Line Integrals

So, I've just learnt line integrals and I understand that

Unparseable latex formula:

\displaystyle \int_{R}{\vec{F}} \cdot d \vec{r} = \int_{a}^{b} \vec{F}(\mathBB{r}(t)) \cdot \frac{d \vec{r}}{dt} dt

, which would be a scalar.

But I thought that a line integral computes the area above the curve defined by the vector function

\displaystyle \vec{r}(t)

and below the function

\displaystyle \vec{F}

. But what happens when the line integral is a vector, for example in the question:

\displaystyle \phi = xyz, \vec{F} = (xy,0,x^2), c : (x,y,z) = (t^2,2t,t^3), t \in (0,1)

the line integral

\displaystyle\int_{c} \phi d\vec{r}

and

\displaystyle \int_{c} \vec{F} \times d\vec{r}

are vectors. Why? and what is this calculating?

(edited 5 years ago)

Can you clarify which terms denote vector quantities please?

Can you clarify which terms denote vector quantities please?

Edited, hopefully it's clearer.

LaTex

What is an LLM and how could it benefit your career?

Powerful questions business students forget to ask their career advisers

Why I chose a law degree and what studying law was like