eta _ {1} ^ {{k-1}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/055b680970d3c6ce198250e0e836fa81aff691d4)
Application in electrodynamics
The case of a magnetic field generated by a stationary current is known from electrodynamics , with the so-called vector potential . This case corresponds , the star-shaped area being the . The vector of the current density is and corresponds to the current shape.
The same applies to the magnetic field : it corresponds to the magnetic flux shape and can be derived from the vector potential:, or . The vector potential corresponds to the potential form.
The closeness of the magnetic flux form corresponds to the absence of sources of the magnetic field ![{\ vec A} ({\ mathbf r}) \ ,.](https://wikimedia.org/api/rest_v1/media/math/render/svg/0bfde060cd08f98a4979cddaa0f5c2154ef75e36)
![k = 2](https://wikimedia.org/api/rest_v1/media/math/render/svg/0bd301789e1f25a3da4be297ff637754ebee5f5d)
![\ mathbb R ^ 3](https://wikimedia.org/api/rest_v1/media/math/render/svg/f936ddf584f8f3dd2a0ed08917001b7a404c10b5)
![{\ vec {j}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8ce1ed1de8493f7cc7d856ca5427cf311b1597f1)
![{\ mathbf I}: = j_ {1} (x, y, z) {{\ rm {d}}} x_ {2} \ wedge {{\ rm {d}}} x_ {3} + j_ {2 } (x, y, z) {{\ rm {d}}} x_ {3} \ wedge {{\ rm {d}}} x_ {1} + j_ {3} (x, y, z) {{ \ rm {d}}} x_ {1} \ wedge {{\ rm {d}}} x_ {2} \ ,.](https://wikimedia.org/api/rest_v1/media/math/render/svg/fb62e5286445a50941801ba81d88a3c4b71d639a)
![\ Phi _ {B}: = B_ {1} {{\ rm {d}}} x_ {2} \ wedge {{\ rm {d}}} x_ {3} + \ dots](https://wikimedia.org/api/rest_v1/media/math/render/svg/419f6e0ea30032ea3dc8ffb03138975c34b9ca96)
![\ textstyle {\ vec B} = \ operatorname {rot} {\ vec A} = \ left ({\ tfrac {\ partial A_ {3}} {\ partial x_ {2}}} - {\ tfrac {\ partial A_ {2}} {\ partial x_ {3}}}, {\ tfrac {\ partial A_ {1}} {\ partial x_ {3}}} - {\ tfrac {\ partial A_ {3}} {\ partial x_ {1}}}, {\ tfrac {\ partial A_ {2}} {\ partial x_ {1}}} - {\ tfrac {\ partial A_ {1}} {\ partial x_ {2}}} \ right) ^ {t}](https://wikimedia.org/api/rest_v1/media/math/render/svg/24728d4a868f499de2e780e38a8596123dd1db3e)
![\ Phi _ {B} = {{\ rm {d}}} {\ mathbf A}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d16703849a54df25dfd5e3ebfe1903c2d88aa8ac)
![{\ vec {A}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/391292ffadc65b0cde3e96f23afcdb811619dd95)
![{\ mathbf A}: = A_ {1} {{\ rm {d}}} x_ {1} + A_ {2} {{\ rm {d}}} x_ {2} + A_ {3} {{\ rm {d}}} x_ {3} \ ,.](https://wikimedia.org/api/rest_v1/media/math/render/svg/55ebc0c781a87aa19b22006a95a62612386e8184)
Using the Coulomb calibration or as appropriate, the following then applies to i = 1,2,3
![\ operatorname {div} {\ vec A} {\ stackrel {!} {=}} 0](https://wikimedia.org/api/rest_v1/media/math/render/svg/2900c1b88bcdb8a38edb51907894fd2c15b6a321)
![\ operatorname {div} {\ vec j} {\ stackrel {!} {=}} 0](https://wikimedia.org/api/rest_v1/media/math/render/svg/ec60c5df920ae37e77ac8d57af2d35b238bb4a66)
![A_ {i} ({\ vec r}) = \ int {\ frac {\ mu _ {0} j_ {i} ({\ vec r} ^ {{\, '}}) \, \, dx_ {1 } 'dx_ {2}' dx_ {3} '} {4 \ pi | {\ vec r} - {\ vec r} ^ {{\,'}} |}} \ ,,](https://wikimedia.org/api/rest_v1/media/math/render/svg/fbaa38be38b01039e6f9911f89223d5f95117b6f)
there is a natural constant , the so-called magnetic field constant .
