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

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Discussion Overview

The discussion revolves around the derivation of the tangent plane equation in calculus, particularly focusing on the implications of assuming the 'c' component of the normal vector is non-zero. Participants explore the conditions under which a tangent plane can be defined, especially in relation to surfaces like spheres and circles, and the differentiability of these shapes.

Discussion Character

  • Exploratory
  • Technical explanation
  • Conceptual clarification
  • Debate/contested
  • Mathematical reasoning

Main Points Raised

  • Some participants question whether 'c' being non-zero implies that the normal vector to a surface must always have a z-component.
  • Others suggest that the tangent plane derived from Stewart's text may only apply to non-vertical planes.
  • There is a discussion about whether a sphere's normal vector can point solely in the x-direction at certain points, raising questions about differentiability at those points.
  • Some participants argue that the definition of differentiability may not apply to geometric objects like circles, which can complicate the understanding of tangent lines.
  • There is a mention of the need to differentiate carefully in 3D, as multiple lines can be chosen for differentiation.
  • Participants discuss the implications of using parametric functions to describe circles and their differentiability across different coordinate systems.

Areas of Agreement / Disagreement

Participants express differing views on the implications of the normal vector's components and the conditions for differentiability. There is no consensus on whether circles can be considered differentiable or how to interpret the relationship between geometric objects and their mathematical representations.

Contextual Notes

Some participants note that the definitions and properties of differentiability may depend on the coordinate system used, and that the terminology surrounding functions and geometric shapes can lead to misunderstandings.

Who May Find This Useful

This discussion may be useful for students and educators in calculus, particularly those interested in the geometric interpretation of derivatives and the subtleties of differentiability in various contexts.

autodidude
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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...
 
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I suspect you'll find that Stewart is giving a method for finding a non-vertical tangent plane?
 
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.
 
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?
 
autodidude said:
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.
 
autodidude said:
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.
 
@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.
 
autodidude said:
@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.
 
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 ;)
 
  • #10
@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!
 
  • #11
Variable relationships can have a derivative depending on their properties: as autodidude pointed out they need not necessarily be functions.
 

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