Newton and Leibniz formula, the most basic formula in unary calculus.
It is proved that the definite integral of a function on an interval can be expressed by the values of the original function at both ends of the interval.
Coincidentally, the double integral in the plane region can also be expressed by the curve integral along the boundary curve of the region, which is the Green's formula we want to introduce.
1, the concept of simply connected region
Let it be a plane region, and if a part of the region surrounded by any closed curve belongs to it, it is called a plane simply connected region; Otherwise it is called a complex connected region.
Generally speaking, a single connected region is a region without "holes" (including "point holes") and "cracks".
2. Positive adjustment of regional boundary curve
Let it be the boundary curve of a plane area, and the specified positive direction is that when the observer walks in this direction, the part close to him is always on his left.
In short: the positive direction of the boundary curve of the region should be suitable for the conditions, people should follow the curve and the region is in the left hand.
3. Green's formula
The theorem assumes that the closed region is surrounded by piecewise smooth curves, and the sum of functions has a first-order continuous partial derivative, then there is
( 1)
Where is the positive boundary curve.
The formula (1) is called Green's formula.
Prove first
Suppose the shape of the region is as follows (a straight line parallel to the axis passes through the region, and the intersection with the boundary curve of the region is at most two points).
Obviously, the area shown in Figure 2 is a special case of the area shown in Figure 1, so we only need to prove the area shown in Figure 1.
On the other hand, according to the properties of curve integral and the calculation method of coordinates, there are
therefore
Assuming that the intersection of a straight line passing through the interior of the region and parallel to the axis and the boundary curve of is at most two points, it can be proved in a similar way.
comprehensive
When the intersection of the boundary curve of the region and any straight line passing through the interior and parallel to the coordinate axis (axis or axis) is at most two points, we have
,
Meanwhile.
Green's formula is obtained by combining two formulas.
Note: If the region does not meet the above conditions, that is, when the intersection of the straight line passing through the region and parallel to the coordinate axis and the boundary curve exceeds two points, one or several auxiliary curves can be introduced into the region and divided into several local regions, so that each local region is suitable for the above conditions, and Green's formula can still be proved.
Green's formula relates the double integral with the curve integral of coordinates, so it is widely used.
If,,, Green's formula is
So the area of this area is
Example 1 Find the graphic area surrounded by the star line.
Solution: When changing from to, the point describes the whole closed curve counterclockwise, so
Example 2 Let it be a smooth closed curve with arbitrary segments, and prove it
Proof: here,
therefore
This is the area surrounded by.
Second, the plane curve integration has nothing to do with the path conditions.
1, the definition of curve integral with coordinates independent of path.
Let's define a hypothesis as an open region, a function, in which there is a first-order continuous partial derivative.
Constant, that is, curve integral, has nothing to do with the path; Otherwise, it is related to the path.
The definition of 1 can also be replaced by the following equivalent statement.
If the curve integral is independent of the path, then
That is to say, on the closed curve formed by in the region, the curve integral is zero. On the other hand, if the curve integral along any closed curve in the region is zero, it can be easily deduced that the curve integral is independent of the path.
The definition of hyperbolic integral is path-independent, which means that for any closed curve, there is always
.
2. Curve integration has nothing to do with path conditions.
The theorem assumes that the open domain is a simply connected domain, and the function has a first-order continuous partial derivative in the open domain, then the necessary and sufficient condition that the curve integral in the open domain is independent of the path is equality.
Neiheng was established.
Evidence: the sufficiency of previous evidence
Take any closed curve, because it is simply connected, so the area surrounded by closed curve is included, so it holds in the world.
According to Green's formula, there are
According to the second definition, the internal curve integral is independent of the path.
The necessity of re-proving (by reducing to absurdity)
Assuming that the internal equation is not always correct, then at least one thing makes
Might as well set up
Because it is continuous inside, there is a circle with a center and a small enough radius in memory, which makes it always exist in the world.
From Green's formula and the properties of double integral, we can know that
This is the positive boundary curve and area.
This contradicts the condition that the curve integral on any closed curve is zero, so it is in the internal equation.
Won Heng was established.
Note: Two conditions required by the theorem
Both are indispensable.
A counterexample is discussed, which is a piecewise smooth curve around the origin and the forward direction is counterclockwise.
here
,
Except for the origin, it exists and is a continuous sum.
Make a circle with a small radius.
In a complex connected domain surrounded by and, Green's formula is used as follows
Third, the total differential quadrature of binary function.
If the curve integral has nothing to do with the path in the open area, it is only related to the coordinates of the starting point and ending point of the curve. Assuming that the starting point and ending point of the curve are both, you can mark it.
or
To express, you don't have to explicitly write out the integral path.
Obviously, this integral form is very similar to definite integral. In fact, we have the following important theorems.
Theorem 1 Suppose that it is a simply connected open domain, a function of the first-order continuous partial derivative, and then
Is a single-valued function, where is the internal fixed point, and
that is
It is proved that the curve integral of any internal curve starting from a point and ending at a point has nothing to do with the path, but only with the coordinates of the starting point and the ending point, that is, it is indeed a single-valued function of the point.
The following facts prove that
Because it can be considered as a curve integral along any path from point to point, take the following path, including
It can also prove that
therefore
Theorem 2 is a simply connected open domain with a first-order continuous partial derivative, so its necessary and sufficient conditions for the total differential of functions are as follows.
Neiheng was established.
Obviously, sufficiency is theorem one.
The following proves the necessity.
If so, then
Because the interior is continuous, the second-order mixed partial derivative conforms to the equation.
therefore
Theorem 3 is a simply connected open domain, and the function has a first-order continuous partial derivative. If there is a binary function, it makes
rule
Among them, there are two arbitrary points.
Prove that the function is known from the theorem 1
suit
So still
So (it's a constant)
that is
but
This is because the point returns to the point along any internal path to form a closed curve, so
Therefore □
Method of determining total differential function
Because the curve integral at the right end has nothing to do with the path, in order to make the calculation simple, a broken line connected by straight lines parallel to the coordinate axis can be taken as the integration path (of course, the broken line should completely belong to the simply connected region).