Special topic I of mathematical physics of the sprint strong foundation program

1. Strong base program mathematical and physical simulation volume

I have worked out a set of mathematical and physical simulation test questions for the strong foundation plan , It is suitable for college entrance examination students with a certain foundation . Due to my limited level , If there are omissions, please criticize and correct them ！

2. Special topic on Gaussian function

Corresponding to The first question in the simulation volume . solve equations ：

Transposition to ：

By the properties of Gaussian function ：
Therefore, comprehensive results can be obtained ：

Solution of inequality ： or

Let's discuss in details ：

1. When when , , At this point, the original equation is , Solution . combination The scope of the , take .
2. When when , , At this point, the original equation is , Solution . combination The scope of the , take .
3. When when , , At this point, we substitute the original equation to get establish . Sum up , or or .

3. Topics on plural numbers and unit roots

Corresponding to The second question of the first question in the simulation volume .

It is known that , And .

It's easy to know It's the equation All roots on a complex field . So the equation It can be written again ：

The coefficient of contrast is not difficult to get ：

therefore ,, Then there are

4. Ellipse area and coordinate system transformation

4.1 Scaling transformation

Corresponding to The first question of the second question in the simulation volume .

Remember ellipse （ Or circle ） The area of is

Method 1 ：

From symmetry ：

Method 2 ：

Use coordinate system transformation ：

Then the ellipse （ Or circle ） Can be reduced to ：

Let the changed area be , There is a corresponding relationship ：

So we can get the ellipse （ Or circle ） The area of is

4.2 Rotation transformation

Corresponding to The second question in the simulation volume .

Any point in the coordinate system Rotate counterclockwise around the coordinate origin The coordinates after are ：

The trigonometric function formula can be expanded into ：

From a matrix perspective , Can be written as ：

In rectangular coordinates , Suppose the coordinates before the transformation are (x,y), Rotate around the coordinate origin The coordinates after are marked as .

Then there are ：

First, the conic Into ：

Recording matrix

Then the matrix It can be diagonalized ：

Then there are ：

Make ：

Then the left side of the equation can be reduced to ：

On the right side of the equation is ：

And the ellipse equation ：

So there is ：

remember It's an ellipse

Thus we can see that , The ellipse Counter clockwise rotation （ Rotate clockwise ） Then we can get ellipse

from The first question of the second question in the simulation volume The resulting ellipse （ Or circle ） The area formula shows , The ellipse The area of is

Because of the rotation invariance of the area , Conic curve can be obtained The area of is also .

To be continued