![\ mu _ {0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/fe2fd9b8decb38a3cd158e7b6c0c6e2d987fefcc)
In this equation u. a. remarkable that it fully corresponds to a well-known formula for the electric field , the Coulomb potential of a given charge distribution with density . At this point it is already assumed that
![\, \ phi (x_ {1}, x_ {2}, x_ {3})](https://wikimedia.org/api/rest_v1/media/math/render/svg/3996d82d7aefdf217074e1548120e0796b11e6d8)
![\ rho (x_ {1}, x_ {2}, x_ {3})](https://wikimedia.org/api/rest_v1/media/math/render/svg/c0c53fc0571256b5c49b8fb8734774bc3e9a86c6)
-
and or![{\ vec {B}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/83ae7d80cab55b606de217162280b2279142bbb4)
-
and as well![{\ vec {j}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8ce1ed1de8493f7cc7d856ca5427cf311b1597f1)
-
and
can be summarized and that the relativistic invariance of Maxwell's electrodynamics results from it, see also electrodynamics .
If you give up the condition of stationarity , the time argument must be added to the space coordinates on the left side of the above equation , while on the right side the so-called "retarded time" must be added. As before, it is integrated using the three spatial coordinates . After all, the speed of light is in a vacuum .
![A_ {i}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1aed3b5def921afbe6cc48aaf8f9b11c6f1c1e2d)
![t](https://wikimedia.org/api/rest_v1/media/math/render/svg/65658b7b223af9e1acc877d848888ecdb4466560)
![j_ {i} '](https://wikimedia.org/api/rest_v1/media/math/render/svg/7df0a2d3b58549ad06d1ba104a00a74a2b100dcb)
![t ': = t - {\ tfrac {| {\ vec r} - {\ vec r} ^ {{\,'}} |} {c}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e473b1ad3cdbe3977f9d64dd19dc778ed7e7d65a)
![{\ vec r} ^ {{\, '}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8f2964540a55f51d72c7b9b50a6e9facfe198ea8)
![c](https://wikimedia.org/api/rest_v1/media/math/render/svg/86a67b81c2de995bd608d5b2df50cd8cd7d92455)
Application in continuum mechanics
In continuum mechanics , the lemma is applied to tensors . B. is needed for the establishment of the compatibility conditions . The starting point is the lemma in the formulation:
|
|
(I)
|
|
The operator "grad" forms the gradient , the vectors are the standard basis of the Cartesian coordinate system with coordinates and Einstein's summation convention was applied, according to which indices occurring twice in a product, here k, are to be summed from one to three, which is also the case in The following should be practiced.
