Q. What are Line Integrals (Flow Integrals) of Vector Fields?
A. Well, they take a vector field and a curve and sort of measure how
much the Vector Field points along the curve. If the Vector Field is
Force and the path is Displacement, they give work.

Q. How to calculate them?
A. If you can't quite see it from the picture, you better parametrize
the curve (like r(t)=x(t)i+ y(t)j + z(t)k) and then integrate this DOT
PRODUCT like this:

         Integral of F.r'(t) dt 

Q. Wait a second. What is that F? Isn't that a vector field? That means
the vector wobbles around quite a bit... Which one of these
many vectors am I using in this integral?
A. Oh, sorry, you are right. Well, that is F calculated at the point
r(t) on the path... So, more explicitly:

        Integral of F(r(t)) . r'(t) dt

where the . is a DOT PRODUCT between two vectors, remember?

Q. And F(r(t)) is F(x(t),y(t),z(t))?
A. Correct. Just plug in the components of the position vector r(t) into
the equation of the vector field F. Everything will depend on t then and
you integrate dt.

Q. Ok, cool, so what is Circulation?
A. Oh, when that curve C is closed, we call the Line Integral "the
Circulation" of the Vector Field around that path. Just a different name
for the Line Integral for this specific case.

Q. And what is this stuff about Gradient Fields, Path-Independent Fields
and Conservative Fields?
A. Ok, let's start with Path-Independent fields. So, you have this
vector field F. Look at it. Admire it. Now, if you pick two points A and
B and a path C1 between them, then the Line Integral of F over C1 is a
certain number. If you pick another path C2 joining the same two points
A and B, it is another Line Integral, another number.

Q. Are they the same?
A. Excellent question! Well, usually not. But there are some vector
fields where the answer is yes! If you have a Vector Field that has this
property (fix two points A and B, no matter what path between them you
get the same Line Integral), then we call this vector field
PATH-INDEPENDENT. Very few Vector Fields seem to have this property.

Q. Hmmm... I see. And the point of Friday's lecture is that these are
Gradient Fields?
A. Yes! Gradient Fields are fields that come from a function f(x,y) by
taking grad f = fx i + fy j, or possibly from f(x,y,z) by taking grad f
= fx i + fy j + fz k. Now the connection is that all PATH-INDEPENDENT
fields are GRADIENT fields and vice-versa! Now, is that amazing or what?

Q. Wowness. That's really weird though... Where does the connection come
from? I mean, they look like such different things...
A. Well, you're right. They look different. But we proved today that
when you integrate the Vector Field F=grad f along a path C joining A to
B then:

        Integral of F along C = Integral of grad f along C = f(B)-f(A)

Look at this expression: the right side DOES NOT DEPEND on the path.
Therefore grad f is path-independent...

Q. And how did we prove that equation?
A. Well, you could look at the book, but I will give you the general
idea. Remember the chain rule? It basically says that

        df/dt = grad f . (dr/dt)

        (that dot is a dot product between two vectors). So when you integrate
F = grad f along C, you are just integrating (df/dt) with relation to t,
that is, you are just measuring how much f is changing along the path!
That is f(B)-f(A).

Q. Hmm. Ok. So GRADIENT VECTOR FIELDS are PATH-INDEPENDENT, that is,
CONSERVATIVE?
A. Yeah. CONSERVATIVE is just another name for PATH-INDEPENDENT.

Q. So no matter what Line Integral I take for a Gradient Field, the
answer is always the same?
A. Well, it still depends on the endpoints A and B. Otherwise, no matter
what path between them, yes, still the same.

Q. Are there other vector fields that are PATH-INDEPENDENT?
A. No, all vector fields that are PATH-INDEPENDENT turn out to be
GRADIENT fields. We didn't prove this in class though... Not yet.

Q. Wow, that is really cool. You are amazing.
A. Oh, gee, thanks. 

Q. No I mean it.
A. Well, you are very smart too. Great questions.

Q. What a shameless plug. I better go now. C ya.
A. Bye. See you on Monday.