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Anyone knows how to apply chain rule to this term $\vec a \cdot \nabla \cdot \nabla \vec b$?

$\nabla$ operator is defined in Cartesian coordinate system $R^3$ with coordinates $(x, y, z)$, see reference.$\nabla \vec b$ is the gradient of a vector, that is "a tensor", so the divergence of a 2nd order tensor $\nabla \cdot \nabla \vec b$ is a vector again, and the final dot product of two vectors $\vec a \cdot \nabla \cdot \nabla \vec b$ would be a scalar.

I'd like to see something similar to $a\nabla b = \nabla (ab) - b\nabla a$, I need this for the purpose of an integration.

Thanks

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    I h$a$ve edited the original question. Thanks2012-05-29

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I guess what you are looking for is a product rule (used for partial integration). For such problems, it is usually very helpful to write the expression explicitly (in coordinates). We have $\vec a \cdot \Delta \vec b= \vec a \cdot (\vec\nabla \cdot \vec\nabla) \vec b = \sum_{i,j}a_i \partial^2_j b_i.$

Now take a look at $ \sum_{i,j}\partial_j (a_i \partial_j b_i)= \sum_{i,j} \left( a_i \partial^2_j b_i +(\partial_j a_i) (\partial_j b_i) \right).$

The formula for partial integration thus reads $\int \!d^dx\,\vec a \cdot \Delta \vec b = \underbrace{\int\!d^dx\,\nabla\left(\sum_i a_i \nabla b_i\right)}_\text{surface term using Gauss} -\int\!d^dx \sum_i(\vec\nabla a_i)(\vec\nabla b_i). $

Some more (potentially) useful formulas:

Interchanging $a$ and $b$, we have $\int \!d^dx\,\vec b \cdot \Delta \vec a = \underbrace{\int\!d^dx\,\nabla\left(\sum_i b_i \nabla a_i\right)}_\text{surface term using Gauss} -\int\!d^dx \sum_i(\vec\nabla a_i)(\vec\nabla b_i). $

Subtracting the two relations yields (this is the vector version of Green's second identity) $\int \!d^dx\,(\vec a \cdot \Delta \vec b- \vec b \cdot \Delta \vec a) =\int\!d^dx\,\nabla\left(\sum_i a_i \nabla b_i - \sum_i b_i \nabla a_i\right).$

Adding the two relations yields $\int \!d^dx\,\vec a \cdot \Delta \vec b = \int\!d^dx\,\nabla\left(\sum_i a_i \nabla b_i + \sum_i b_i \nabla a_i\right)- \int \!d^dx\,\vec b \cdot \Delta \vec a-2\int\!d^dx \sum_i(\vec\nabla a_i)(\vec\nabla b_i). $

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    So, if I understand correctly, you do not suggest using notation like "$\left( {\nabla \vec b} \right) \cdot \left( {\nabla \vec a} \right)$", because they are not well defined and is not powerful enough to represent the tensor calculus. Please correct me if I am wrong. But if I do use the notation of $\left( {\nabla \vec b} \right) \cdot \left( {\nabla \vec a} \right)$ to represent $\vec\nabla a_i)(\vec\nabla b_i$, then it is OKAY to write $\vec a\cdot\nabla\cdot\nabla\vec b=\nabla \cdot \left({\vec a\cdot\nabla\vec b}\right) - \left({\nabla \vec b} \right)\cdot\left({\nabla\vec a}\right)$?2012-05-29