![{\ hat {e}} _ {1,2,3}](https://wikimedia.org/api/rest_v1/media/math/render/svg/91ec2fd7cc46b4427cfd09bab244026d27d81519)
![x _ {{1,2,3}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b67eda9b9b24758489f6004e13d51444f494e207)
Let us now be given a tensor field whose row vectors are combined with the dyadic product “ ” to form the tensor. The rotation of the transposed tensor vanishes
![{\ mathbf {T}} = {\ hat {e}} _ {i} \ otimes {\ vec {t}} _ {i}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5132182ca1b00d6224f35ee974a18ed1da8f6617)
![{\ vec {t}} _ {{1,2,3}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/fe7f1757c9f5c657288d666b8b62eb4ddc43d12d)
![\ otimes](https://wikimedia.org/api/rest_v1/media/math/render/svg/de29098f5a34ee296a505681a0d5e875070f2aea)
![{\ displaystyle \ operatorname {rot} (\ mathbf {T} ^ {\ top}): = {\ hat {e}} _ {k} \ times {\ frac {\ partial} {\ partial x_ {k}} } ({\ vec {t}} _ {i} \ otimes {\ hat {e}} _ {i}) = \ left ({\ hat {e}} _ {k} \ times {\ frac {\ partial {\ vec {t}} _ {i}} {\ partial x_ {k}}} \ right) \ otimes {\ hat {e}} _ {i} = \ mathbf {0} \ quad \ rightarrow \ quad { \ hat {e}} _ {k} \ times {\ frac {\ partial {\ vec {t}} _ {i}} {\ partial x_ {k}}} = {\ vec {0}} \ ,, \ quad i = 1,2,3 \ ,,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/dd9ce8624763941952c031c8c0dcc6ff4680058a)
so that every row vector is rotation-free. Then there is a scalar field for every row vector , the gradient of which it is:
![u_ {i}](https://wikimedia.org/api/rest_v1/media/math/render/svg/14f13cb025ff2e136dcbd2fc81ddf965b728e3d7)
![{\ displaystyle {\ vec {t}} _ {i} = \ operatorname {grad} u_ {i} \ quad \ rightarrow \ quad \ mathbf {T} = {\ hat {e}} _ {i} \ otimes { \ vec {t}} _ {i} = {\ hat {e}} _ {i} \ otimes \ operatorname {grad} u_ {i} = \ operatorname {grad} {\ vec {u}} \ ,,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6725cfa11a7a7bb15254d31fef6dc70206c0bc08)
because the gradient of the vector is formed according to:
![{\ displaystyle {\ vec {u}}: = u_ {i} {\ hat {e}} _ {i}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8fd645895eab9325205f24342ee7846a1f7ec084)
![{\ displaystyle \ operatorname {grad} {\ vec {u}}: = {\ frac {\ partial u_ {i}} {\ partial x_ {k}}} {\ hat {e}} _ {i} \ otimes {\ hat {e}} _ {k} = {\ hat {e}} _ {i} \ otimes {\ frac {\ partial u_ {i}} {\ partial x_ {k}}} {\ hat {e }} _ {k} = {\ hat {e}} _ {i} \ otimes \ operatorname {grad} u_ {i} \ ,.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/923aa292a8956ae22cf3a3b40a188c6abe613d58)
Thus the second form of the lemma applies:
|
|
(II)
|
|
If the trace of the tensor also disappears, then the vector field is free of divergence:
![{\ displaystyle \ operatorname {Sp} (\ mathbf {T}) = \ operatorname {Sp} ({\ hat {e}} _ {i} \ otimes \ operatorname {grad} u_ {i}) = {\ hat { e}} _ {i} \ cdot {\ frac {\ partial u_ {i}} {\ partial x_ {k}}} {\ hat {e}} _ {k} = {\ frac {\ partial u_ {i }} {\ partial x_ {i}}} = \ operatorname {div} {\ vec {u}} = 0 \ ,.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1c0da1d5c7151064eda82f109f51dbba4942f476)
In this case, the Kronecker delta δ ij is used to calculate :
