Conic sections vs multivariable functions

[leehufford]

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 dependent on x? Are ellipses just single variable functions slightly rearranged? Thanks in advance,

Lee

 

[Vorde]

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)).

Are ellipses just single variable functions slightly rearranged?

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.

 

[Simon Bridge]

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 dependent on x? Are ellipses just single variable functions slightly rearranged? Thanks in advance,

Lee

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.

 

[WannabeNewton]

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

 

[micromass]

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

f:\mathbb{R}\rightarrow \mathbb{R}^2: t\rightarrow (a \cos(t), b\sin(t))

 

[WannabeNewton]

We can easily write an ellipse as a function.

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.

 

[leehufford]

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

 

[Simon Bridge]

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

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.