The Integration-By-Parts Formula for Vector Products

[Jhenrique]

We know that: $$\frac{d}{dx}(\vec{f} \cdot \vec{g}) = \frac{d\vec{f}}{dx} \cdot \vec{g} + \vec{f} \cdot \frac{d\vec{g}}{dt}$$ and: $$\frac{d}{dx}(\vec{f} \times \vec{g}) = \frac{d\vec{f}}{dx} \times \vec{g} + \vec{f} \times \frac{d\vec{g}}{dt}$$ But, exist some formula (some expansion) for: $$\int \vec{f} \times \vec{g}\;\;dx$$ and for: $$\int \vec{f} \cdot \vec{g}\;\;dx$$ ?

 

[HallsofIvy]

The analogy with integration by parts would be
$$\int\vec{f}\times\vec{g}dx= \vec{f}\times\int\vec{g}dx- \int \frac{d\vec{f}}{dx}\times \vec{g}dx$$
and
$$\int\vec{f}\cdot\vec{g}dx= \vec{f}\cdot\int\vec{g}dx- \int \frac{d\vec{f}}{dx}\cdot \vec{g}dx$$

can you use your derivative formulas to verify those?

 

[vanhees71]

I don't understand this conclusion. You get the integration-by-parts formula from integrating the differential expressions given in #1, e.g.,
$$\frac{\mathrm{d}}{\mathrm{d} x} (\vec{f} \times \vec{g}) = \frac{\mathrm{d} \vec{f}}{\mathrm{d} x} \times \vec{g} + \vec{f} \times \frac{\mathrm{d} \vec{g}}{\mathrm{d} x}$$
wrt. $x$
$$\vec{f} \times \vec{g}=\int \mathrm{d} x \frac{\mathrm{d} \vec{f}}{\mathrm{d} x} \times \vec{g} + \int \mathrm{d} x \vec{f} \times \frac{\mathrm{d} \vec{g}}{\mathrm{d} x}$$
or bringing one term to the other side
$$\int \mathrm{d} x \frac{\mathrm{d} \vec{f}}{\mathrm{d} x} \times \vec{g} = \vec{f} \times \vec{g} - \int \mathrm{d} x \vec{f} \times \frac{\mathrm{d} \vec{g}}{\mathrm{d} x}.$$