如何得到映射方程斜率的向量场

时间：2021-03-04 04:23:16

Hey so I'm reading this article by Chris Hecker where he has an image of a Parabola surrounded by the a vector field of it's derivative:

我正在读克里斯·赫克的这篇文章他有一个抛物线的图像被它的导数的向量场包围着

如何得到映射方程斜率的向量场

However he never mentions how exactly he got the vector field equation, and never even states it. He does say he overlayed the vector field of the slopes in Figure 1, by drawing the solution to the slope equation, dy/dx = 2x, as a short vector at each coordinate on the grid.

但是他从来没有提到他是如何得到向量场方程的，甚至从来没有说过。他确实说过，他在图1中画出了斜率方程的解，dy/dx = 2x，作为网格上每个坐标的一个短向量。

How do you create a vector field of the slopes of an equation in the vector field syntax of
V = xi + yj

如何在V = xi + yj的向量场语法中创建方程斜率的向量场

2 个解决方案

#1

2  

The Figure title would be clearer if it read:

如果数字标题为:

  • The curve y = x^2, and the vector field dy/dx = 2x for the general case y = x^2 + C
  • 曲线y = x ^ 2,向量场dy / dx = 2 x为一般情况下y = x ^ 2 + C

There are three equations at work in the graph above:

上图中有三个方程:

  1. y = x^2 - The equation for the parabola drawn - This is the one long solid curve
  2. y = x ^ 2 -抛物线的方程所吸引——这是一个坚实的曲线
  3. y = x^2 + C -The equation for all parabolas that fit on the vector field - C is a constant. This is the equation for all parabolas that fit on that vector field
  4. y = x ^ 2 + C——所有抛物线方程适用于向量场- C是一个常数。这是适用于这个向量场的所有抛物线的方程
  5. dy/dx = 2x The equation for the slope field. - This is the slope or derivative of the both the curve drawn and all the possible curves that can be drawn with y = x^2 + C for all constant Cs.
  6. dy/dx = 2x斜率场的方程。——这是斜率或导数曲线绘制和所有可能的曲线可以得出,y = x ^ 2 +所有常数C Cs。

Note that C is a constant, since the derivative of y = x^2 + C with any C is 2x. So the vector field shows how to draw all the different parabolas with different Cs.

注意,C是一个常数,因为y = x ^ 2 + C的导数与任何C 2 x。所以向量场展示了如何用不同的Cs表示所有不同的抛物线。

So there are two ways to calculate the vector field:

所以有两种方法来计算向量场:

  1. Iterate over your desired range of x and y and calculate the slope, dy/dx- 2x independent of y in this case - at each point. This is how the author did it.
  2. 遍历x和y的期望范围，计算每个点的斜率，dy/dx- 2x与y无关。作者就是这样做的。
  3. Draw a bunch of parabolas by slowly varying C in y = x^2 + C over a desired range of - let's say - x calculating y.
  4. 画一堆抛物线的慢变在y = x ^ 2 + C / C所需的范围——比方说——x计算y。

#2

1  

For a differential equation dy/dx = f(x,y) (e.g., dy/dx = 2x in this case, with f(x,y) = 2x), the vector field (F) will be F = i + f(x,y)j (so in your case, F = i + 2x j )

对于微分方程dy/dx = f(x,y)(例如，dy/dx = 2x, f(x,y) = 2x)，向量场f = i + f(x,y)j(在这里，f = i + 2x j)

#1

2  

The Figure title would be clearer if it read:

如果数字标题为:

  • The curve y = x^2, and the vector field dy/dx = 2x for the general case y = x^2 + C
  • 曲线y = x ^ 2,向量场dy / dx = 2 x为一般情况下y = x ^ 2 + C

There are three equations at work in the graph above:

上图中有三个方程:

  1. y = x^2 - The equation for the parabola drawn - This is the one long solid curve
  2. y = x ^ 2 -抛物线的方程所吸引——这是一个坚实的曲线
  3. y = x^2 + C -The equation for all parabolas that fit on the vector field - C is a constant. This is the equation for all parabolas that fit on that vector field
  4. y = x ^ 2 + C——所有抛物线方程适用于向量场- C是一个常数。这是适用于这个向量场的所有抛物线的方程
  5. dy/dx = 2x The equation for the slope field. - This is the slope or derivative of the both the curve drawn and all the possible curves that can be drawn with y = x^2 + C for all constant Cs.
  6. dy/dx = 2x斜率场的方程。——这是斜率或导数曲线绘制和所有可能的曲线可以得出,y = x ^ 2 +所有常数C Cs。

Note that C is a constant, since the derivative of y = x^2 + C with any C is 2x. So the vector field shows how to draw all the different parabolas with different Cs.

注意,C是一个常数,因为y = x ^ 2 + C的导数与任何C 2 x。所以向量场展示了如何用不同的Cs表示所有不同的抛物线。

So there are two ways to calculate the vector field:

所以有两种方法来计算向量场:

  1. Iterate over your desired range of x and y and calculate the slope, dy/dx- 2x independent of y in this case - at each point. This is how the author did it.
  2. 遍历x和y的期望范围，计算每个点的斜率，dy/dx- 2x与y无关。作者就是这样做的。
  3. Draw a bunch of parabolas by slowly varying C in y = x^2 + C over a desired range of - let's say - x calculating y.
  4. 画一堆抛物线的慢变在y = x ^ 2 + C / C所需的范围——比方说——x计算y。

#2

1  

For a differential equation dy/dx = f(x,y) (e.g., dy/dx = 2x in this case, with f(x,y) = 2x), the vector field (F) will be F = i + f(x,y)j (so in your case, F = i + 2x j )

对于微分方程dy/dx = f(x,y)(例如，dy/dx = 2x, f(x,y) = 2x)，向量场f = i + f(x,y)j(在这里，f = i + 2x j)

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