Changing variables or coordinates?












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$begingroup$


While posing another question I got stuck on the distinction of the following two concepts;



The components of a vector is usually referred to as it's coordinates, while trying to understand what "change of variables" means when it comes to systems of ODE's, I saw several sourse talked about change of variables as "a change of coordiinates".



Sure a function has a graph $(x,f(x))$



 where if we do some kind of transformation we get a new graph. But this should be the same kind of coordinates as in a linear space.



Is this the thing that one refers to when one says "change of coordinates" as in change to polar coordiantes?










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  • $begingroup$
    If you edit the question to provide direct quotes from the "several sources" we may be able to help. As posed now the too vague to answer.
    $endgroup$
    – Ethan Bolker
    Dec 8 '18 at 14:17










  • $begingroup$
    @Ethan Bolker so there is no consensus how to use these words? It is just a system of ODEs that should undergo change to polar "coordiates"
    $endgroup$
    – user7534
    Dec 8 '18 at 14:27






  • 1




    $begingroup$
    Both are general phrases whose precise meaning depends on context. If you have an ODE vocabulary question just ask it directly. That's all the help I can give you.
    $endgroup$
    – Ethan Bolker
    Dec 8 '18 at 14:40
















2












$begingroup$


While posing another question I got stuck on the distinction of the following two concepts;



The components of a vector is usually referred to as it's coordinates, while trying to understand what "change of variables" means when it comes to systems of ODE's, I saw several sourse talked about change of variables as "a change of coordiinates".



Sure a function has a graph $(x,f(x))$



 where if we do some kind of transformation we get a new graph. But this should be the same kind of coordinates as in a linear space.



Is this the thing that one refers to when one says "change of coordinates" as in change to polar coordiantes?










share|cite|improve this question









$endgroup$












  • $begingroup$
    If you edit the question to provide direct quotes from the "several sources" we may be able to help. As posed now the too vague to answer.
    $endgroup$
    – Ethan Bolker
    Dec 8 '18 at 14:17










  • $begingroup$
    @Ethan Bolker so there is no consensus how to use these words? It is just a system of ODEs that should undergo change to polar "coordiates"
    $endgroup$
    – user7534
    Dec 8 '18 at 14:27






  • 1




    $begingroup$
    Both are general phrases whose precise meaning depends on context. If you have an ODE vocabulary question just ask it directly. That's all the help I can give you.
    $endgroup$
    – Ethan Bolker
    Dec 8 '18 at 14:40














2












2








2





$begingroup$


While posing another question I got stuck on the distinction of the following two concepts;



The components of a vector is usually referred to as it's coordinates, while trying to understand what "change of variables" means when it comes to systems of ODE's, I saw several sourse talked about change of variables as "a change of coordiinates".



Sure a function has a graph $(x,f(x))$



 where if we do some kind of transformation we get a new graph. But this should be the same kind of coordinates as in a linear space.



Is this the thing that one refers to when one says "change of coordinates" as in change to polar coordiantes?










share|cite|improve this question









$endgroup$




While posing another question I got stuck on the distinction of the following two concepts;



The components of a vector is usually referred to as it's coordinates, while trying to understand what "change of variables" means when it comes to systems of ODE's, I saw several sourse talked about change of variables as "a change of coordiinates".



Sure a function has a graph $(x,f(x))$



 where if we do some kind of transformation we get a new graph. But this should be the same kind of coordinates as in a linear space.



Is this the thing that one refers to when one says "change of coordinates" as in change to polar coordiantes?







analysis multivariable-calculus






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share|cite|improve this question











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share|cite|improve this question










asked Dec 8 '18 at 14:11









user7534user7534

666




666












  • $begingroup$
    If you edit the question to provide direct quotes from the "several sources" we may be able to help. As posed now the too vague to answer.
    $endgroup$
    – Ethan Bolker
    Dec 8 '18 at 14:17










  • $begingroup$
    @Ethan Bolker so there is no consensus how to use these words? It is just a system of ODEs that should undergo change to polar "coordiates"
    $endgroup$
    – user7534
    Dec 8 '18 at 14:27






