Is the output of a line integral over a scalar field a vector?

Question

In my physics book of "mathematical methods for physics", the author writes that line integral of a scalar function $\phi$ over a curve $C$ can be written as the following: $$\int_C\phi\,\text d{\textbf r}=\textbf{i}\int_C\phi(x,y,z)\,\text dx\,+\textbf{j}\int_C\phi(x,y,z)\,\text dy\,+\textbf{k}\int_C\phi(x,y,z)\,\text dz\,$$

Is it not supposed to be a numerical value? If we parametrize the curve $𝐶=𝑟(𝑡)$ with $𝑎≤𝑡≤𝑏$ then evaluate $\int_a^𝑏𝜙(𝑟(𝑡))|𝑣(𝑡)|𝑑𝑡$ we get such a number, not a vector. $𝑣(𝑡)$ is the derivative of $𝑟(𝑡)$ wrt $𝑡$

I posted this on MSE too: https://math.stackexchange.com/q/3479481/328520 but it was unhelpful.

It might come down to understanding the difference between $\int_C\phi\,\text d{\textbf r}$ and $\int_C\phi\,\text ds$. I have never seen a differential in the form of a vector in any of my calculus books.

Any help is greatly appreciated :)

Answer 1

Remember, you can think of integrals as just a continuous, infinite sum. You are adding up elements of $\phi\,\text d\mathbf r$ along the specified path. $\phi$ is a scalar multiplying a vector quantity $\text d\mathbf r$. So you are essentially just adding up a bunch of vectors. Thus, your integral gives you a vector quantity.