Dimension problems and contour lines

1. Ask questions

Here I list three formulas , Let's take a look at ：

y=x Is a unary function ,z=y+x It's a function of two variables ,f=z+y+x It's a ternary function . So one dollar is one dimension , Duality is two-dimensional , Ternary is three-dimensional ？

In fact, it can be understood like this , Then if you understand us so , There is one dimension we are wrong , What's wrong ？

One dimension is wrong , Obviously , We drew it directly when drawing y=x In a straight line , Straight lines are two-dimensional . One dimension can only represent points , There can be no wire , Then why did we draw a line here ？

Therefore, it is predicted for us , Predicted y=x What happened next .

Compare with two dimensions , There is no prediction in two dimensions , It makes a line calmly , It doesn't predict how it will change next .

Compare with 3D , Three dimensional is also normal , It is placed there as a surface , I didn't predict what would happen next .

2. Line 、 Mathematical expression of surface

Now let's give the mathematical expression of line and surface , This helps us understand the contour lines behind .

2.1 Mathematical expression of line

For lines , stay xoy In plane , There are two ways of expression in Mathematics ：

• The first one is ：
• The second kind ：

It's actually quite understandable , In the first expression y Move to and x One side is the second form .

2.2 Mathematical expression of surface

Through the extension of mathematical expression of lines , stay xyz In the space , You can write the mathematical expression of the surface ：

• The first one is ：
• The second kind ：

3. contour

Now suppose you have a surface ：, We use a parallel to xoy The plane of the To intercept

, The resulting curve is the contour , Its mathematical expression is as follows ：

 

Further, it can be written as ：

We will c By moving items, you can get ：

Here's a special explanation ： From the definition of contour , This curve is a “ floating ” Curve in midair , The height of floating is c, Like three-dimensional . From the mathematical definition , This is the same as the general mathematical expression of the straight line we talked about earlier It's the same in form , It's a two-dimensional curve . These two statements are not contradictory , We can understand it from two perspectives ：

The first angle ： We will have a height of c The plane of is regarded as xoy Plane , So this curve is in this plane c The equation in is

The second angle ： We can also take contour lines Projection to xoy The plane of the , The process of projection is to reduce both sides at the same time c, The result is . This method is better understood , We draw contour lines by projection i On going .

in general , From a higher dimension , Contour lines are floating in the air . But from the mathematical dimension , It's two-dimensional , It's not three-dimensional .

Let me take a look at the relationship between the gradient of the surface and the tangent of the contour line .

about We can get ：

We write it in the form of vector multiplication ：

among It's a gradient , It's very understandable . What do you mean ？ In fact, it is related to the curve Get the tangent direction consistent .

If it's hard to understand , Think about the slope of a function of one variable , We will Turn into , Turn into , Then the slope of the curve at a certain point is ：, because They are all directional , therefore It's actually the direction of the slope , That is, the direction of the tangent .

because And Multiply is equal to 0, So there is ： The gradient direction of a point on the surface is perpendicular to the tangent of the contour line at that point .