![{\ displaystyle {\ begin {aligned} \ operatorname {rot} ({\ vec {u}} \ times \ mathbf {I}) = & {\ hat {e}} _ {k} \ times {\ frac {\ partial} {\ partial x_ {k}}} [u_ {i} {\ hat {e}} _ {i} \ times ({\ hat {e}} _ {j} \ otimes {\ hat {e}} _ {j})] = {\ frac {\ partial u_ {i}} {\ partial x_ {k}}} [{\ hat {e}} _ {k} \ times ({\ hat {e}} _ {i} \ times {\ hat {e}} _ {j})] \ otimes {\ hat {e}} _ {j} = {\ frac {\ partial u_ {i}} {\ partial x_ {k} }} (\ delta _ {jk} {\ hat {e}} _ {i} - \ delta _ {ik} {\ hat {e}} _ {j}) \ otimes {\ hat {e}} _ { j} \\ = & {\ frac {\ partial u_ {i}} {\ partial x_ {j}}} {\ hat {e}} _ {i} \ otimes {\ hat {e}} _ {j} - {\ frac {\ partial u_ {i}} {\ partial x_ {i}}} {\ hat {e}} _ {j} \ otimes {\ hat {e}} _ {j} = \ operatorname {grad } {\ vec {u}} \ end {aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/583a54d46a79f817a3878e20ce508abffbc87e8d)
and the tensor is skew symmetric :
![{\ mathbf {W}}: = {\ vec {u}} \ times {\ mathbf {I}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/635204ed06aa3eb1c458ab9e976e3dfa6c095653)
![{\ displaystyle ({\ vec {u}} \ times \ mathbf {I}) ^ {\ top} = (u_ {i} {\ hat {e}} _ {i} \ times {\ hat {e}} _ {j} \ otimes {\ hat {e}} _ {j}) ^ {\ top} = \ epsilon _ {ijk} u_ {i} {\ hat {e}} _ {j} \ otimes {\ hat {e}} _ {k} = - u_ {i} {\ hat {e}} _ {i} \ times {\ hat {e}} _ {k} \ otimes {\ hat {e}} _ {k } = - {\ vec {u}} \ times \ mathbf {I} \ ,.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4b4799b2e38aad6fa0f376f05b0191d4c4074ddc)
Inside is the permutation symbol . The third form of the lemma follows:
![\ epsilon _ {{ijk}} = ({\ hat {e}} _ {i} \ times {\ hat {e}} _ {j}) \ cdot {\ hat {e}} _ {k}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b9708bc8338e426d34cdbb4260c63d63fab14c7c)
|
|
(III)
|
|
In the literature, a rotation operator is also used, which directly forms the rotation of the row vectors:
![{\ displaystyle \ operatorname {\ tilde {red}} \ mathbf {T} = \ operatorname {red} (\ mathbf {T} ^ {\ top}) \ ,.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ee9a0d51ce3d9c54a5edfeb8515779e0880b793e)
With this operator:
![{\ displaystyle {\ begin {array} {rrcll} {\ textsf {II}}: & \ operatorname {\ tilde {red}} \ mathbf {T} = \ mathbf {0} & \ rightarrow & \ exists {\ vec {u}} \ colon & \ mathbf {T} = \ operatorname {grad} {\ vec {u}} \\ {\ textsf {III}}: & \ operatorname {\ tilde {red}} \ mathbf {T} = \ mathbf {0} \ quad {\ text {and}} \ quad \ operatorname {Sp} (\ mathbf {T}) = 0 & \ rightarrow & \ exists \ mathbf {V} \ colon & \ mathbf {T} = \ operatorname {\ tilde {rot}} \ mathbf {V} \ quad {\ text {with}} \ quad \ mathbf {V} ^ {\ top} = - \ mathbf {V} \ end {array}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/fffd596d09fd5d0ff0f88e5b4db3ba6c2d4151f7)
literature
-
Otto Forster : Analysis. Volume 3: Integral calculus in R n with applications. 4th edition. Vieweg + Teubner, Braunschweig et al. 2007, ISBN 978-3-528-37252-1 .
- John M. Lee: Introduction to Smooth Manifolds (= Graduate Texts in Mathematics 218). Springer-Verlag, New York NY et al. 2003, ISBN 0-387-95448-1 .
- C. Truesdell: Solid Mechanics II in S. Flügge (Ed.): Handbook of Physics , Volume VIa / 2. Springer-Verlag, 1972, ISBN 3-540-05535-5 , ISBN 0-387-05535-5 .