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Transport theorem
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<h2 id="derivation">Derivation</h2> <p>Let b i := T B E e i {\displaystyle {\boldsymbol {b}}_{i}:=T_{B}^{E}{\boldsymbol {e}}_{i}} be the <a href="/facts/Basis_vectors/89IPoN6c">basis vectors</a> of B {\displaystyle B} , as seen from the reference frame E {\displaystyle E} , and denote the components of a vector f {\displaystyle {\boldsymbol {f}}} in B {\displaystyle B} by just f i {\displaystyle f_{i}} . Let </p> G := T ′ ⋅ T − 1 {\displaystyle G:=T'\cdot T^{-1}} <p>so that this <a href="/facts/Coordinate_transformation/Mtd0q4SP">coordinate transformation</a> is <a href="/facts/C0-semigroup/GrcTHhj0">generated</a>, in time, according to T ′ = G ⋅ T {\displaystyle T'=G\cdot T} . Such a generator <a href="/facts/Differential_equation/9MGbE3Ca">differential equation</a> is important for trajectories in <a href="/facts/Lie_group/a9jCQyjF">Lie group theory</a>. Applying the <a href="/facts/Product_rule/MyKxsP5t">product rule</a> with implict <a href="/facts/Summation_convention/UXwS43wU">summation convention</a>, </p> f ′ = ( f i b i ) ′ = ( f i T ) ′ e i = ( f i ′ T + f i G ⋅ T ) e i = ( f i ′ + f i G ) b i = ( ( d d t ) B + G ) f {\displaystyle {\boldsymbol {f}}'=(f_{i}{\boldsymbol {b}}_{i})'=(f_{i}T)'{\boldsymbol {e}}_{i}=(f_{i}'T+f_{i}G\cdot T)\,{\boldsymbol {e}}_{i}=(f_{i}'+f_{i}G)\,{\boldsymbol {b}}_{i}=\left(\left({\tfrac {\mathrm {d} }{\mathrm {d} t}}\right)_{B}+G\right){\boldsymbol {f}}} <p>For the <a href="/facts/Rotation_group/jKWM1KDM">rotation groups</a> S O ( n ) {\displaystyle {\mathrm {SO} }(n)} , one has T E B := ( T B E ) − 1 = ( T B E ) T {\displaystyle T_{E}^{B}:=(T_{B}^{E})^{-1}=(T_{B}^{E})^{T}} . In three dimensions, n = 3 {\displaystyle n=3} , the generator G {\displaystyle G} then equals the <a href="/facts/Cross_product/uRBJzkcW">cross product</a> operation from the left, a <a href="/facts/Skew-symmetric_matrix/nw5CQeLN">skew-symmetric</a> <a href="/facts/Linear_map/Q1PBKNVV">linear map</a> [ Ω E ] × g := Ω E × g {\displaystyle [{\boldsymbol {\Omega }}_{E}]_{\times }{\boldsymbol {g}}:={\boldsymbol {\Omega }}_{E}\times {\boldsymbol {g}}} for any vector g {\displaystyle {\boldsymbol {g}}} . As a matrix, it is also related to the vector as seen from B {\displaystyle B} via </p> [ Ω E ] × = [ T B E Ω B ] × = T B E ⋅ [ Ω B ] × ⋅ T E B {\displaystyle [{\boldsymbol {\Omega }}_{E}]_{\times }=[T_{B}^{E}{\boldsymbol {\Omega }}_{B}]_{\times }=T_{B}^{E}\cdot [{\boldsymbol {\Omega }}_{B}]_{\times }\cdot T_{E}^{B}} <h2 id="references">References</h2> <ol> <li id="fn:1"><p>Rao, Anil Vithala (2006). Dynamics of particles and rigid bodies: a systematic approach. New York: Cambridge University Press. pp. 47, eq. (2–128). ISBN 978-0-511-34840-2. <a href="978-0-511-34840-2" target="_blank">978-0-511-34840-2</a> <a href="#fnref:1" class="footnote-back-ref">↩</a></p></li> <li id="fn:2"><p>Harithuddin, A.S.M. (2014). Derivative Kinematics in Relatively Rotating Coordinate Frames: Investigation on the Razi Acceleration. RMIT University. p. 22. <a href="https://researchrepository.rmit.edu.au/esploro/outputs/doctoral/Derivative-kinematics-in-relatively-rotating-coordinate-frames-investigation-on-the-Razi-acceleration/9921864038601341" target="_blank">https://researchrepository.rmit.edu.au/esploro/outputs/doctoral/Derivative-kinematics-in-relatively-rotating-coordinate-frames-investigation-on-the-Razi-acceleration/9921864038601341</a> <a href="#fnref:2" class="footnote-back-ref">↩</a></p></li> <li id="fn:3"><p>Baruh, H. (1999). Analytical Dynamics. McGraw Hill. <a href="/wiki/McGraw_Hill" target="_blank">/wiki/McGraw_Hill</a> <a href="#fnref:3" class="footnote-back-ref">↩</a></p></li> <li id="fn:4"><p>"Course Notes MIT" (PDF). <a href="https://ocw.mit.edu/courses/mechanical-engineering/2-003sc-engineering-dynamics-fall-2011/newton2019s-laws-vectors-and-reference-frames/MIT2_003SCF11Kinematic.pdf" target="_blank">https://ocw.mit.edu/courses/mechanical-engineering/2-003sc-engineering-dynamics-fall-2011/newton2019s-laws-vectors-and-reference-frames/MIT2_003SCF11Kinematic.pdf</a> <a href="#fnref:4" class="footnote-back-ref">↩</a></p></li> <li id="fn:5"><p>Harithuddin, A.S.M. (2014). Derivative Kinematics in Relatively Rotating Coordinate Frames: Investigation on the Razi Acceleration. RMIT University. p. 22. <a href="https://researchrepository.rmit.edu.au/esploro/outputs/doctoral/Derivative-kinematics-in-relatively-rotating-coordinate-frames-investigation-on-the-Razi-acceleration/9921864038601341" target="_blank">https://researchrepository.rmit.edu.au/esploro/outputs/doctoral/Derivative-kinematics-in-relatively-rotating-coordinate-frames-investigation-on-the-Razi-acceleration/9921864038601341</a> <a href="#fnref:5" class="footnote-back-ref">↩</a></p></li> </ol>