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What does Equidimensional equation mean
The Next CEO of Stack Overflowwhat does 'arbitrary' mean?What does this equation mean?What does this mean? Second order DEWhat does compactness actually meanWhat does d(something) mean?What does |AM|=|AC| mean?What does this mean??equidimensional equationWhat is the rigorous definition of $dy$ and $dx$?What does $[a + b]_+$ mean?
$begingroup$
What does the adjective equidimensional mean ?
I encountered it while studying Cauchy-euler equation in differential equations with applications and historical notes by gf simmons :
http://matematicaeducativa.com/foro/download/file.php?id=2247&sid=f867258ad39908e15b331d33e763a8b0
Kindly see page no. 126, q5
Wikipedia says :
"A topological space X is said to be equidimensional if for all points p in X the dimension at p that is, dim p(X) is constant. The Euclidean space is an example of an equidimensional space."
But isn't this always the case and how is an ODE related with dimensions of a point ?
I am a first year undergraduate , please explain in simple or heuristic way.
general-topology ordinary-differential-equations euclidean-geometry
asked Mar 16 at 8:20
Shlok VaibhavShlok Vaibhav
83
$endgroup$
add a comment |
$begingroup$
What does the adjective equidimensional mean ?
I encountered it while studying Cauchy-euler equation in differential equations with applications and historical notes by gf simmons :
http://matematicaeducativa.com/foro/download/file.php?id=2247&sid=f867258ad39908e15b331d33e763a8b0
Kindly see page no. 126, q5
Wikipedia says :
"A topological space X is said to be equidimensional if for all points p in X the dimension at p that is, dim p(X) is constant. The Euclidean space is an example of an equidimensional space."
But isn't this always the case and how is an ODE related with dimensions of a point ?
I am a first year undergraduate , please explain in simple or heuristic way.
general-topology ordinary-differential-equations euclidean-geometry
asked Mar 16 at 8:20
Shlok VaibhavShlok Vaibhav
83
$endgroup$
$begingroup$
Welcome to MSE. Could you elaborate more on the context? Because it seems to me that these "equidimensionalities" have nothing to do with each other, except the name.
$endgroup$
– user539887
Mar 16 at 8:36
$begingroup$
Take the union of the x-axis and the unit disk. There you have points of dimension one and two, thus not "of the same dimension", not equidimensional.
$endgroup$
– LutzL
Mar 16 at 8:42
$begingroup$
But isn't x-axis also subset of a two dimensional plane
$endgroup$
– Shlok Vaibhav
Mar 16 at 9:02
$begingroup$
Sorru, due to no reputation , i can't upload picture of the concerned pages
$endgroup$
– Shlok Vaibhav
Mar 16 at 9:05
$begingroup$
Added the link.
$endgroup$
– Shlok Vaibhav
Mar 16 at 9:21
add a comment |
$begingroup$
What does the adjective equidimensional mean ?
I encountered it while studying Cauchy-euler equation in differential equations with applications and historical notes by gf simmons :
http://matematicaeducativa.com/foro/download/file.php?id=2247&sid=f867258ad39908e15b331d33e763a8b0
Kindly see page no. 126, q5
Wikipedia says :
"A topological space X is said to be equidimensional if for all points p in X the dimension at p that is, dim p(X) is constant. The Euclidean space is an example of an equidimensional space."
But isn't this always the case and how is an ODE related with dimensions of a point ?
I am a first year undergraduate , please explain in simple or heuristic way.
general-topology ordinary-differential-equations euclidean-geometry
asked Mar 16 at 8:20
Shlok VaibhavShlok Vaibhav
83
$endgroup$
What does the adjective equidimensional mean ?
I encountered it while studying Cauchy-euler equation in differential equations with applications and historical notes by gf simmons :
http://matematicaeducativa.com/foro/download/file.php?id=2247&sid=f867258ad39908e15b331d33e763a8b0
Kindly see page no. 126, q5
Wikipedia says :
"A topological space X is said to be equidimensional if for all points p in X the dimension at p that is, dim p(X) is constant. The Euclidean space is an example of an equidimensional space."
But isn't this always the case and how is an ODE related with dimensions of a point ?
I am a first year undergraduate , please explain in simple or heuristic way.
general-topology ordinary-differential-equations euclidean-geometry
general-topology ordinary-differential-equations euclidean-geometry
asked Mar 16 at 8:20
Shlok VaibhavShlok Vaibhav
83
asked Mar 16 at 8:20
Shlok VaibhavShlok Vaibhav
83
edited Mar 16 at 9:21
Shlok Vaibhav
asked Mar 16 at 8:20
Shlok VaibhavShlok Vaibhav
83
asked Mar 16 at 8:20
Shlok VaibhavShlok Vaibhav
83
asked Mar 16 at 8:20
Shlok VaibhavShlok Vaibhav
83
83
$begingroup$
Welcome to MSE. Could you elaborate more on the context? Because it seems to me that these "equidimensionalities" have nothing to do with each other, except the name.
