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Conic sections vs multivariable functions

  1. Jun 8, 2013 #1
    Hello,

    We just started to learn about functions of several variables in my Calculus class and my question is simple:

    Are conic sections, like ellipses, multivariable functions or is y still dependant on x? Are ellipses just single variable functions slightly rearranged? Thanks in advance,

    Lee
     
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  3. Jun 8, 2013 #2
    I don't like the term dependent variable vs independent variable. I prefer to say that "y is a function of x" or vice versa; meaning that if y is a function of x then it can be put into the form y=f(x).

    Remember that an ellipse is not a function; it doesn't pass the vertical line rule (in my lingo; you cannot put an ellipse equation into the form y=f(x)).

    As per the above, ellipses aren't really functions, but I understand the meaning of the question you are asking and the answer is yes.
     
  4. Jun 8, 2013 #3

    Simon Bridge

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    You cannot write the equation of an ellipse as a single y=f(x) because each value of x has two values of y. That is why it isn't a function.

    Sometimes it is best to consider the relation to be what has to be true about point (x,y).
    An ellipse is the set of all points which satisfy ##b(x-x0)^2 + a(y-y0)^2 = ab## where a,b > 0.
    In this sense, x and y depend on each other.
     
  5. Jun 8, 2013 #4

    WannabeNewton

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    The implicit function theorem allows you to describe the ellipse locally as the graph of some function ##f: \mathbb{R} \rightarrow \mathbb{R}##. That's not an issue. Your question is really more of a geometric nature. An ellipse is just a regular curve ##\gamma :J \rightarrow \mathbb{R}^{3}## (which is obviously a function) that will always be contained in some plane. There are many regular curves like this (specifically the ones with vanishing torsion). On the other hand surfaces such as ##S^{2}## cannot be contained in a single plane (although it too can be described locally as the graph of a function as per the implicit function theorem).

    EDIT: See here to add on to what micromass said: http://en.wikipedia.org/wiki/Ellipse#General_parametric_form
     
    Last edited: Jun 9, 2013
  6. Jun 8, 2013 #5

    micromass

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    We can easily write an ellipse as a function. For example, we can do it as follows:

    [tex]f:\mathbb{R}\rightarrow \mathbb{R}^2: t\rightarrow (a \cos(t), b\sin(t))[/tex]
     
  7. Jun 9, 2013 #6

    WannabeNewton

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    I think the misconception here was that a function is necessarily a map ##f: \mathbb{R} \rightarrow \mathbb{R}## which is a gross restriction on what a function is but it seems to be a common misunderstanding at the calculus / pre-calculus level.
     
  8. Jun 9, 2013 #7
    Thanks for the replies.

    So essentially, we can say an ellipse is an expression with one input and one output, so if it were a rectangular function, it would be of one variable. Or, we can say its domain is of one dimension, where expressions or functions of two variables have a domain of two dimensions, with two inputs and one output.

    This is at least the general consensus I've interpreted. Thanks again for the replies.

    Lee
     
  9. Jun 10, 2013 #8

    Simon Bridge

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    You can write an expression which describes an ellipse which has a real-number input and two real-numbers as output - the two real-numbers output describing a point in a plane.

    If you consider only x and y and take a value of x as the input, though, you get two possible outputs for y (and vice versa).

    You could treat the ellipse equation as taking two inputs (x,y) and returning one output z, whose value tells you which ellipse, of a particular a/b, x and y are on.

    In other words - it is not going to be the kind of function you were used to with the single-variable stuff.

    This part of the course is trying to get you to broaden your understanding of math - if you try to treat multi-variable functions as a special case of single variable functions, you are going to miss out.
     
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