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Surface Normal and Parametric Surface?

  1. Jan 3, 2014 #1
    I've been working on Multivariable Calculus with a few books. In Vector Analysis, I've had some parts which made some questions come up in my mind. I have two questions about them.

    1)Can we think of surface normal vector ([itex]\vec{n}[/itex]) as a vector field ([itex]\vec{F}[/itex]) or just a position vector ([itex]\vec{r}[/itex]) which doesn't make sense to me?

    2)Do we always have to parametrize a surface with two variables ([itex]\vec{r}(u,v)[/itex]) which I haven't seen any of ([itex]\vec{r}(t)[/itex]) giving us a single variable parameterized surface? Why?

    Thanks..
     
  2. jcsd
  3. Jan 3, 2014 #2
    I really don't understand question 1, but with regard to question 2: A surface is a 2D entity, and requires two independent variables to establish position within the surface.

    Chet
     
  4. Jan 4, 2014 #3

    HallsofIvy

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    We certainly cannot think of a single vector as a vector field since a vector field is a set of vectors. The set of all normal vectors to a given surface form a vector field.

    Personally, I would recommend that you forget about "position vectors" entirely. They only make sense, to begin with, in Cartesian coordinates, in a given coordinate system, so are not really general. (Suppose we set up a coordinate system on the surface of a sphere. Do "position vectors", from (0,0) to [itex](\theta, \phi)[itex], go through the sphere or do they curve around the sphere? I used to worry a lot about that! Of course the answer is that there are NO "position vector" on such a surface.)

    A surface is, pretty much by definition, two dimensional. That means that we need two numbers, whether parameters or not, to identify a point on a surface.

     
  5. Jan 4, 2014 #4
    I respectfully disagree. If [itex]\vec{r}(θ,\phi)[/itex] represents a position vector from the origin (i.e., the center of the sphere) to a point on the surface of the sphere, then [itex]\vec{r}(θ,\phi)=r\vec{i_r}(θ,\phi)[/itex], where [itex]\vec{i_r}(θ,\phi)[/itex] is the unit vector in the radial direction. A differential position vector within the surface is then given by:
    [itex]d\vec{r}(θ,\phi)=\frac{\partial \vec{r}(θ,\phi)}{\partial θ}dθ+\frac{\partial \vec{r}(θ,\phi)}{\partial \phi}d\phi=\vec{a_θ}dθ+\vec{a_{\phi}}d\phi[/itex]
    where [itex]\vec{a_θ}[/itex] and [itex]\vec{a_{\phi}}[/itex] represent the coordinate basis vectors in the θ and [itex]\phi[/itex] coordinate directions, respectively.

    In short, even though there are no position vectors within such a surface, position vectors from an arbitrary origin to points on the surface can be used to establish differential position vectors within the surface.
     
  6. Jan 4, 2014 #5
    Anyway I know it doesn't mean a single vector. My question is considering whether its notation refers to a vector field or a position vector. Okay, I've considered it correctly then.

    Can I generalize the state so that;

    In [itex]R^{2}[/itex], [itex]\vec{r}[/itex] must be at least single variable.
    In [itex]R^{3}[/itex], [itex]\vec{r}[/itex] must be at least two variables.
    In [itex]R^{n}[/itex], [itex]\vec{r}[/itex] must be at least [itex]n-1[/itex] variables. ??
     
  7. Jan 4, 2014 #6
    In [itex]R^{2}[/itex], [itex]\vec{r}[/itex] must be at least single variable.

    This is a line.
    You can also use r in this case as a function of a single variable to define a curved line in space.

    Even in higher dimensional spaces, r as a function of two variables is still a 2D surface. In this case, r as a function of m=1,...,n-1 variables is an m dimensional "surface." Note that this only works if R^n is a flat space. However, some people (myself included) are comfortable with assuming that the m dimensional curved "surface" is immersed in a higher dimensional flat space. For example, in general relativity, I have no qualms about imagining that curved 4D space-time is immersed in a higher dimensional flat space.
     
  8. Jan 4, 2014 #7
    Okay, I guess I got it. Thanks for help..
     
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