$endgroup$
– user539887
Mar 16 at 8:36
$begingroup$
Take the union of the x-axis and the unit disk. There you have points of dimension one and two, thus not "of the same dimension", not equidimensional.
$endgroup$
– LutzL
Mar 16 at 8:42
$begingroup$
But isn't x-axis also subset of a two dimensional plane
$endgroup$
– Shlok Vaibhav
Mar 16 at 9:02
$begingroup$
Sorru, due to no reputation , i can't upload picture of the concerned pages
$endgroup$
– Shlok Vaibhav
Mar 16 at 9:05
$begingroup$
Added the link.
$endgroup$
– Shlok Vaibhav
Mar 16 at 9:21
add a comment |
$begingroup$
Welcome to MSE. Could you elaborate more on the context? Because it seems to me that these "equidimensionalities" have nothing to do with each other, except the name.
$endgroup$
– user539887
Mar 16 at 8:36
$begingroup$
Take the union of the x-axis and the unit disk. There you have points of dimension one and two, thus not "of the same dimension", not equidimensional.
$endgroup$
– LutzL
Mar 16 at 8:42
$begingroup$
But isn't x-axis also subset of a two dimensional plane
$endgroup$
– Shlok Vaibhav
Mar 16 at 9:02
$begingroup$
Sorru, due to no reputation , i can't upload picture of the concerned pages
$endgroup$
– Shlok Vaibhav
Mar 16 at 9:05
$begingroup$
Added the link.
$endgroup$
– Shlok Vaibhav
Mar 16 at 9:21
$begingroup$
Welcome to MSE. Could you elaborate more on the context? Because it seems to me that these "equidimensionalities" have nothing to do with each other, except the name.
$endgroup$
– user539887
Mar 16 at 8:36
$begingroup$
Welcome to MSE. Could you elaborate more on the context? Because it seems to me that these "equidimensionalities" have nothing to do with each other, except the name.
$endgroup$
– user539887
Mar 16 at 8:36
$begingroup$
Take the union of the x-axis and the unit disk. There you have points of dimension one and two, thus not "of the same dimension", not equidimensional.
$endgroup$
– LutzL
Mar 16 at 8:42
$begingroup$
Take the union of the x-axis and the unit disk. There you have points of dimension one and two, thus not "of the same dimension", not equidimensional.
$endgroup$
– LutzL
Mar 16 at 8:42
$begingroup$
But isn't x-axis also subset of a two dimensional plane
$endgroup$
– Shlok Vaibhav
Mar 16 at 9:02
$begingroup$
But isn't x-axis also subset of a two dimensional plane
$endgroup$
– Shlok Vaibhav
Mar 16 at 9:02
$begingroup$
Sorru, due to no reputation , i can't upload picture of the concerned pages
$endgroup$
– Shlok Vaibhav
Mar 16 at 9:05
$begingroup$
Sorru, due to no reputation , i can't upload picture of the concerned pages
$endgroup$
– Shlok Vaibhav
Mar 16 at 9:05
$begingroup$
Added the link.
$endgroup$
– Shlok Vaibhav
Mar 16 at 9:21
$begingroup$
Added the link.
$endgroup$
– Shlok Vaibhav
Mar 16 at 9:21
add a comment |
1 Answer
1
active
oldest
votes
$begingroup$
A topological space is called equi-dimensional, if every irreducible component of X has the same dimension - see here. This is not always the case.
The disjoint union of two spaces $X$ and $Y$ (as topological spaces) of different dimension is an example of a non-equidimensional space.
Edit: Now there is a link in the question, which shows that we talk about Euler's equidimensional equation. So what is the meaning of "equidimensional" for this equation?
"Say for example the variable $y$ is a distance measured in meters $(m)$ and
the variable $x$ is time, measured in seconds $(s)$. Then $y'$ is a velocity
$(m/s)$ and $y''$ is an acceleration $(m/s^2)$. Thus $x^2 y$ is a distance $(m)$
and $x y'$ is a distance $(m)$. All three terms have the same dimension
$(m)$. This we call "equidimensional".
answered Mar 16 at 9:16
Dietrich BurdeDietrich Burde
81.5k648106
$endgroup$
add a comment |
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1 Answer
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1 Answer
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$begingroup$
A topological space is called equi-dimensional, if every irreducible component of X has the same dimension - see here. This is not always the case.
