Is the following line-integral valid and defined?

[whole rivet]

So there's your typical line integral of a scalar field, which I'll send a picture of the definitions for in a page.

And then there are line integrals of a vector field. Typically line integrals of a vector field are done with dot products, so technically speaking, what you're doing is integrating a scalar to receive a scalar by the end.

But what if you did a line integral of a vector field where the end result is a vector? I'll send a second picture of one sentence for what I mean, using many of the same terms for what's in the picture of the scalar field example

I've seen examples where a vector valued function of a scalar variable (e.g. $\boldfont{F}(t)) in physics is integrated, and the end result is a vector. So I'm thinking that this should generalise to doing a line integral of a vector valued function where the input itself is multivariable/(a vector), and the end result of integration is a vector.

[hollow fern]

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