In electrostatic field, the flux of electric field intensity passing through any closed surface is only related to the algebraic sum of closed surface charges, which is equal to the algebraic sum of closed surface charges divided by the dielectric constant in vacuum. This law shows the relationship between the charge distribution on a closed surface and the generated electric field. The integral of a function or vector-valued function defined on a surface with respect to the surface.
Surface integral is generally divided into the first kind of surface integral and the second kind of surface integral. The physical meaning of the first kind of surface integral comes from calculating the quality of spatial surface with a given density function. The physical meaning of the second kind of surface integral comes from calculating the total flow through the surface per unit time for a given spatial surface and fluid velocity.
The electric flux passing through any closed surface (called Gaussian surface) S in electrostatic field is equal to the algebraic sum of all charges in the closed surface divided by the dielectric constant in vacuum, regardless of out-of-plane charges. The physical meaning of the first kind of surface integral comes from calculating the quality of spatial surface with a given density function.
The difference between surface integral and double integral;
The integral area of double integral is a two-dimensional plane, and the integral area of the first kind of surface integral is a three-dimensional surface. The second kind of surface integral plus direction. This causes the calculation of the first kind of curve integral to be transformed into the calculation of definite integral, while the calculation of the first kind of surface integral to be transformed into the calculation of double integral.
The first kind has no direction, the second kind of curve integral and the second kind of surface integral introduce direction. With the direction, the calculation of hard steel will be more complicated, so we introduce the universal Green's formula to the second kind of integral and transform the second kind of curve integral into double integral calculation. Gauss formula is to convert the second kind of surface integral into triple integral calculation.
Surface integral, as the name implies, the integral on the surface, regardless of the first and second types, is the integral done on the surface. You straighten this surface (mathematically, by making appropriate parameter transformation and expressing it in an appropriate parameter form), it becomes a "straight" space (that is, it becomes a regular form), and finally it can be converted into a multiple integral for calculation.