[advanced mathematics] from normal vector to surface integral of the second kind

From normal vector to surface integral of the second kind

One 、 introduction

When I see the formula of the second kind of curve integral , I'm confused about the sign , The textbook gives a conclusion ： When the included angle between the normal and the corresponding coordinate axis is an acute angle, it is taken as “+”, Otherwise, take the minus sign . However , Not knowing the root cause of its sign will make you uneasy , So I explored a little .

Facet projection method ：

$$\begin{array}{rl}{\iint }_S\mathbf{F}(x,y,z)\cdot d\mathbf{S}& ={\iint }_{D_{yz}}\pm P[x(y,z),y,z]dydz+{\iint }_{D_{xz}}\pm Q[x,y(x,z),z]dxdz+{\iint }_{D_{xy}}\pm R[x,y,z(x,y)]dxdy\end{array}$$

Oneness projection method ：

$$\begin{array}{rl}{\iint }_S\mathbf{F}(x,y,z)\cdot d\mathbf{S}& ={\iint }_S\mathbf{F}(x,y,z)\cdot {\mathbf{n}}_0(x,y,z)dS\\ & =\pm {\iint }_{D_{xy}}\mathbf{F}(x,y,z)\cdot (-\frac{\partial z}{\partial x},-\frac{\partial z}{\partial y},1)dxdy\end{array}$$

Two 、 Normal and tangent plane

1. The representation of a plane

First , Need to know , The explicit and implicit representation of a plane ：

• Implicit ： $F(x,y,z)=0$;

• Explicit ：$z=z(x,y)$.

actually , Explicit representation is a geometric parameterization , The inverse process is implicit , Can be expressed as ：$F(x,y,z)=z-z(x,y)=z(x,y)-z=0$

2. The normal vector

Now find any point on the plane $M(x_0,y_0,z_0)$

that , The normal vector at this point can be

• Expressed as a gradient ：$\nabla \mathbf{F}(x,y,z)$;

• Or the outer product of the partial derivative is expressed as ：$\frac{\partial \mathbf{X}}{\partial x}\times \frac{\partial \mathbf{X}}{\partial y}$ perhaps $\frac{\partial \mathbf{X}}{\partial y}\times \frac{\partial \mathbf{X}}{\partial x}$, there X Is the surface coordinates , For example, in three-dimensional space , Use z(x,y) Parameterize , So the surface coordinates are **(x,y,z(x,y))**.

For the normal vector of the plane represented by the display , Can be expressed as ：$\mathbf{n}(x,y)=(-\frac{\partial z}{\partial x},-\frac{\partial z}{\partial y},1)$ perhaps $\mathbf{n}(x,y)=(\frac{\partial z}{\partial x},\frac{\partial z}{\partial y},-1)$, At this point, the value of the normal vector is only the sum of x、y of

I have deliberately written the normal vector in two forms , This is because for a directed surface there are two sides . Now the two most critical sentences are ：
1. If the z(x,y) Parameterize , Its normal vector $\mathbf{n}(x,y)=(-\frac{\partial z}{\partial x},-\frac{\partial z}{\partial y},1)$ It must be pointing to God ,$\mathbf{n}(x,y)=(\frac{\partial z}{\partial x},\frac{\partial z}{\partial y},-1)$ It must point to The earth ;
1. If the z(x,y) Parameterize , Make sure that the function $z=z(x,y)$ Is a single valued function .

Now, if we look at a surface , It's a right hemisphere , So using **y(x,z)** Parameterize , It is more convenient , At this point, the surface coordinates are only related to x、z of . Then its normal vector $\mathbf{n}(x,z)=(-\frac{\partial y}{\partial x},1,-\frac{\partial y}{\partial z})$ It must be pointing to In the east ,$\mathbf{n}(x,z)=(\frac{\partial y}{\partial x},-1,\frac{\partial y}{\partial z})$ It must point to In the west .

Now, if we look at a surface , It's a front hemisphere , So using **x(y,z)** Parameterize , It is more convenient , At this point, the surface coordinates are only related to y、z of . Then its normal vector $\mathbf{n}(y,z)=(1,-\frac{\partial x}{\partial y},-\frac{\partial x}{\partial z})$ It must be pointing to south ,$\mathbf{n}(y,z)=(-1,\frac{\partial x}{\partial y},\frac{\partial x}{\partial z})$ It must point to north .

3、 ... and 、 The second kind of surface integral

$$\begin{array}{rl}{\iint }_S\mathbf{F}(x,y,z)\cdot d\mathbf{S}& ={\iint }_S\mathbf{F}(x,y,z)\cdot {\mathbf{n}}_0(x,y,z)dS\\ & ={\iint }_S\{P(x,y,z){\mathbf{n}}_0(x,y,z)\cdot [1,0,0]+Q(x,y,z){\mathbf{n}}_0(x,y,z)\cdot [0,1,0]+R(x,y,z){\mathbf{n}}_0(x,y,z)\cdot [0,0,1]\}dS\\ & ={\iint }_S[P(x,y,z)\cos \alpha +Q(x,y,z)\cos \beta +P(x,y,z)\cos \gamma ]dS\end{array}$$

The area element of a directed surface has a direction , Its orientation is determined by the side of the surface you select , For example, if a surface is selected Front side Then its normal vector must point to south , The normal vector at this point will be $\mathbf{n}(y,z)=(1,-\frac{\partial x}{\partial y},-\frac{\partial x}{\partial z})$.

By the way Oneness projection method and Facet projection method ：

1. Oneness projection method ： Is the first step of the above equation , The result is projected onto a plane , Such as choice **z=z(x,y)** The parameterization of , Then the result will be projected to xOy In the plane .
2. Facet projection method ： Is the second step of the above equation , The three normal vectors can be parameterized in different ways , For example, for x In the axial direction P(x,y,z), Just use x=x(y,z), Then project to yOz Plane ; about z In the direction of the axis R(x,y,z), Just use z=z(x,y), Then project to xOy In the plane .

here , Through vectors [1,0,0] 、[0,1,0]、 [0,0,1] In fact, the partial derivative of the normal vector is partially zeroed . We found that , Understand... From this perspective , There is no essential difference between the unified projection method and the faceted projection method .