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*<math>(\mathbf{A} \cdot \nabla)\mathbf{B} = \frac{1}{2}\bigg[\nabla(\mathbf{A} \cdot \mathbf{B}) - \nabla\times(\mathbf{A} \times \mathbf{B}) - \mathbf{B}\times(\nabla \times \mathbf{A}) - \mathbf{A}\times(\nabla \times \mathbf{B}) - \mathbf{B}(\nabla \cdot \mathbf{A}) + \mathbf{A}(\nabla \cdot\mathbf{B})\bigg]</math><ref name=KuoAcharya>{{cite book |last1=Kuo |first1=Kenneth K. |last2=Acharya |first2=Ragini |title=Applications of turbulent and multi-phase combustion |date=2012 |publisher=Wiley |location=Hoboken, N.J. |isbn=9781118127575 |page=520 |doi=10.1002/9781118127575.app1 |url=https://onlinelibrary.wiley.com/doi/pdf/10.1002/9781118127575.app1|access-date=19 April 2020 |archive-url=https://web.archive.org/web/20210419221120/https://onlinelibrary.wiley.com/doi/pdf/10.1002/9781118127575.app1|archive-date=19 April 2020 |url-status=live }}</ref>
*<math>(\mathbf{A} \cdot \nabla)\mathbf{A} = \frac{1}{2}\nabla |\mathbf{A}|^2-\mathbf{A}\times(\nabla\times\mathbf{A}) = \frac{1}{2}\nabla |\mathbf{A}|^2 + (\nabla\times\mathbf{A})\times \mathbf{A}</math>
====Second derivatives====
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