In the three-dimensional surface, the normal vector of any point is obtained. The formula is very simple, that is. But I can't figure out why the formula is like this.
In fact, I have some faint feeling that this and the extremum of the Lagrange multiplier method is somewhat related. The solution can be either the maximum or the minimum value, because it is also a list of conditions that are satisfied. Look at the machine learning public class, which refers to the multiplier can be regarded as a super-plane, the parameters of the partial guide all zero is the normal direction, for the Extremum Balabala ... But I also can not understand Lagrange multiplier method, only know the back formula ... Cry
Please know friends help, although I know this is difficult, thank you here First ~ Add comments to share Arwen Qing , halfway academic, halfway learn 4 people agree that the surface in three-dimensional space can be understood as the equivalent of a scalar field in three-dimensional space, is the gradient at each point, That is, the direction of the maximum change in value, which is intuitively the normal direction of the equivalent plane.
Considering the formula of full differential
If the and both are on the surface, then
That is, perpendicular to any minimum segment near the surface, i.e. it is the normal vector of the surface. It's not strictly written, it's roughly what it means.
is not directly related to Lagrange multiplier, even if there is a relationship, but also through the indirect relationship. Operators of various operations and meanings, need to be understood in some examples, in any of the calculus textbook will be involved. Beginners are basically going to die to remember some, after ripe before they will understand.
How to understand normal vector formulas of three-dimensional surfaces
