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 | | From: | Alex Hunsley | | Subject: | denoting the dimension an implicit equation is plotted in | | Date: | Sun, 23 Jan 2005 12:17:32 GMT |
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 | Normally an implicit function like
f(x, y) = x^2 + y^2 - 1
is thought of as corresponding to a figure in the amount of dimensions
mentioned in the equation, where we find the zero points:
0 = f(x,y)
(So here we have a circle in 2 dimensions.)
I need some notation, however, to denote the figure produced by solving
the implicit equation in a certain amount of dimensions, since changing
the number of dimensions changes the final figure... e.g. the f(x, y)
given above would be a circle if you plotted it in 2 dimensions, but it
would be a cylinder/tube if you plotted it in three dimensions, and so
on...
Does such a notation exist?
If not, what would be a reasonable way to represent it?
First thing that comes to mind is something like:
f(x, y, ...)
N
which denotes the figure corresponding to the solution of f(x, y, ...) =
0 in dimension N.
cheerio,
alex
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| | From: | William Elliot | | Subject: | Re: denoting the dimension an implicit equation is plotted in | | Date: | Sun, 23 Jan 2005 04:27:58 -0800 |
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| On Sun, 23 Jan 2005, Alex Hunsley wrote:
> Normally an implicit function like
>
> f(x, y) = x^2 + y^2 - 1
>
> is thought of as corresponding to a figure in the amount of dimensions
> mentioned in the equation, where we find the zero points:
>
> 0 = f(x,y)
> (So here we have a circle in 2 dimensions.)
>
> I need some notation, however, to denote the figure produced by solving the
> implicit equation in a certain amount of dimensions, since changing the
> number of dimensions changes the final figure... e.g. the f(x, y) given above
> would be a circle if you plotted it in 2 dimensions, but it would be a
> cylinder/tube if you plotted it in three dimensions, and so on...
f(x,y,z) = x^2 + y^2 - 1
f(x,y,z) = 0 implies a cylinder
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| | From: | Alex Hunsley | | Subject: | Re: denoting the dimension an implicit equation is plotted in | | Date: | Sun, 23 Jan 2005 17:21:59 GMT |
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| William Elliot wrote:
> On Sun, 23 Jan 2005, Alex Hunsley wrote:
>
>> Normally an implicit function like
>>
>> f(x, y) = x^2 + y^2 - 1
>>
>> is thought of as corresponding to a figure in the amount of dimensions
>> mentioned in the equation, where we find the zero points:
>>
>> 0 = f(x,y)
>> (So here we have a circle in 2 dimensions.)
>>
>> I need some notation, however, to denote the figure produced by
>> solving the implicit equation in a certain amount of dimensions, since
>> changing the number of dimensions changes the final figure... e.g. the
>> f(x, y) given above would be a circle if you plotted it in 2
>> dimensions, but it would be a cylinder/tube if you plotted it in three
>> dimensions, and so on...
>
>
> f(x,y,z) = x^2 + y^2 - 1
> f(x,y,z) = 0 implies a cylinder
I see what you mean, but the point is that I want to attach the amount
of dimensions solved for to either just the expression f(x,y,z) or just
the expression x^2+y^2, and not mention both.
thanks,
alex
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| | From: | Travis Willse | | Subject: | Re: denoting the dimension an implicit equation is plotted in | | Date: | Sun, 23 Jan 2005 17:21:58 -0800 |
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| Alex,
William's example is well-guided: It turns out that the dimension of
the domain of a function is almost always clear from context. That is,
if you're writing down x^2+y^2-1 = 0, you ought to have already made
clear whether the object represented is in R^2 or R^3 or some other space.
Cheers,
Travis
Alex Hunsley wrote:
> William Elliot wrote:
>
>> On Sun, 23 Jan 2005, Alex Hunsley wrote:
>>
>>> Normally an implicit function like
>>>
>>> f(x, y) = x^2 + y^2 - 1
>>>
>>> is thought of as corresponding to a figure in the amount of
>>> dimensions mentioned in the equation, where we find the zero points:
>>>
>>> 0 = f(x,y)
>>> (So here we have a circle in 2 dimensions.)
>>>
>>> I need some notation, however, to denote the figure produced by
>>> solving the implicit equation in a certain amount of dimensions,
>>> since changing the number of dimensions changes the final figure...
>>> e.g. the f(x, y) given above would be a circle if you plotted it in 2
>>> dimensions, but it would be a cylinder/tube if you plotted it in
>>> three dimensions, and so on...
>>
>>
>>
>> f(x,y,z) = x^2 + y^2 - 1
>> f(x,y,z) = 0 implies a cylinder
>
>
> I see what you mean, but the point is that I want to attach the amount
> of dimensions solved for to either just the expression f(x,y,z) or just
> the expression x^2+y^2, and not mention both.
> thanks,
> alex
>
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| | From: | William Elliot | | Subject: | Re: denoting the dimension an implicit equation is plotted in | | Date: | Sun, 23 Jan 2005 19:18:48 -0800 |
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| On Sun, 23 Jan 2005, Alex Hunsley wrote:
> William Elliot wrote:
>> On Sun, 23 Jan 2005, Alex Hunsley wrote:
>>
>>> Normally an implicit function like
>>>
>>> f(x, y) = x^2 + y^2 - 1
>>>
>>> is thought of as corresponding to a figure in the amount of dimensions
>>> mentioned in the equation, where we find the zero points:
>>>
>>> 0 = f(x,y)
>>> (So here we have a circle in 2 dimensions.)
>>>
>>> I need some notation, however, to denote the figure produced by solving
>>> the implicit equation in a certain amount of dimensions, since changing
>>> the number of dimensions changes the final figure... e.g. the f(x, y)
>>> given above would be a circle if you plotted it in 2 dimensions, but it
>>> would be a cylinder/tube if you plotted it in three dimensions, and so
>>> on...
>>
>> f(x,y,z) = x^2 + y^2 - 1
>> f(x,y,z) = 0 implies a cylinder
>
> I see what you mean, but the point is that I want to attach the amount of
> dimensions solved for to either just the expression f(x,y,z) or just the
> expression x^2+y^2, and not mention both.
> thanks,
To describe a function, two things are needed, it's domain, and it's value
thruout the domain. x^2 + y^2 doesn't describe the domain.
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