TITLE: Can a vector be integrated with respect to another vector?
QUESTION [10 upvotes]: I know it is possible to differentiate a vector with respect to another vector. However this left me wondering if it is possible to integrate a vector with respect to another vector. Such as for example:
$$ \int \vec{v} \space d\vec{v} $$
Where $\vec{v}$ is a unknown vector. Is this or something similar possible? If yes under what conditions?

REPLY [2 votes]: As said by others in the comments, to answer this question we must first specify what kind of product is intended between the differential $d\vec{v}$ and the integrand. If the product we have in mind is the dot product, then this integral indeed exists.
Let $\vec{F}:\mathbb R^n\to\mathbb R^n$ be a vector field. Then the integral
$$\int_\gamma\vec{F}(\vec{v})\cdot d\vec{v}$$
where $\gamma$ is a path in $\mathbb R^n$, can be defined. If we interpret $\gamma$ as the path of a particle and we interpret $\vec{F}$ as a force, then an intuitive interpretation of this integral is as the continuous sum of forces acting on the particle as it travels along its path. This is a physical quantity known as work, which is extremely useful in physics.