$$\frac{\partial \vec{A}\cdot \vec{B}}{\partial \vec{x}}$$
What does the above mean? How do you compute it? All vectors are three vectors and the $\vec{x}$ is the position vector.
What does this mean: $\frac{\partial \vec{A}\cdot \vec{B}}{\partial \vec{x}}$?
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$\begingroup$
notation
partial-derivative
vector-analysis
2 Answers
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It seems that $A$ and $B$ are vector fields depending on $x$. Then $\phi(x):=A(x)\cdot B(x)$ is a scalar function and has a gradient $$\nabla\phi(x)=\nabla\bigl(A(x)\cdot B(x)\bigr)\ .$$ Writing a vector in the denominator of a fraction is bad practice. You cannot divide by a vector.
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I agree with Blatter that this is a bad practice, and it is unclear.
It could also mean the Jacobian matrix. It is a matrix that contains partial derivatives, in such a way that the notation preserves "chain rule" for differentials, sort of. E.g. $\dot{\mathbf{f}}(\mathbf{x},t)=\frac{\partial \mathbb f}{\partial \mathbb x} \dot{\mathbb{x}}$.
One example of this notation in use is http://www.cs.huji.ac.il/%7Ecsip/tirgul3_derivatives.pdf, page 2.
In this case, a scalar function is differentiated, so the Jacobian is just a row vector instead of a matrix (though I'm not very sure, I don't see parentheses around $\vec A \cdot \vec B$.)