The disjoint union of two spaces $X$ and $Y$ (as topological spaces) of different dimension is an example of a non-equidimensional space.
Edit: Now there is a link in the question, which shows that we talk about Euler's equidimensional equation. So what is the meaning of "equidimensional" for this equation?
"Say for example the variable $y$ is a distance measured in meters $(m)$ and
the variable $x$ is time, measured in seconds $(s)$. Then $y'$ is a velocity
$(m/s)$ and $y''$ is an acceleration $(m/s^2)$. Thus $x^2 y$ is a distance $(m)$
and $x y'$ is a distance $(m)$. All three terms have the same dimension
$(m)$. This we call "equidimensional".
answered Mar 16 at 9:16
Dietrich BurdeDietrich Burde
81.5k648106
$endgroup$
add a comment |
$begingroup$
A topological space is called equi-dimensional, if every irreducible component of X has the same dimension - see here. This is not always the case.
The disjoint union of two spaces $X$ and $Y$ (as topological spaces) of different dimension is an example of a non-equidimensional space.
Edit: Now there is a link in the question, which shows that we talk about Euler's equidimensional equation. So what is the meaning of "equidimensional" for this equation?
"Say for example the variable $y$ is a distance measured in meters $(m)$ and
the variable $x$ is time, measured in seconds $(s)$. Then $y'$ is a velocity
$(m/s)$ and $y''$ is an acceleration $(m/s^2)$. Thus $x^2 y$ is a distance $(m)$
and $x y'$ is a distance $(m)$. All three terms have the same dimension
$(m)$. This we call "equidimensional".
answered Mar 16 at 9:16
Dietrich BurdeDietrich Burde
81.5k648106
$endgroup$
add a comment |
$begingroup$
A topological space is called equi-dimensional, if every irreducible component of X has the same dimension - see here. This is not always the case.
The disjoint union of two spaces $X$ and $Y$ (as topological spaces) of different dimension is an example of a non-equidimensional space.
Edit: Now there is a link in the question, which shows that we talk about Euler's equidimensional equation. So what is the meaning of "equidimensional" for this equation?
"Say for example the variable $y$ is a distance measured in meters $(m)$ and
the variable $x$ is time, measured in seconds $(s)$. Then $y'$ is a velocity
$(m/s)$ and $y''$ is an acceleration $(m/s^2)$. Thus $x^2 y$ is a distance $(m)$
and $x y'$ is a distance $(m)$. All three terms have the same dimension
$(m)$. This we call "equidimensional".
answered Mar 16 at 9:16
Dietrich BurdeDietrich Burde
81.5k648106
$endgroup$
A topological space is called equi-dimensional, if every irreducible component of X has the same dimension - see here. This is not always the case.
The disjoint union of two spaces $X$ and $Y$ (as topological spaces) of different dimension is an example of a non-equidimensional space.
Edit: Now there is a link in the question, which shows that we talk about Euler's equidimensional equation. So what is the meaning of "equidimensional" for this equation?
"Say for example the variable $y$ is a distance measured in meters $(m)$ and
the variable $x$ is time, measured in seconds $(s)$. Then $y'$ is a velocity
$(m/s)$ and $y''$ is an acceleration $(m/s^2)$. Thus $x^2 y$ is a distance $(m)$
and $x y'$ is a distance $(m)$. All three terms have the same dimension
$(m)$. This we call "equidimensional".
answered Mar 16 at 9:16
Dietrich BurdeDietrich Burde
81.5k648106
edited Mar 16 at 9:31
answered Mar 16 at 9:16
Dietrich BurdeDietrich Burde
81.5k648106
answered Mar 16 at 9:16
Dietrich BurdeDietrich Burde
81.5k648106
answered Mar 16 at 9:16
Dietrich BurdeDietrich Burde
81.5k648106
81.5k648106
add a comment |
add a comment |
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$begingroup$
Welcome to MSE. Could you elaborate more on the context? Because it seems to me that these "equidimensionalities" have nothing to do with each other, except the name.
$endgroup$
– user539887
Mar 16 at 8:36
$begingroup$
Take the union of the x-axis and the unit disk. There you have points of dimension one and two, thus not "of the same dimension", not equidimensional.
$endgroup$
– LutzL
Mar 16 at 8:42
$begingroup$
But isn't x-axis also subset of a two dimensional plane
$endgroup$
– Shlok Vaibhav
Mar 16 at 9:02
$begingroup$
Sorru, due to no reputation , i can't upload picture of the concerned pages
$endgroup$
– Shlok Vaibhav
Mar 16 at 9:05
$begingroup$
Added the link.
$endgroup$
– Shlok Vaibhav
Mar 16 at 9:21