如何获得映射等式斜率的矢量场 [英] How to get a vector field that maps the slopes of an equation

问题描述

嘿，所以我正在阅读克里斯·赫克尔(Chris Hecker)撰写的这篇文章，其中他有一个抛物线的图像，被抛物线的向量场包围:

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:

但是，他从未提及他如何精确地获得矢量场方程，甚至从未陈述过.他确实说他通过绘制斜率方程dy/dx = 2x的解作为网格上每个坐标的短矢量来覆盖图1中斜率的矢量场.

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.

如何使用
的矢量场语法创建方程斜率的矢量场 V = xi + yj

推荐答案

如果图形标题显示为:

• 一般情况y = x^2 + C的曲线y = x^2和矢量场dy/dx = 2x

• The curve y = x^2, and the vector field dy/dx = 2x for the general case y = x^2 + C

上图中有三个方程在工作:

1. y = x^2-绘制的抛物线方程-这是一条长实线
2. y = x^2 + C-适用于矢量场的所有抛物线的方程-C是一个常数. 这是适合该矢量场的所有抛物线的方程式
3. dy/dx = 2x斜率场的方程式. -这是两者的斜率或 导数
所有常数C s 绘制的曲线以及所有可用y = x^2 + C绘制的可能曲线.

1. y = x^2 - The equation for the parabola drawn - This is the one long solid curve
2. 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
3. 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.

请注意，C是常数，因为y = x^2 + C与任何C的导数是2x.因此，矢量场显示了如何绘制具有不同C的所有不同抛物线.

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.

因此，有两种计算向量场的方法:

So there are two ways to calculate the vector field:

1. 在所需的x和y范围内进行迭代，并在每个点上独立于y来计算斜率dy/dx-2x.这就是作者的做法.
2. 通过在所需范围内逐渐改变y = x^2 + C中的C来绘制一串抛物线-假设-x计算y.

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. 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.

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