tangents, gradients,normals

切向量、梯度向量和法向量是微积分中描述曲面性质的重要概念。切向量表示曲面在某点的局部直线方向,梯度向量是函数在该点的法向量,指向函数增加最快的方向。法向量垂直于曲面,与切向量正交。梯度的模长表示函数在该点的变化率,而方向指示函数增大的方向。在等值线上,梯度即为法向量,与切向量垂直,用于定义切平面和法线。这些概念在求解极值问题和理解曲面结构中起关键作用。

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tangents, gradients,normals  all are Directional derivative.

tangents, 切向量

gradients,梯度向量 ∇f

normals,法向量

这是三个容易混淆的概念,a wishy washy argument.


Tangent vectors , https://en.wikipedia.org/wiki/Tangent_space
Gradient      https://en.wikipedia.org/wiki/Gradient
Normal (geometry), https://en.wikipedia.org/wiki/Normal_(geometry)

Directional derivative ,https://en.wikipedia.org/wiki/Directional_derivative

方向导数(directional derivative)与梯度,https://blog.youkuaiyun.com/win_in_action/article/details/52729075

Level set,https://en.wikipedia.org/wiki/Level_set#Level_sets_versus_the_gradient

==========

https://mathoverflow.net/questions/1977/why-is-the-gradient-normal

The gradient of a function is normal to the level sets because it is defined that way. The gradient of a function is not the natural derivative.

When you have a function, f, defined on some Euclidean space (more generally, a Riemannian manifold) then its derivative at a point, say x, is a function dxf on tangent vectors.

The intuitive way to think of it is that dxf(v) answers the question:

If I move infinitesimally in the direction v, what happens to f?

So dxf is not itself a tangent vector. However, as we have an inner product lying around, we can convert it into a tangent vector which we call ∇f. This represents the question:

What tangent vector u at x best represents dxf?

What we mean by "best represents" is that u should satisfy the condition:

<u,v> = dxf(v) for all tangent vectors v

Now we look at the level set of f through x. If v is a tangent vector at x which is tangent to the level set then dxf(v) = 0 since f doesn't change if we go (infinitesimally) in the direction of v.

Hence our vector ∇f (aka u in the question) must satisfy <∇f, v> = 0. That is, ∇f is normal to the set of tangent vectors at x which are tangent to the level set.

For a generic x and a generic f (i.e. most of the time), the set of tangent vectors at x which are tangent to the level set of f at x is codimension 1 so this specifies ∇f up to a scalar multiple.

The scalar multiple can be found by looking at a tangent vector v such that f does change in the v-direction. If no such v exists, then ∇f = 0, of course.

 

==============

 

tangents 曲面平行方向;

gradients 曲面垂直方向,也就是normals法向量。

由于方向垂直,tangents和gradients的内积为零。

对于函数(曲面),梯度就是在某点的法向量,并指向数值更高的等值线,

这也解释了为什么求最小值的时候要用负方向梯度。

 

======================================================

Video3157 - Gradient Vector Equation Tangent Plane and Equation Normal Line - Example

https://www.youtube.com/watch?v=db1UTj6-QMc

Calculus 3 Lecture 13.7: Finding Tangent Planes and Normal Lines to Surfaces

https://www.youtube.com/watch?v=yLbqHfuWsr8

Tangent Plane and Normal Line to Surface

https://www.youtube.com/watch?v=ePuePJvVyPc

Gradients, Tangents and Normals

https://www.youtube.com/watch?v=DDLBFM2fL5M

Gradients, Tangents & Normals

https://www.savemyexams.co.uk/notes/a-level-maths-edexcel-pure/7-differentiation/7-2-applications-of-differentiation/7-2-1-gradients-tangents-normals/

这里有简单证明:切向量,法向量及梯度

http://longzxr.blog.sohu.com/200492096.html

切向量和梯度的关系

https://blog.youkuaiyun.com/silence1214/article/details/8875809

切线、法线、梯度之间的关系

https://blog.youkuaiyun.com/Queen0911/article/details/100611797

小谈导数、梯度和极值

https://www.cnblogs.com/jerrylead/archive/2011/03/09/1978280.html

 

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