1
99:59:59,999 --> 99:59:59,999
Now that we know a little bit about how to construct a unit normal vector

2
99:59:59,999 --> 99:59:59,999
at any point in a curve--that's what we did in the last video--

3
99:59:59,999 --> 99:59:59,999
I wanna start exploring and interesting expression.

4
99:59:59,999 --> 99:59:59,999
So the expression is: The line integral around a closed loop, and we're gonna go in the positive counter

5
99:59:59,999 --> 99:59:59,999
clockwise orientation

6
99:59:59,999 --> 99:59:59,999
of a vector field F dotted with the unit normal vector at any point on that curve ds

7
99:59:59,999 --> 99:59:59,999
(I'll write ds in magenta)

8
99:59:59,999 --> 99:59:59,999
ds.

9
99:59:59,999 --> 99:59:59,999
So first lets conceptualize what this is even saying

10
99:59:59,999 --> 99:59:59,999
and then let's manipulate it a little bit to see if we can come up with

11
99:59:59,999 --> 99:59:59,999
and interesting conclusion, we actually will manipulate it

12
99:59:59,999 --> 99:59:59,999
we'll use green's theorem

13
99:59:59,999 --> 99:59:59,999
and we're actually gonna come up with a 2 dimensional

14
99:59:59,999 --> 99:59:59,999
version of the divergence theorem

15
99:59:59,999 --> 99:59:59,999
which all sounds very complicated

16
99:59:59,999 --> 99:59:59,999
but hopefully we can get a little bit of an intuition

17
99:59:59,999 --> 99:59:59,999
for it as to why it actually a little bit of common sense.

18
99:59:59,999 --> 99:59:59,999
So, first lets just think about this.

19
99:59:59,999 --> 99:59:59,999
So let me draw- let me draw a coordinate plane here

20
99:59:59,999 --> 99:59:59,999
so we do it in white

21
99:59:59,999 --> 99:59:59,999
So this right over here thats our y axis

22
99:59:59,999 --> 99:59:59,999
that over there is our x axis

23
99:59:59,999 --> 99:59:59,999
let me draw ourselves my curve so my curve might look

24
99:59:59,999 --> 99:59:59,999
something like--I'll do it in a blue color

25
99:59:59,999 --> 99:59:59,999
so my curve might look something like this- my contour

26
99:59:59,999 --> 99:59:59,999
and its going in the positive, counterclockwise direction

27
99:59:59,999 --> 99:59:59,999
just like that

28
99:59:59,999 --> 99:59:59,999
and now we have our vector field, and just a reminder

29
99:59:59,999 --> 99:59:59,999
we've seen this multiple times

30
99:59:59,999 --> 99:59:59,999
I can- my vector field will associate a vector with any

31
99:59:59,999 --> 99:59:59,999
point on the x-y plane

32
99:59:59,999 --> 99:59:59,999
and it will-- it can be defined as some function

33
99:59:59,999 --> 99:59:59,999
of x and y, which I'll call that P

34
99:59:59,999 --> 99:59:59,999
some function of x and y times the i unit vector

35
99:59:59,999 --> 99:59:59,999
so it says what the i component of the vector field is for any x and y point

36
99:59:59,999 --> 99:59:59,999
and then what the j component, or what we multiply

37
99:59:59,999 --> 99:59:59,999
the j component by

38
99:59:59,999 --> 99:59:59,999
or the vertical component by for any x and y point

39
99:59:59,999 --> 99:59:59,999
so some function of x and y times i

40
99:59:59,999 --> 99:59:59,999
plus some other scalar function of x and y

41
99:59:59,999 --> 99:59:59,999
times j

42
99:59:59,999 --> 99:59:59,999
and so if you give me any point

43
99:59:59,999 --> 99:59:59,999
there will be an associated vector with it

44
99:59:59,999 --> 99:59:59,999
any point there is an associated vector depending on

45
99:59:59,999 --> 99:59:59,999
how we define this function.

46
99:59:59,999 --> 99:59:59,999
But this expression right over here

47
99:59:59,999 --> 99:59:59,999
we're taking a line integral

48
99:59:59,999 --> 99:59:59,999
we care specifically about the points about along this curve

49
99:59:59,999 --> 99:59:59,999
along this contour

50
99:59:59,999 --> 99:59:59,999
right over here and so lets think about what this is actually

51
99:59:59,999 --> 99:59:59,999
this piece right over here is actually telling us

52
99:59:59,999 --> 99:59:59,999
before we sum up all of these infinitesimally small pieces

53
99:59:59,999 --> 99:59:59,999
so if we just take f dot n

54
99:59:59,999 --> 99:59:59,999
so lets just think about a point on this curve

55
99:59:59,999 --> 99:59:59,999
so a point on this curve may be this point right over here

56
99:59:59,999 --> 99:59:59,999
so associated with that point there is a vector

57
99:59:59,999 --> 99:59:59,999
that's what the vector field does

58
99:59:59,999 --> 99:59:59,999
So f might look something like that right over there

59
99:59:59,999 --> 99:59:59,999
so that might be f at that point

60
99:59:59,999 --> 99:59:59,999
and we're gonna dot it with the unit normal vector at that point

61
99:59:59,999 --> 99:59:59,999
so the unit normal vector might look something like that

62
99:59:59,999 --> 99:59:59,999
that would be n hat at that point

63
99:59:59,999 --> 99:59:59,999
this is the vector field at that point