The equation of curve C is x + Y-1 = 2

(1) When. X > = 0, Y > = 1, the curve is x + Y-1 = 2, x + y = 3
It is a line segment between points (0,3) and (2,1) on the line x + y = 3
(2).x>=0,y

The area of the area enclosed by the curve expressed by the equation | XY | + 1 = | x | + | y |, is

First of all, after square the two sides of the original formula, we get x ^ 2 * y ^ 2 + 1 = x ^ 2 + y ^ 2. For the first time, we get x ^ 2 * (y ^ 2-1) = y ^ 2-1, x ^ 2 = 1 according to x ^ 2. For the second time, we get y ^ 2 * (x ^ 2-1) = x ^ 2-1, y ^ 2 = 1 according to y ^ 2

Cube of x-square of x-square of y-xy

x³-x²y-xy²
=x(x²-xy-y²)

Questions about Green's formula
Recently, I learned the second kind of curve integral and Green's formula. I'm puzzled. Is the partial derivative in Green's formula equivalent to the horizontal component integral of X for coordinate integral and X integral?
I'm sorry, maybe I'm wrong. I mean, what's the role of the partial derivative in Green's formula? What's the meaning? And the partial derivative of x minus the partial derivative of Y, what's the meaning here?
Do I mean to integrate the horizontal component of a function?

I think you'd better take a look at the overturning process of Green's formula In fact, the process of pushing down in the textbook uses the patchwork method, and the partial derivative is used to obtain the original function when integrating X and Y respectively Only when both sides of the equation are equal, this trial is green's formula, and the integral of the two is the component integral respectively. Through his physical meaning, it is well understood that the work done by the variable force is the sum of the work done by the component forces, and so is green's formula

Questions about the concept of "Green's formula"
There are two functions P (x, y) and Q (x, y) in Green's formula, and what does the difference between the partial derivative of Q to X and the partial derivative of P to Y represent,

Personal understanding is that they represent two different functions ~ the difference between partial derivatives is just the formula used, which has no special significance

Let the equation of Cartesian leaf line be x ^ 3 + y ^ 3 = 3axy (a > 0), and find the area of the figure enclosed by it

The parameter equation of Cartesian leaf line is
x=3at/(1+t^3),y=3at^2/(1+t^3)(t=tanθ)
The area of the figure enclosed by it is in the first quadrant, namely 0

Cartesian cardioid formula

Polar expression:
Horizontal direction: r = a (1-cos θ) or r = a (1 + cos θ) (a > 0)
Or vertical direction: r = a (1-sin θ) or r = a (1 + sin θ) (a > 0)
The expressions of plane rectangular coordinates are as follows:
X ^ 2 + y ^ 2 + A * x = a * sqrt (x ^ 2 + y ^ 2) and
x^2+y^2-a*x=a*sqrt(x^2+y^2

Why is it that for a curve integral of the second kind, if we use Green's formula to do it equal to 0, but directly use the parameter to solve the curve integral, we get 2 π (PIE), for example
(XDY YDX) / (the square of X + the square of Y) the second kind of curve integral of L on (with R as the radius and the origin as the center of the circle). If Green's formula is used to convert it into double integral, then the result is 0!

What is green's formula used to calculate?

