Deriving tangent plane equation - assuming 'c' is non-zero

[autodidude]

In Stewart's calculus text, the way he derives the tangent plane equation at some point is to divide the general plane equation

$$a(x-x_0)+b(y-y_0)+c(z-z_0)=0$$ by c

This must mean c is always non-zero right? But isn't c is the 'z'-component of the normal vector to the surface at some point? If it's non-zero, does that mean that for some surface in 3D, the normal vector always has a component in the z-direction?

The one counter-example I can think of is the case of a sphere of radius 1. At (1,0,0), wouldn't the normal vector be pointing in just the x-direction? But then I'm also not sure that the derivative of z with respect to x is defined at this point...

 

[Simon Bridge]

I suspect you'll find that Stewart is giving a method for finding a non-vertical tangent plane?

 

[Vargo]

It could be that he is referring to the tangent plane of the graph of a function z=f(x,y). Assuming the function is differentiable, then the tangent plane is not vertical.

 

[autodidude]

Ah ok, thanks. So, for a sphere - or a circle in 2D, I know we can take derivatives to figure out the slope at some point but the derivative functions aren't defined at x=r and x=-r, does this mean it's not differentiable?

 

[HallsofIvy]

In Stewart's calculus text, the way he derives the tangent plane equation at some point is to divide the general plane equation

$$a(x-x_0)+b(y-y_0)+c(z-z_0)=0$$ by c

This must mean c is always non-zero right? But isn't c is the 'z'-component of the normal vector to the surface at some point? If it's non-zero, does that mean that for some surface in 3D, the normal vector always has a component in the z-direction?

The one counter-example I can think of is the case of a sphere of radius 1. At (1,0,0), wouldn't the normal vector be pointing in just the x-direction? But then I'm also not sure that the derivative of z with respect to x is defined at this point...

If the problem is to write the equation of the tangent plane (of some surface) in the form z= Ax+ By+ C, then, yes, of course, z must be a function of x and y and that will be true if and only if the normal vector to the surface is NOT parallel to the xy-plane. And that is true if and only if it is possible to write the surface itself in the form z= f(x, y) in some neighborhood of that point.

I don't have a copy of Stewart right here, but I suspect you are misquoting since it does not make sense to talk about "deriving the tangent plane" without talking about the surface. What you are talking about is NOT "deriving the tangent plane" of any surface but deriving it in a specific, limited, form.

The "general plane equation", ax+ by+ cz= d, is itself a perfectly good form for a tangent plane to some surface.

 

[Simon Bridge]

Ah ok, thanks. So, for a sphere - or a circle in 2D, I know we can take derivatives to figure out the slope at some point but the derivative functions aren't defined at x=r and x=-r, does this mean it's not differentiable?

depends... eg.

In the case of the circle $x^2+y^2=r^2$ - the slope of the tangent to the circle at x=r is what?

What is dy/dx there?

You have answered (in intent) your own question.

Note, however, it is equally valid to differentiate the other way ... finding dx/dy.
In 3D you have a choice of lines to differentiate against so you need to be more careful with what you mean.

 

[autodidude]

@HallsOfIvy: Yeah, I did misunderstand him (of course I did!)

@Simon_Bridge: Undefined? I just looked up the 'differentiable function' in wikipedia and it says that there must be a non-vertical tangent line at every point so I guess circles wouldn't be considered differentiable then (but then they're not functions either are they?). These details weren't emphasised in class, all we had to do was just be able to differentiate different functions and relations and be able to use them to solve applied problems.

 

[HallsofIvy]

@HallsOfIvy: Yeah, I did misunderstand him (of course I did!)

@Simon_Bridge: Undefined? I just looked up the 'differentiable function' in wikipedia and it says that there must be a non-vertical tangent line at every point so I guess circles wouldn't be considered differentiable then (but then they're not functions either are they?). These details weren't emphasised in class, all we had to do was just be able to differentiate different functions and relations and be able to use them to solve applied problems.

It is a misuse of terminology to say that "circles are not differentiable". Functions are or are not differentiable, not geometric objects. Given a specific coordinate system, we can write an equation describing a circle in that coordinate system and the fuction involved may not be differentiable at specific points. But that is an artifact of the coordinate system. Even given a coordinate system you could write parametric functions describing the circle that are differentiable for all points.

 

[Simon Bridge]

What HallsofIvy said - you'll notice that I was being specific about what I meant in my questions ... I was hoping to get you thinking about the relationship between the geometrical object and the math - i.e. the circle is not the function though a function can be found which describes a circle.

What are are learning about differentiation is a set of techniques for finding out relationships ... you will also be starting to figure out how to describe different things in terms of functions so that you can use the tools of calculus to analyse them.

I hope you've just discovered that, not only do you have to read your references carefully and be careful about generalizing from them, but that you need to be more careful with the terminology too. Especially while you are still getting used to it ;)

 

[autodidude]

@HallsOfIvy: So in the cartesian coordinate system, if the equation $$x^2+y^2=r^2$$ describes the circle, what would be the functions involved? If it's in terms of x, would it be the two functions describing the top and bottom half of the circle? Also, when we are implicitly differentiating the equation of a circle...are we treating y as a function of x (if differentiating with respect to x)?
I'll have to try the differentiating the parametric representation!

@Simon_Bridge: Ah, ok...there's lot of subtleties that make it all much more interesting than when I took the course. Yeah, it seems like there are a lot of important points I missed!

 

[chiro]

Variable relationships can have a derivative depending on their properties: as autodidude pointed out they need not necessarily be functions.