The integral of a function defined on a surface with respect to that surface. The physical meaning of the first kind of curve integral comes from calculating the quality of space surface with a given density function.

2. The second kind of surface integral is about the surface integral projected on the coordinate plane, and its physical background is the calculation of flow.

The second kind of curve integral is related to the integration path, the second kind of surface integral also depends on the direction of the surface, and the second kind of surface integral is related to the edge of the surface. If you change the edge of the surface (that is, the normal vector changes from pointing to one side to pointing to the other), it is obvious that the sign of the surface integral will change. Note that the tag above does not specify which side.

It must be pointed out that the second kind of surface integral has some properties similar to the second kind of curve integral.

Mathematically, symmetry is expressed by group theory. Groups correspond to Galileo group, Lorenz group and U( 1) group respectively. The cases where symmetric groups are continuous groups and discrete groups are called continuous symmetry and discrete symmetry respectively. German mathematician Herman Weil was the first to apply this mathematical method to physics and realized the importance of gauge symmetry.

4. Integer rotational symmetry refers to the rotational symmetry of coordinates. Simply put, the coordinate axis is renamed. If the function expression of the integral interval is unchanged, the integral value will remain unchanged after the x, y, z, y and z in the integrand function are also changed.

Extended data:

1, symmetric operation:

When a molecule has a center of symmetry, extending a second line from any atom in the molecule to the center of symmetry, you can find another same atom on the other side equidistant from the center of symmetry, that is, every point is symmetrical about the center. Symmetry operation according to the center of symmetry is inversion operation, and it is inversion according to the center of symmetry, which is denoted as I; In=E when n is even, and in=i when n is odd.

Anti-axis:

The basic operation of the anti-axis In is to rotate 360/n around the axis, and then the inversion is carried out according to the center point on the axis, followed by the joint operation of C 1n and I: I1n = IC1n; Rotate 360/n around the In axis, and then reverse according to the center.

Mirror axis:

The basic operation of the reflection axis Sn is to rotate 360/n around the axis, and then reflect according to the plane perpendicular to the axis, followed by the joint operation of C 1n and σ: s1n = σ c1n; Rotate 360/n around the Sn axis, and then reflect according to the plane perpendicular to the axis.

2. The difference between the first kind of surface integral and the second kind of surface integral.

1, the first one has no direction and has geometric and physical significance; The second category has direction and only physical meaning.

2. One curve is the length of the curve, and the other is the x and y coordinates. For example, if the linear density of a line is known, then the quality of this line must be calculated by one method. If the path curve equation is known and the forces in X and Y directions are informed, the second kind of curve can also separate X and Y, and there is a cosine ratio between the integrals of the first and second kinds of curves.

There are differences between two kinds of surface integrals, one is area integral and the other is coordinate integral. If the surface density is known, one is used to calculate the surface quality. If the velocity and surface equations in the x, y and z directions are known, the second method is used to calculate the flow. Similarly, the directions of x, y and z can also be separated.

References:

Baidu encyclopedia-surface integral of the first kind

References:

Baidu encyclopedia-second surface integral

References:

Baidu encyclopedia-symmetry

References:

Encyclopedia-100 pairs of integer rotation symmetry