Why is it that a line integral of a vector field takes the dot product of the vector field with the tangent? This results in us taking the component of the vector field in the direction of the tangent of the curve we are integrating over. This can give us work done as a physical interpretation.
If this curve encloses an area, we can also use Green's theorem over that area instead of the line integral to get the same result. Green's theorem measures flux ALONG the boundary.
Now the analogue to that in a higher dimension is the surface integral of a vector field. However here the definition of the surface integral is taking the dot product of the vector field with the normal (more specifically the cross product of the first partial derivatives), with its interpretation as flux across the surface. Here, we can use the divergence theorem also if the surface encloses a volume. Divergence theorem measures the flux ACROSS the boundary. Why is there a difference?
So why is a surface integral measuring something across the boundary and the line integral something along?
EDIT: We notice the curl is actually a vector field itself.
Looking through some notes, if we take a vector field in two dimensions, we have components $$f = (P,Q,0)$$
Hence, curl is equal to $$curl f.k$$ which is just $$\frac{\partial Q}{\partial x} – \frac{\partial P}{\partial y}$$ and an integral with this as its integrand is exactly Green's theorem.
The curl in this case is integrated over a flat surface on the xy plane and the normal to the curl is just the k component (standard basis of the z axis). By applying the definition of surface integrals on Stoke's theorem, we get Green's theorem. This shows that Green's theorem is just a special case of Stoke's theorem.
So the exact definition of the surface integral can be used to result in Green's theorem, and Green's theorem can be obtained directly by considering circulation over an area which links to our line integral, so there is definitely a link here that I can't seem to grasp.
EDIT 2: I would like to add the fact that I mention in my comments. The line integral and surface integral over scalar valued functions makes sense. One could say that the line integral uses a tangential component (the norm of our the derivative of our parametrisation), while the surface integral a normal (the norm of the cross product of the first partial derivatives), but this is only because of its geometrical interpretation. The tangent used in line integral approximates the line, while the norm of the cross product used in surface integrals approximate the surface using the area of a parallelogram – which is exactly the norm of the cross product between arbitrary vectors.
So why is the link in vector valued functions less obvious? What am I missing in my thinking?
Best Answer
You are mixing up two different things; the surface integral is not a generalization of the line integral.* This is easiest to see in two dimensions, where everything is an integral along curves and yet you will still find a difference.
For simplicity, let's consider a constant wind field blowing to the right, $\mathbf f(x,y)=(1,0)$. Also consider two curves, $A$ a horizontal line segment from $(0,0)$ to $(1,0)$, and $B$ a vertical line segment $B$ from $(0,0)$ to $(0,1)$. In 2D, given a vector field and a curve there are two different kinds of integral you can consider.
Interpret the curve as a wire on which a bead is threaded. If you move the bead from one end to the other, how much does the wind help or hinder the motion of the bead? This is the usual line integral $\int \mathbf f\cdot\mathrm d\mathbf r$. It is large for curve $A$ and zero for curve $B$.
Interpret the curve as a butterfly net being held stationary while the wind blows through it. How much air passes through it per unit time? This is the flux integral $\int \mathbf f\cdot\mathbf n\,\mathrm d\ell$, where $\mathbf n$ is the unit vector perpendicular to the curve tangent. It is zero for curve $A$ and large for curve $B$.
These two notions generalize to higher dimensions in different ways. The line integral remains an integral over a $1$-dimensional object, i.e. a curve. The flux integral becomes an integral over a over an $(n-1)$-dimensional object, i.e. a surface.
*In an abstract sense one could argue that they are both specializations of the same thing, but that will take us too far into the theory of differential forms.