  • 1




    $begingroup$
    Both are general phrases whose precise meaning depends on context. If you have an ODE vocabulary question just ask it directly. That's all the help I can give you.
    $endgroup$
    – Ethan Bolker
    Dec 8 '18 at 14:40


















  • $begingroup$
    If you edit the question to provide direct quotes from the "several sources" we may be able to help. As posed now the too vague to answer.
    $endgroup$
    – Ethan Bolker
    Dec 8 '18 at 14:17










  • $begingroup$
    @Ethan Bolker so there is no consensus how to use these words? It is just a system of ODEs that should undergo change to polar "coordiates"
    $endgroup$
    – user7534
    Dec 8 '18 at 14:27






  • 1




    $begingroup$
    Both are general phrases whose precise meaning depends on context. If you have an ODE vocabulary question just ask it directly. That's all the help I can give you.
    $endgroup$
    – Ethan Bolker
    Dec 8 '18 at 14:40
















$begingroup$
If you edit the question to provide direct quotes from the "several sources" we may be able to help. As posed now the too vague to answer.
$endgroup$
– Ethan Bolker
Dec 8 '18 at 14:17




$begingroup$
If you edit the question to provide direct quotes from the "several sources" we may be able to help. As posed now the too vague to answer.
$endgroup$
– Ethan Bolker
Dec 8 '18 at 14:17












$begingroup$
@Ethan Bolker so there is no consensus how to use these words? It is just a system of ODEs that should undergo change to polar "coordiates"
$endgroup$
– user7534
Dec 8 '18 at 14:27




$begingroup$
@Ethan Bolker so there is no consensus how to use these words? It is just a system of ODEs that should undergo change to polar "coordiates"
$endgroup$
– user7534
Dec 8 '18 at 14:27




1




1




$begingroup$
Both are general phrases whose precise meaning depends on context. If you have an ODE vocabulary question just ask it directly. That's all the help I can give you.
$endgroup$
– Ethan Bolker
Dec 8 '18 at 14:40




$begingroup$
Both are general phrases whose precise meaning depends on context. If you have an ODE vocabulary question just ask it directly. That's all the help I can give you.
$endgroup$
– Ethan Bolker
Dec 8 '18 at 14:40










1 Answer
1






active

oldest

votes


















1












$begingroup$

"Change of coordinates" comes up in many contexts in mathematics. Even in calculus, there are multiple contexts. For example, the graph $x^2+y^2=k^2$ is not a function in x. However, when we change from rectangular to polar coordinates, we get a function r = k. Then it's easier to compute tangent lines and area. In integral calculus, u-substitution is a "change of coordinates" to make integration easier.



In linear algebra, a change of coordinates comes up in linear transformations and diagonalization. In many contexts, we change the coordinates to make calculations easier. In exchange, there's a little bit of work to change the coordinates, either by using the Jacobian or finding a change of basis matrix.






share|cite|improve this answer









$endgroup$













  • $begingroup$
    So my example with the function is also called a change of coordinates?
    $endgroup$
    – user7534
    Dec 9 '18 at 8:08










  • $begingroup$
    Not quite. A transformation in general changes the image to another image. A change of coordinates, in my mind, keeps the image the same, but the elements have different "names."
    $endgroup$
    – Joel Pereira
    Dec 9 '18 at 16:03











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1 Answer
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active

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1












$begingroup$

"Change of coordinates" comes up in many contexts in mathematics. Even in calculus, there are multiple contexts. For example, the graph $x^2+y^2=k^2$ is not a function in x. However, when we change from rectangular to polar coordinates, we get a function r = k. Then it's easier to compute tangent lines and area. In integral calculus, u-substitution is a "change of coordinates" to make integration easier.



