Property 1? Set to a constant, and then
Nature 2? If a complete arc segment can be divided into two smooth curved arcs, then
Nature 3? If it is online, there is.
In particular, there are
Theorem? Assuming that the curve arc is defined and continuous, the parameter equation of is
In which there is a first-order continuous derivative, and then the curve integral exists, and
If the curve is given by an equation, then we can regard this situation as a special parametric equation.
Then the formula is
For coordinate curve integration, we must pay attention to the direction of the integration arc segment.
The parameter equation of is defined to be continuous on the directed curve arc.
When the parameter changes monotonously from to, the point moves from the starting point to the end point, and there is a first-order continuous derivative in the closed interval with sum as the end point, then the curve integral exists, and
Where is the direction angle of the tangent vector of the directed curve arc at that point.
The expression in vector form is as follows
Where is the unit tangent vector of the directed curve arc at this point, which is called directed curve element.
(1) For the boundary curve of a plane area, the prescribed positive direction is as follows: When the observer walks in this direction, the part close to him is always on his left. As shown in the figure, the forward direction is counterclockwise and the forward direction is clockwise.
(2) Let it be a plane region. If all the parts surrounded by any closed curve belong to it, it is a plane simply connected region, otherwise it is called a complex connected region.
Let the closed region be surrounded by piecewise smooth curves, and the sum of functions has a first-order continuous partial derivative, then there is
Where is the positive boundary condition.
(1) For a complex connected region, the right end of Green's formula should include the curve integrals of all the boundaries in this region, and the directions of the boundaries are positive for this region.
(2) From the formula.
The left end of the above formula is twice the enclosed area, so there is
Theorem 1 If the region is a simply connected region and the sum of functions has a first-order continuous partial derivative, the necessary and sufficient conditions for the curve integral to be independent of road stiffness (or the curve integral along any closed curve is zero) are as follows.
Naiheng was established.
Theorem 2? Let the region be a simply connected region, and the sum of functions has a first-order continuous partial derivative, then the necessary and sufficient conditions for curve integration to be a total differential of functions are as follows.
Naiheng was established.
Inference? If the region is simply connected and the sum of functions has the first-order continuous partial derivative, then the necessary and sufficient condition that curve integral has nothing to do with road stiffness is that there is a function in it, so that
The function is bounded on the surface, and it will be divided into small pieces. Let the area of the smallest block be any point on the surface, and the bending area of the function on the surface is divided into
Where is the projection on the surface?
Let it be a smooth directed surface, the function is bounded at the top, and it can be arbitrarily divided into fast small surfaces (also representing the area of the fastest small surface), and the projection on the surface is any point on the top, then the function can divide the coordinates and the surface area on the directed surface into
Similarly, a function can be defined to divide the coordinates and the bending area on the directed surface into
This function divides the bending region of the coordinate sum on the directed surface into
Let the integral surface be given by the equation, then there is
When the integral surface is taken as the upper side of (i.e.), it is taken as positive, otherwise it is taken as negative.
Where is the direction cosine of the normal vector of the directed surface at that point.
Write in vector form
Where is the unit normal vector of the directed surface at that point, which is called directed bin.
Let the closed region of the space be surrounded by piecewise smooth closed surfaces, and if the functions,, and have first-order continuous partial derivatives, then there are.
or
Where is the outside of the whole boundary surface and is the direction cosine of the normal vector of the point.
Let a piecewise smooth spatial directed closed curve be a piecewise smooth directed surface with boundaries, where the positive direction and the right-hand side conform to the right-hand rule, functions, and have a first-order continuous partial derivative (together with the boundary) on the surface, then there are
It can also be written as
Using the relationship between two kinds of surface integrals, it can also be written as
Where is the unit normal vector of the directed surface at that point.