The analytical expression of the straight line obtained by rotating the straight line y = - x + 2 90 ° anticlockwise around the origin o

The analytical expression of the straight line obtained by rotating the straight line y = - x + 2 90 ° anticlockwise around the origin o

The intersection of the line y = - x + 2 and the X axis is (2,0), and the intersection of the line and the Y axis is (0,2). The triangle formed by the line and the coordinate axis is an isosceles right triangle. The line rotates 90 degrees anticlockwise around the o point, which can be regarded as the rotation of the triangle. The analytical formula of the new line is y = x + 2

Find the curve xy = - 1 and rotate 45 ° counterclockwise around the origin of the coordinate to get the curve,

The image of inverse scale function is obtained by hyperbolic trajectory rotation, so this curve is hyperbolic. It's better to rotate the coordinate system clockwise for 45 degrees. The corresponding relationship between the new coordinate and the old coordinate is: X '= xcos45 + ysin45y' = ycos45-xsin45 simplification √ 2x '= x + y √ 2Y' = y-xy = √ 2 / 2 (x '+ y') x = √ 2 / 2 (x '- y')

The expression of the curve xy = 1 after rotating 45 ° clockwise around the origin

Change the XY coordinate system into the length and angle coordinate system, and then the angle minus pi / 4 is the new expression

It is known that the center of circle C is on the curve xy = 2 and passes through the coordinate origin o
It is known that the center of circle C is on the curve xy = 2, and it passes through the coordinate origin o, intersects with the straight line y = - 2x + 1 at two points a and B. when OA = AB, the equation of circle C is solved

The equation of a line passing through o perpendicular to the line y = - 2x + 1: y = 1 / 2x
Y = 1 / 2x and xy = 2 simultaneous x = - 2 x = 2 (rounding off) y = - 1 y = 1 (rounding off)
Center coordinates: (- 2, - 1) radius = √ 5
The equation of circle C: (x + 2) ^ 2 + (y + 1) ^ 2 = 5

The difference between the first type curve integral and the second type curve integral
It is not clear what these two types of curve integrals require

The second type of curvilinear integral is directional, and the results of AB and Ba are different, because their directions are different; while the first type of curvilinear integral is directional, and the second type can be transformed into the first type

What exactly do the first type curve integral and the second type curve integral seek
For example, the binary integral is the area, and the ternary integral is the volume

The first type of curvilinear integral and the second type of curvilinear integral seek the same thing,
It's all one-dimensional, so it's some one-dimensional correlations, if the mass, length, etc. of the line,
The above answer: after the transformation of Green's formula, it becomes two-dimensional, and then it is to calculate the area, which is the area surrounded by the curve

What is the relationship between definite integral, double integral, curve integral and Green's formula?
I always feel that these problems can solve each other, but I can't solve them. What's the relationship between them?

It's a good question. It's very helpful to understand this problem. The operation of integral involves two elements, namely integrand and integral region. According to the different integral region (shape, dimension, etc.), the integral is classified as those things

Double integral, we may use Green's formula or Gauss's formula
Let f (x, y) Let f (x, y) be a second-order differentiable in X (x, y) in X \\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\+ y × &; F / &; y) DXDY, the integral region is the unit circle

(x × &; F / &; X + y × &; F / &; y) in geometric sense, is the inner product of position vector (x, y) and gradient vector (&; F / &; X, &; F / &; y)
∂ & # 178; F / & # 8706; X & # 178; + & # 8706; & # 178; F / & # 8706; Y & # 178; is a Laplace operator. It looks strange, and the name is even more strange. In fact, it is the gradient of gradient, which is the fastest growing direction. In fact, it is a vector. Don't think about complexity, it's embarrassing
∫ (x × &; F / &; X + y × &; F / &; y) DXDY is actually summation. How many things are there to be added? There are so many elements in a unit circle. Then I will add a circle, a circle, from radius 1 to radius 0
For a small circle with radius r, the vector (x, y) is perpendicular to the circle, so ∫ (x × &; F / &; X + y × &; F / &; y) is essentially an integral of the circle along radius R. it represents the inner product of two vectors, one is the normal vector n = (x, y), the other is the gradient vector f = (&; F / &; X, &; F / &; y)
In this paper, we use the Gauss theorem, for any vector function g, G, ∫ = 8747theorem, for any vector function g, for any vector function g, ∫ = 8747874787478747\\\8747\\\\\\\\\\\\\\\dxdy, and then for each small circle of radius r, you add up all the results, which is ∫ ∫ re ^ (- x & # 178; -The DXDY, which is 8747\\\8747\87478747\\\8747\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\ ∫ (1 / E) DXDY = 3.14/2e

What is the condition of using Green's formula to solve double integral

The first partial derivative is a continuous function in the integral region bounded by the integral curve

The geometric meaning of the first type and the second type curve integral in the advanced mathematics class,

Geometric problems: length, area, volume, etc. have nothing to do with the direction of the curve