In linear algebra, a change of coordinates comes up in linear transformations and diagonalization. In many contexts, we change the coordinates to make calculations easier. In exchange, there's a little bit of work to change the coordinates, either by using the Jacobian or finding a change of basis matrix.






share|cite|improve this answer









$endgroup$













  • $begingroup$
    So my example with the function is also called a change of coordinates?
    $endgroup$
    – user7534
    Dec 9 '18 at 8:08










  • $begingroup$
    Not quite. A transformation in general changes the image to another image. A change of coordinates, in my mind, keeps the image the same, but the elements have different "names."
    $endgroup$
    – Joel Pereira
    Dec 9 '18 at 16:03
















1












$begingroup$

"Change of coordinates" comes up in many contexts in mathematics. Even in calculus, there are multiple contexts. For example, the graph $x^2+y^2=k^2$ is not a function in x. However, when we change from rectangular to polar coordinates, we get a function r = k. Then it's easier to compute tangent lines and area. In integral calculus, u-substitution is a "change of coordinates" to make integration easier.



In linear algebra, a change of coordinates comes up in linear transformations and diagonalization. In many contexts, we change the coordinates to make calculations easier. In exchange, there's a little bit of work to change the coordinates, either by using the Jacobian or finding a change of basis matrix.






share|cite|improve this answer









$endgroup$













  • $begingroup$
    So my example with the function is also called a change of coordinates?
    $endgroup$
    – user7534
    Dec 9 '18 at 8:08










  • $begingroup$
    Not quite. A transformation in general changes the image to another image. A change of coordinates, in my mind, keeps the image the same, but the elements have different "names."
    $endgroup$
    – Joel Pereira
    Dec 9 '18 at 16:03














1












1








1





$begingroup$

"Change of coordinates" comes up in many contexts in mathematics. Even in calculus, there are multiple contexts. For example, the graph $x^2+y^2=k^2$ is not a function in x. However, when we change from rectangular to polar coordinates, we get a function r = k. Then it's easier to compute tangent lines and area. In integral calculus, u-substitution is a "change of coordinates" to make integration easier.



In linear algebra, a change of coordinates comes up in linear transformations and diagonalization. In many contexts, we change the coordinates to make calculations easier. In exchange, there's a little bit of work to change the coordinates, either by using the Jacobian or finding a change of basis matrix.






share|cite|improve this answer









$endgroup$



"Change of coordinates" comes up in many contexts in mathematics. Even in calculus, there are multiple contexts. For example, the graph $x^2+y^2=k^2$ is not a function in x. However, when we change from rectangular to polar coordinates, we get a function r = k. Then it's easier to compute tangent lines and area. In integral calculus, u-substitution is a "change of coordinates" to make integration easier.



In linear algebra, a change of coordinates comes up in linear transformations and diagonalization. In many contexts, we change the coordinates to make calculations easier. In exchange, there's a little bit of work to change the coordinates, either by using the Jacobian or finding a change of basis matrix.







share|cite|improve this answer












share|cite|improve this answer



share|cite|improve this answer










answered Dec 8 '18 at 14:35









Joel PereiraJoel Pereira

74519




74519












  • $begingroup$
    So my example with the function is also called a change of coordinates?
    $endgroup$
    – user7534
    Dec 9 '18 at 8:08










  • $begingroup$
    Not quite. A transformation in general changes the image to another image. A change of coordinates, in my mind, keeps the image the same, but the elements have different "names."
    $endgroup$
    – Joel Pereira
    Dec 9 '18 at 16:03


















  • $begingroup$
    So my example with the function is also called a change of coordinates?
    $endgroup$
    – user7534
    Dec 9 '18 at 8:08










  • $begingroup$
    Not quite. A transformation in general changes the image to another image. A change of coordinates, in my mind, keeps the image the same, but the elements have different "names."
    $endgroup$
    – Joel Pereira
    Dec 9 '18 at 16:03
















$begingroup$
So my example with the function is also called a change of coordinates?
$endgroup$
– user7534
Dec 9 '18 at 8:08




$begingroup$
So my example with the function is also called a change of coordinates?
$endgroup$
– user7534
Dec 9 '18 at 8:08












$begingroup$
Not quite. A transformation in general changes the image to another image. A change of coordinates, in my mind, keeps the image the same, but the elements have different "names."
$endgroup$
– Joel Pereira
Dec 9 '18 at 16:03




$begingroup$
Not quite. A transformation in general changes the image to another image. A change of coordinates, in my mind, keeps the image the same, but the elements have different "names."
$endgroup$
– Joel Pereira
Dec 9 '18 at 16:03


















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