Graphical symmetry of curve integrals

1. Illustrating the symmetry of curve integrals

1.1 General symmetries of curve 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
plane curves and space curves are the same. The following content takes space curves as an example.

The figure shows the integration region $\Gamma$，函数$f(x,y,z)$ represents point$(x,y,The\; density\; at\; z)$ can be expressed by color depth, but drawing is too cumbersome, so the integrand is not visualized. The

integral area space curve$\Gamma About$ xx_even function of$x$$yoz$ plane symmetry)

integral area space curve$\Gamma About$ yy_$Even\; function\; of\; y$ (i.e. xoz with respect to$xoz$ plane symmetry)

integral area space curve$\Gamma About$ zz_$Even\; function\; of\; z$ (i.e. xoy with respect to$xoy$ plane symmetry)

1.2 Rotation symmetry of curve integrals of the first kind

Rotation symmetry means that the integration region LLThe expression of$L$ is in x , y , zx , y , zThe form remains unchanged after$x$$,$$y$$and$$z$
are interchanged, that is, the integral has nothing to do with the integral variable. Example:
$$L:\begin{array}{c}\{\begin{array}{ll}{x}^2+y^2+z^2& =a^2\\ x+y+z& =0\end{array}\end{array}$$
求${\oint }_L{x}^{2}ds$、${\oint }_L$
The intersection of the$x$$d$$s$$L$

changex, yx, yAfter exchanging$x$$and$$y$$y^2+x^2+z^2=a^2$ , the expression remains unchanged,
variablesy, zy, zThe expression after swapping$y$$and$$z$$x^2+z^2+y^2=a^2$ , the expression remains unchanged,
variablesx, zx, zThe expression after exchanging$x$$and$$z$$z^2+y^2+x^2=a^2$ , the expression remains unchanged.xx
in the integrand willReplace$x$$ywazz$ _$z$后积分大小不变
$$\begin{gathered} {\oint }_L{x}^{2d}s\_={\oint }_L{y}^{2d}s\_={\oint }_L{z}^{2d}s\_ \\ {\oint }_L{x}^{2d}s\_=\frac{1}{3}({\oint }_L{x}^{2d}s\_+{\oint }_L{y}^{2d}s\_+{\oint }_L{z}^{2ds})\_\_ \\ {\oint }_L{x}^{2d}s\_=\frac{1}{3}({\oint }_L{x}^2+y^2+z^{2ds})\_\_ \\ {\oint }_L{x}^{2d}s\_=\frac{a^2}{3}\left({\oint }_{L}ds\right)=\frac{a^2}{3}\cdot 2\pi a=\frac{2}{3}p{a}^3 \end{gathered}$$
Change amount x, yx, yThe expression after exchanging$x$$and$$y$$y+x+z=0$ , the expression remains unchanged.
Variablesy, zy, zThe expression after swapping$y$$and$$z$$x+z+y=0$ , the expression remains unchanged.
Variables$The\; expression\; after\; exchanging\; xandz$ is:$z+y+x=0$ , the expression remains unchanged.xx
in the integrand will beReplace$x$$ywazz$ _$z$后积分大小不变
$$\begin{gathered} {\oint }_{L}xds={\oint }_{L}yds={\oint }_{L}zds \\ {\oint }_{L}xds=\frac{1}{3}({\oint }_{L}xds+{\oint }_{L}yds+{\oint }_{L}zds) \\ {\oint }_{L}xds=\frac{1}{3}({\oint }_{L}x+y+zds)=\frac{1}{3}\left({\oint }_{L}0ds\right)=0 \end{gathered}$$

1.3 General symmetry of curve integrals of the second kind

When the integration area is a plane curve
Plane curve $LAbout$ yy_$y-$ axis symmetry

Plane curve $LAbout$ xx_$x-$ axis symmetry

When the integration area is a space curve
Space curve $\Gamma is$ aboutxxeven function of$x$$yoz$ plane symmetry)

Space curve $\Gamma$ is about$Even\; function\; of\; y$ (i.e. xoz with respect to$xoz$ plane symmetry)

Space curve $\Gamma is$ aboutzz$Even\; function\; of\; z$ (i.e. xoy with respect to$xoy$ plane symmetry)