What integral is used to calculate the electric field generated by a continuous charged curve?

Question

I'm studying Multivariable Mathematics, by Ted Shifrin, in which one reads that ''the gravitational force exerted by a continuous mass distribution $\Omega$ with density function $\delta$ is

$$\mathbf{F} = G \int_{\Omega} \delta \frac{\mathbf{x}}{||\mathbf{x}||^3} \mathrm{d}V \,."$$ A few paragraphs earlier, the meaning of the multiple integral of a vector-valued function $\mathbf{f}\colon \Omega (\subseteq \mathbf{R}^n) \to \mathbf{R}^m$ is defined: "Assuming each of the component functions $f_1,\dots, f_m$ is integrable on $\Omega$, we set

$$\int_\Omega \mathbf{f} \mathrm{d}V = \begin{bmatrix} \int_\Omega f_1 \mathrm{d}V\\ \vdots \\ \int_\Omega f_m \mathrm{d}V \end{bmatrix} \,."$$

My question is: how do I proceed if I want to calculate the gravitational force exerted by a curve $C$ with linear mass density $\lambda(\mathbf{x})$? My guess is that I should compute the path integral for each component function, so that $$\mathbf{F} = G \int_{C} \lambda \frac{\mathbf{x}}{||\mathbf{x}||^3} \mathrm{d}\ell = G \int_C \mathbf{f} \mathrm{d}\ell = G \begin{bmatrix} \int_C f_1 \mathrm{d}\ell\\ \vdots \\ \int_C f_m \mathrm{d}\ell \end{bmatrix} = G \begin{bmatrix} \int_{a}^b \, f_1(\mathbf{g}(t)) \, ||\mathbf{g}'(t)|| \, \mathrm{d}t\\ \vdots \\ \int_{a}^b \, f_m(\mathbf{g}(t)) \, ||\mathbf{g}'(t)|| \, \mathrm{d}t \end{bmatrix} \,,$$ given a parametrization $\mathbf{g} \colon [a, b] \to \mathbf{R}^n$ for $C$. Is that correct?

Answer 1

Yes, you want to compute a line integral (which comes in the next chapter of the book). More generally, you can do the same thing with any submanifold once you've learned how to integrate over a submanifold; there is a subtlety that you will need an "area form" for that submanifold. The cases of curves and surfaces are done explicitly in the next chapter, and you can generalize based on the discussion of surface area in section 4.2 of Chapter 8.