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I was studying Multi-variable Calculus, and I got confused with the definition of the Gradient. The definition that I learned was this: enter image description here

But, doing some examples, and searching in Google I saw that the Gradient vector can be normal to every tangent plane to the surface given. The picture below gives the tangent plane and the gradient vector normal to that tangent plane. enter image description here

So, I got confused because I do not understand is the Gradient vector normal only to the level curves or also to other tangent planes? And if they are tangent to other planes, why there are a lot definition that define it differently?

Update:

Based on the comments what I understood was that a function of two variables has a level CURVE, and gradient is normal to that level curve. But, functions of three variables don’t have level curves but level SURFACES, that is why the gradient is normal to every point in that level surface. So, the second picture is the gradient vector normal at the level surface of some function with three variables. This is what I understood. If you believe that is not correct, than please provide with some explanation.

The sources for the 1st and 2nd pictures:

http://tutorial.math.lamar.edu/Classes/CalcIII/GradientVectorTangentPlane.aspx https://bvmtc.math.tamu.edu/~glahodny/Math251/Section%2012.6.pdf

Kasiopea
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  • Notice the level curves one refers to functions of $2$ variables, but the tangent planes are functions of $3$ variables. – Ninad Munshi May 10 '20 at 12:37
  • @NinadMunshi the definition of gradient vector talks about functions of 3 variables also, not only about 2 variables – Kasiopea May 10 '20 at 12:39
  • ...that's my point. It refers to different objects, Functions of $3$ variables do not have "level curves" – Ninad Munshi May 10 '20 at 12:40
  • I think I see your confusion a bit better now. A tangent plane is defined to be as "parallel as possible" to a surface at a given point. That means its normal vector will be the gradient of the surface because the gradient will be perpendicular to the surface there. – Ninad Munshi May 10 '20 at 12:42
  • @NinadMunshi But, are we allowed to use gradient vector to tangent planes also? Or am I missing something? – Kasiopea May 10 '20 at 12:44
  • I searched on Google about the "level curve" of a function with three variables and it said that: For a function of two variables, we saw that a level set was a curve in two dimensions that we called a level curve. For a function of three variables, a level set is a surface in three-dimensional space that we will call a level surface. Is gradient normal to the level surfaces also? and maybe that is the reason that I am getting confused – Kasiopea May 10 '20 at 12:46
  • Yes, gradient is perpendicular to all level sets, no matter the dimension. – Ninad Munshi May 10 '20 at 12:46
  • But, is the gradient vector normal to tangent planes of a function also except for the level sets? – Kasiopea May 10 '20 at 12:47
  • That is the definition of a tangent plane – Ninad Munshi May 10 '20 at 12:49
  • Level sets slice the curve of a function as I learned they are not tangent to the curve? I do not think I am understanding your point of view? – Kasiopea May 10 '20 at 12:51
  • When you have to find the equation of the tangent hyperplane to a curve that describes a function from $\mathbb{R^2}$ to $\mathbb{R}$, for example, the orthogonal vector of the hyperplane is a vector whose components are the partial derivatives of your function ,except for the last component which is $-1$. So the vector is $\underline{u}=(\frac{\partial f(x_0,y_0)}{\partial x},\frac{\partial f(x_0,y_0)}{\partial y}-1)$, where $(x_0,y_0)$ is the point of tangency. – Vajra May 10 '20 at 13:08
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    @psidaga yes, but the second picture is suggesting that is not always normal to the level set, but is also normal to any tangent plane to any point of the surface, because we use gradient vector to find the tangent plane? – Kasiopea May 10 '20 at 13:33
  • In general we can say that if $f$ is differentiable in $a$ the tangent hyperplane in $(a,f(a))$ is the set ${(x,y)\in \mathbb{R^{n+1}}:y=f(a)+<\nabla f(a),(x-a)>}$ – Vajra May 10 '20 at 13:39
  • You need to be clear on which function is is you’re talking about when you speak of its gradient. See this question, of which this one is more or less a duplicate. – amd May 10 '20 at 20:04

1 Answers1

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The first picture is a part of the definition of gradient and the other picture shows one usage of gradient. The surface there is given by $f(\vec{x})=c$ where $c$ is a constant. The surface there is a level curve (or surface curve if you want) and it is the same level curve as the one which is talked about in the first picture.

Hume2
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