1: Because it is directional, it is.

The nature of "even zero odd degree" is contrary to the general situation.

When F(x) is an even function, if σ is symmetric about the corresponding surface, one part takes+and the other part takes-

The result is F(x)-F(- x) = F(x)-F(x) = 0, and the two parts cancel each other out.

F(x) odd function, in the same case, one part takes+the other part takes-

The result is F(x)-F(- x) = F(x)+F(x) = 2F(x), and the integrals of the two parts are equal and can be superimposed.

2. Three-in-one formula

Because the form of σ is z = z(x, y).

Normal vector n = {-z 'x, -z'y, 1}

∫∫ _ (σ) pdydz+qdzdx+rdxdy

=∫∫_(D){ P(-z ' x)+Q(-z ' y)+ 1 } dxdy

Take the+sign when taking the upper/right/front side.

Please mark the removal/left/rear side.

3. Gauss formula

∫∫_(σ)pdy dz+Qdzdx+Rdxdy

= ∫∫∫_(Ω) (? P/？ x+？ Q/？ y+？ R/？ z) dxdydz

-∫ _ (σ and) Pdydz+Qdzdx+Rdxdy

In the latter part (σ and σ), if the given surface can't be enclosed in a closed space, the Gaussian formula can't be used directly. You need to add several surfaces to close the region. For example, you can use Gaussian formula by adding several surfaces (σ and σ). Finally, you should pay attention to reducing the corresponding integrals of those surfaces (σ and σ).

Step 4: Dig a hole

If the integrand on σ has singular points, the Gaussian formula cannot be used directly.

It is necessary to fill a small space r=ε, which is enough to contain all the internal singularities, and then the radius ε tends to 0.

When using Gaussian formula, this part of the corresponding integral should also be subtracted.

So there is ∫∫ _ (σ) = ∫∫ _ (ω)-∫ _ (ε).

5: Substitution

If the equation of the integrand function f is on σ, the σ equation can be substituted into F first.

For example, give the σ equation: x? +y？ +z？ =a？

Then ∫∫ _ (σ) (pdydz+qdzdx+rdxdy)/√ (x? +y？ +z？ )

=∫∫_(σ)(pdy dz+qddx+Rdxdy)/a

=( 1/a)∫∫_(σ)pdy dz+Qdzdx+Rdxdy

So this can avoid the situation of 4: 0, and there is no need to dig a hole.

After removing the singularity, we can continue to use Gaussian formula to patch the surface.

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