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IEVref: | 102-05-32 | ID: | |

Language: | en | Status: Standard | |

Term: | first Green formula | ||

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Definition: | identity resulting from the divergence theorem applied to the vector field ${f}_{1}\text{\hspace{0.17em}}\mathrm{grad}\text{\hspace{0.17em}}{f}_{2}$, where f_{1} and f_{2} are two scalar fields given at each point of a three-dimensional domain V limited by a closed surface S$\underset{\text{V}}{\iiint}\left(\mathrm{grad}\text{\hspace{0.17em}}{f}_{1}\cdot grad\text{\hspace{0.17em}}{f}_{2}+{f}_{1}\text{\hspace{0.17em}}\Delta \text{\hspace{0.05em}}{f}_{2}\right)}dV={\displaystyle \underset{\text{S}}{\u222f}{f}_{1}}\text{\hspace{0.17em}}grad\text{\hspace{0.17em}}{f}_{2}\cdot {e}_{\text{n}}dA$ where d Note 1 to entry: In English the first Green formula is sometimes called “first Green theorem” or “first Green identity”. | ||

Publication date: | 2008-08 | ||

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Internal notes: | 2017-02-20: Editorial revisions in accordance with the information provided in C00019 (IEV 102) - evaluation. JGO | ||

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$\underset{\text{V}}{\iiint}\left(\mathrm{grad}\text{\hspace{0.17em}}{f}_{1}\cdot grad\text{\hspace{0.17em}}{f}_{2}+{f}_{1}\text{\hspace{0.17em}}\Delta \text{\hspace{0.05em}}{f}_{2}\right)}dV={\displaystyle \underset{\text{S}}{\u222f}{f}_{1}}\text{\hspace{0.17em}}grad\text{\hspace{0.17em}}{f}_{2}\cdot {e}_{\text{n}}dA$

where d*V* is the volume element, *e*_{n}d*A* is the vector surface element and Δ is the Laplacian operator

Note 1 to entry: In English the first Green formula is sometimes called “first Green theorem” or “first Green identity”.