In contexts where the fact that the quantity we are measuring via a line integral is best measured via a dot product (such as calculating work), the notation we have used thus far for line integrals is fairly common. However, sometimes the vector field is such that the units on \(x\text{,}\) \(y\text{,}\) and \(z\) are not distances. In this case, a dot product may not have quite the same physical meaning, and an alternative notation using differentials can be common. Specifically, if \(\vF(x,y,z) = F_1(x,y,z)\vi + F_2(x,y,z)\vj + F_3(x,y,z)\vk\text{,}\) then

\begin{equation*}
\int_C\vF\cdot d\vr = \int_C \langle F_1, F_2, F_3 \rangle \cdot \langle dx, dy, dz \rangle= \int_C F_1\, dx + F_2\, dy + F_3\, dz\text{.}
\end{equation*}

A line integral in the form of \(\int_C F_1\, dx + F_2\, dy + F_3\, dz\) is called the differential form of a line integral. (If \(\vF\) is a vector field in \(\R^2\text{,}\) the \(F_3\, dz\) term is omitted.) For example, if \(\vF(x,y,z) = \langle x^2y,z^3,x\cos(z)\rangle\) and \(C\) is some oriented curve in \(\R^3\text{,}\) then

\begin{equation*}
\int_C\vF\cdot d\vr = \int_C x^2y\, dx + z^3\, dy + x\cos(z)\,dz
\end{equation*}