Hi.
Imagine a random curve in the xy-plane.
z = f(x,y) defines a surface in 3-space. The part of that surface directly above the curve in the xy-plane is a curve on the surface.
The integral of a scalar function along this curve is the area of the fence rising straight up from the curve to the surface above the curve.
Now, for the vector function:
Take the same random curve in the xy-plane. At each point on this curve there is a vector emanating in some direction.
At each point, there is also the Tangent vector to the curve.
If I project each vector of the vector field at a given point ONTO the Tangent vector at that point, I get a scalar -- this is the magnitude of the projection. I then add up (Integrate) all of these magnitudes along the entire curve. This is the Line Integral of a Vector Field ALONG a curve. It is a numerical measure of the net movement of the vector field ALONG the curve.
Note that you can also project each vector of the field ONTO the Normal vector at a given point, instead of onto the Tangent vector. In this case, we call it the Line Integral of the Vector Field ACROSS the curve. This is a numerical measure of the net movement of the vector field ACROSS the curve.
The first is called a Circulation Integral
The second is called a Flux (Flow) Integral
Hope that helps.
Best wishes.
