Mathematical analysis_spatial geometry-the relationship between normal vector and gradient

The relationship between normal vector and gradient

First, give a conclusion: the normal vector of the surface is the gradient of the ternary function, and the normal vector of the curve is the gradient of the binary function.

$w=F(x,and,z)$ is a ternary function$w=0$ means a contour surface (curved surface), and its two sides are fully differentiated to obtain$$\frac{\partial F}{\partial x}dx+\frac{\partial F}{\partial y}dy+\frac{\partial F}{\partial z}dz=0$$ can be further written as$$(\frac{\partial F}{\partial x},\frac{\partial F}{\partial y},\frac{\partial F}{\partial z})\cdot (dx,dy,dz)=0\; The$$ two vectors are orthogonal to each other, because$(dx,dy,dz)$ is the tangent vector of the surface, then$(\frac{\partial F}{\partial x},\frac{\partial F}{\partial y},\frac{\partial F}{\partial z})$ Must beSurface normal vector, And it is still $w=F(x,and,z)$ gradient.

Notably, the gradient is a multi-function fastest changing direction (gradient or direction when taking the maximum derivative), in the ternary function, the gradient Grad $w$ points towwThe direction where$w$ changes the most, notzzThe direction in which$z$ changes the most.

Many beginners do not understand the concept of gradient well, and mistake the gradient here for pointing to $In$ the direction where$z$ changes the most, draw a curved surface and draw its normal vector at the same time. It is found that the normal vector is not pointing tozzThe direction of$z$ change.
This is because the gradient is for the function, the ternary function is the three independent variables, and the function value is$w$ , thusgrad $w$ points towwThe direction in which$w$ changes the most.

But! If it is a binary function$with=F(x,y)$ , its gradientgrad $z$ points tozzThe direction in which$z$ changes the most, it also serves as$with=0$ ofCurve normal vector(Because $with=0$ means a contour line)

The normal vector is intuitive, and the gradient is abstract. From the above analysis, we should understand the gradient more intuitively from the relationship between the normal vector and the gradient:

the gradient of the binary function is the normal vector of the curve (contour), the normal vector It is two-dimensional and lies on the plane, so the gradient is also on the plane, while the binary function is three-dimensional, and the gradient is one dimension lower than the function.
The gradient of the ternary function is the normal vector of the curved surface (climbing surface). The normal vector is three-dimensional and located in space, so the gradient is also located in space. The ternary function is four-dimensional, and the gradient is one dimension lower than the function.