Newton Leibniz formula, the most basic formula in Calculus of one variable, shows that the definite integral of a function in an interval can be expressed by the values of the original function at the two 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. This is green's formula
1. The concept of simply connected region
Let d be a planar region. If any region enclosed by a closed curve in D belongs to D, then D is called a planar simply connected region; otherwise, it is called a complex connected region. Generally speaking, a simply connected region is a region without "holes" (including "point holes") and "cracks"
2. Positive regulation of the boundary curve of the region
Let it be the boundary curve of a plane region. The prescribed positive direction is: when the observer walks in this direction, the part near him is always on his left side. In short, the positive direction of the boundary curve of the region should be suitable for the condition. When the observer walks along the curve, the region is on his left side
3. Green's formula
[theorem] let a closed region be bounded by piecewise smooth curves and a function and have first order continuous partial derivatives on it, then there is (1) ∮ CP (x, y) DX + Q (x, y), y) Dy = ∫∫ D (DQ / DX DP / dy) DXDY, where is the positive boundary curve. Formula (1) is called Green's formula. [proof] it is easy to see that the shape of the region is assumed as follows (a line parallel to the axis passes through the region, and the intersection point with the boundary curve of the region is at most two points). The region shown in Figure 2 is a special case of the region shown in Figure 1, On the other hand, according to the difference between the curvilinear integral property and the calculation method of coordinates, it is assumed that the intersection point of the straight line passing through the interior of the region and parallel to the axis and the boundary curve of the region is at most two points. By using a similar method, it can be proved that the boundary curve of the region and the boundary curve passing through the interior and parallel to the axis are synthesized（ We have green's formula when the point of intersection of any straight line (axis or axis) is at most two points
Note: if the region does not meet the above conditions, that is, when the intersection of the line passing through the region and parallel to the coordinate axis and the boundary curve exceeds two points, one or more auxiliary curves can be introduced into the region to divide it into several parts, so that each part is suitable for the above conditions, Green's formula can still be proved to be true. Green's formula communicates the relationship between double integral and curvilinear integral of coordinates, so it is widely used
The condition that plane curve integral has nothing to do with path
1. The definition of path independent curve integral of coordinates
[Definition 1] let a function be an open region with first order continuous partial derivative. If the equality holds for any two points in the region and any two curves from point to point in the region, the curve integral is said to be path independent; otherwise, it is said to be path dependent. Definition 1 can also be changed into the following equivalent statement: if the curve integral is path independent, the curve integral is said to be path independent, On the other hand, if the curve integral along any closed curve in the region is zero, it can be easily derived that the curve integral is path independent. [definition 2] curve integral is path independent, which means that for any closed curve in the region, there is always a path independent
2. The condition that curve integral is independent of path
[theorem] let an open domain be a simply connected domain and a function have a first order continuous partial derivative in it, then the necessary and sufficient condition for the path independence of the integral of the inner curve is that the equation holds for the inner constant. It is proved that the sufficiency of the proof is that if any closed curve in the inner domain is simply connected, the region bounded by the closed curve is all included, so it holds for the upper constant, According to definition 2, the inner curve integral has nothing to do with the path, This is in contradiction with the condition that the integral of a curve on any inner closed curve is zero. Therefore, the inner equation should always hold. Note: one of the two conditions required by the theorem is missing, Where is a piecewise smooth curve surrounding the origin, and the positive direction is counterclockwise. Here, except for the origin, there is a continuous circle in the enclosed area, and. Inside, make a circle with sufficiently small radius, and use Green's formula in the complex connected area surrounded by and
Total differential quadrature of functions of two variables
If the curve integral is independent of the path in the open region, it is only related to the coordinates of the starting point and the ending point of the curve. Suppose that the starting point of the curve is, and the ending point is, it can be represented by a sign or, and there is no need to explicitly write the integral path, We have the following important theorem [theorem 1] let a function be a simply connected open domain with a continuous partial derivative of first order, and then Green's formula
It is a single valued function, here is an internal fixed point, and that is to say, according to the conditions, for any curve with a point as the starting point and a point as the ending point, the curve integral 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 a point, Let's take the following path, which can be proved similarly. So [theorem 2] let's be a simply connected open domain with a continuous partial derivative of the first order on it, then the necessary and sufficient condition for the total differential of a certain function is that the inner constant holds, Then the mixed partial derivative of the second order is suitable for the equation, so that [Theorem 3] let be a simply connected open domain, where a function has a continuous partial derivative of the first order. If there is a function of two variables such that, where, is any two points in, The function is suitable for then or therefore (a constant), that is, a closed curve is formed by a point returning to a point along an arbitrary inner path. Therefore, because the curve integral at the right end has nothing to do with the path, in order to simplify the calculation, A broken line connected by a straight line segment parallel to the coordinate axis can be taken as the integral path (of course, the broken line should belong to the simply connected region). - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -, The region shown in Figure 2 is a special case of the region shown in Figure 1. We can only prove the region shown in Figure 1. On the other hand, according to the difference between the curvilinear integral property of coordinates and the calculation method, it is assumed that the intersection point of the straight line passing through the interior of the region and parallel to the axis and the boundary curve of is at most two points, By using a similar method, we can prove that if the intersection of the boundary curve of the region and any line passing through the interior and parallel to the coordinate axis (axis or axis) is at most two points, we have and hold at the same time, That is to say, when the intersection point of the line passing through the interior of 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 to divide it into several parts, so that each part of the region is suitable for the above conditions, and the Green's formula can still be proved

On the conditions of using Green's formula
In definition, it means that its integral curve is along the positive direction of D, and D is a simply connected region. In some problems, D contains the origin, which makes the partial derivation meaningless. After D digs out the origin, D1 can apply green's formula. When is D1 not a simply connected region? How can it be applied?

A simply connected region is one in which the part surrounded by any closed curve in D belongs to

The equation of curve C is x + Y-1 = 2