Graphing the symmetry of surface integrals

1. Diagram the symmetry of surface integrals

1.1 General symmetries of surface integrals of the first kind

The principles of double integrals, triple integrals, curve integrals of the first kind, and general symmetries of surface integrals of the first kind are similar. The principles of planes
and space surfaces are the same. The following content takes space surfaces as an example.
The figure shows the integration area $\Sigma$，函数$f(x,y,z)$ represents point$(x,y,The\; density\; at\; z)$ can be represented by color depth, but drawing is too cumbersome, so the integrand is not visualized. The

integrated area space surface$\Sigma About$ xx_even function of$x$$yoz$ plane symmetry)

Integration region space surface $\Sigma About$ yy_$Even\; function\; of\; y$ (i.e. xoz with respect to$xoz$ plane symmetry)

integration area space surface$\Sigma About$ zz_$Even\; function\; of\; z$ (i.e. xoy with respect to$xoy$ plane symmetry)

1.2 Rotation symmetry of surface integrals of the first kind

Rotation symmetry means that the integration region $The\; expression\; of\; \Sigma$ is in$The\; form\; remains\; unchanged\; after\; x,yandz$ are interchanged, that is, the integral has nothing to do with the integral variable.
Example:
Assume the surface$\Sigma$：$|x|+|y|+|z|=1$，求$\underset{S}{\iint }(x+|y|)dS$
surface$\Sigma About$ xoz$xoz$ plane symmetry, that is, about$x$ is an even function, and the integrand$x+|y|About$ xx_$x$ is an odd function, so$\underset{S}{\iint }xdS=0$
$$\underset{S}{\iint }(x+|y|)dS=\underset{S}{\iint }|y|dS$$
changex , yx, yThe expression after exchanging$x$$and$$y$$|y|+|x|+|z|=1$ , the expression remains unchanged,
variablesy, zy, zThe expression after swapping$y$$and$$z$$|x|+|z|+|y|=1$ , the expression remains unchanged,
variablesx, zx, zThe expression after exchanging$x$$and$$z$$|z|+|y|+|x|=1$ , the expression remains unchanged.
After verification, the expression in the integral area has rotation symmetry, thenyyReplace$y$$xwazz$ _$z$ ∣
$$\begin{gathered} \underset{S}{\iint }|y|dS=\underset{S}{\iint }|x|dS=\underset{S}{\iint }|z|dS \\ \underset{S}{\iint }|y|dS=\frac{1}{3}(\underset{S}{\iint }|y|dS+\underset{S}{\iint }|x|dS+\underset{S}{\iint }|z|dS) \\ \underset{S}{\iint }|y|dS=\frac{1}{3}(\underset{S}{\iint }|y|+|x|+|z|dS) \\ \underset{S}{\iint }|y|dS=\frac{1}{3}\underset{S}{\iint }dS=\frac{1}{3}\cdot 8\cdot \frac{\sqrt{3}}{4}(\sqrt{2}{)}^2=\frac{4\sqrt{3}}{3} \end{gathered}$$
The picture below shows the surface $\Sigma$ (consisting of 8 sides of length$\sqrt{2}$composed of equilateral triangles), $\underset{S}{\iint }dS$ represents the surface area

1.3 General symmetries of surface integrals of the second kind

Integration region space surface $\Sigma About$ xx_even function of$x$$yoz$ plane symmetry)
is to highlight the various directions after vector decomposition to avoid confusion with the surface. The surface$S$

Integration region space surface $\Sigma About$ yy_$Even\; function\; of\; y$ (i.e. xoz with respect to$xoz$ plane symmetry)
is to highlight the various directions after vector decomposition to avoid confusion with the curved surface. The curved surface$The\; \Sigma$ principle is similar to the above, and the plot integration area space surface Σ \Sigma
is no longer performed.

$\Sigma About$ zz_$Even\; function\; of\; z$ (i.e. xoy with respect to$xoy$ plane symmetry)
is to highlight the various directions after vector decomposition to avoid confusion with the surface. The surface$The\; \Sigma$
principle is similar to the above, and no drawing is required.