Geometric object carrying integration: oriented curve and surface.
Definition 4.5.1Directional curve (path). Given (a kind of motion), represents the position
( is the time), .
Note is not a parametric representative of ! Curve is a static geometric object, while path is a
dynamic motion (meaning it could turn back along ).
Example 4.5.1 Work done by force. (continuous) is a vector field where represents the position.
Work
where represents a forward-direction unit tangent vector field of .
Flow velocity field (continuous). Assuming represents a forward-direction unit tangent vector field of
closed curve , then the circulation is
Assuming
Then we have
where has the form of first-order derivative.
Example 4.5.2 Assuming is a simple closed curve in plain (boundary of plain), represents a
forward-direction unit outer-norm vector field of , then the flux is
The relation between circulation and flux. Assuming the norm vector of the plain (direction
determined by the forward direction of and right-hand rule), then construct where is derived by
rotating by counterclockwise. Since , naturally we have
Assuming
then we have
It also has a form of first-order derivative.
Example 4.5.3 Assuming
and its direction is determined by right-hand rule circling around axis. Seek .
Firstly, seek the parametric equation of . Eliminate by to get , so
Let , then .
The range of . It is trivial that .
Is the increasing direction of the same as the direction of ? Take , we can get that and
. Then we can verify that these directions are the same.
Then on , we have
Then it’s left as exercise. :)
Example 4.5.4 Assuming is a simple closed (Jordan) oriented (counterclockwise, natural positive
direction) plain curve of , then
where represents the area of closed domain bounded with .
Assuming where , then we have
Notice that
So the former 2 equalities are proved.
Assuming , then has inverse function , we have
Theorem is proved.
Definition 4.5.2 is a field with potential if such that , . is called a potential function of .
Theorem 4.5.3For any path (starting from , ending at ) of in , we have
Assuming
then we have
where is called the total differential of . Then (Newton-Leibniz formula).
Definition 4.5.4If the integral of a vector field on any path only relates to the starting
point and the destination, i.e. it has nothing to do with the path itself, then the vector field is
conservative.
Corollary 4.5.5Field with potential is conservative.
Theorem 4.5.6Assuming is a connected open set (domain), then any conservative field on
has potential function.
ProofMark as the conservative field, assuming is a path from to . Mark , it is to be proven
that .
Notice that
Repeat this process, we can get that .
Example 4.5.5 Assuming , seek its potential function.
Notice that
hence .
Example 4.5.6 Seek
where is part of , circling from to in counterclockwise direction.
Notice that
hence
Example 4.5.7 Seek
where , , .
Notice that
hence
Assuming (), then . Assuming , we have .
Definition 4.5.7Assuming is a vector field. If
then is irrotational.
Corollary 4.5.8Conservative field is irrotational.
Example 4.5.8 Irrotational may not be conservative. Assuming is defined on , it is trivial that it is
irrotational since
Yet after transformation ,
In fact, the integral reflects the geometric property of – how many turns does circle around the
origin by.
Assuming where , then
Since we can’t define a singular-value continuous function , so the vector field above is not
conservative! If the domain of definition , then the vector field is conservative. It relates to the
topological structure of the domain.
Definition 4.5.9Assuming is a vector field of , mark
respectively as the rotation and divergence of .
Theorem 4.5.10(Green formula in physical form) Assuming is a bounded closed domain, is
piecewisely of , is a vector field of , then
1.
.
2.
.
where the forward direction of is the natual positive direction.
Mark , we could mark
Notice that
indicating that the divergence has nothing to do with the selection of the coordinate system, since the
similarity transformation of a matrix doesn’t change its trace.
Definition 4.5.11Wedge (exterior) product. Define a formal product satisfying
Binary linearity.
Anti-symmetry.
It is trivial that . Actually, represents a oriented area, yet represents a non-oriented area. When
moving forward along , the outside of is on the right, i.e. the forward direction of is the natual
positive direction, we have .
Definition 4.5.12Exterior differential. Assuming a form of first-order derivative , define
Then the third form of Green formula is
Theorem 4.5.13(Green formula in mathematical form) .
Notice that
so we have
verifying the second form of Green formula.
Note The physical meaning of divergence. Assuming is a vector field of , , then we
have
Note
If , is called a non-source vector field.
If , is called a irrotational field.
For a linear vector field , is non-source, is symmetric is irrotational.
Corollary 4.5.14If is a single-connected domain, i.e. any continuous closed loop in could
become a single point after continuous transformation in , then any irrotational vector field of
on is conservative.