Changing variables or coordinates?
$begingroup$
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?
analysis multivariable-calculus
asked Dec 8 '18 at 14:11
user7534user7534
645
$endgroup$
add a comment |
$begingroup$
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?
analysis multivariable-calculus
asked Dec 8 '18 at 14:11
user7534user7534
645
$endgroup$
$begingroup$
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
add a comment |
$begingroup$
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?
analysis multivariable-calculus
asked Dec 8 '18 at 14:11
user7534user7534
645
$endgroup$
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
analysis multivariable-calculus
asked Dec 8 '18 at 14:11
user7534user7534
645
asked Dec 8 '18 at 14:11
user7534user7534
645
asked Dec 8 '18 at 14:11
user7534user7534
645
asked Dec 8 '18 at 14:11
user7534user7534
645
asked Dec 8 '18 at 14:11
user7534user7534
645
645
$begingroup$
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
add a comment |
$begingroup$
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
add a comment |
1 Answer
1
active
oldest
votes
$begingroup$
"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.
answered Dec 8 '18 at 14:35
Joel PereiraJoel Pereira
74519
$endgroup$
$begingroup$
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
add a comment |
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1 Answer
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1 Answer
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active
oldest
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oldest
votes
$begingroup$
"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.
answered Dec 8 '18 at 14:35
Joel PereiraJoel Pereira
74519
$endgroup$
$begingroup$
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
add a comment |
$begingroup$
"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.
answered Dec 8 '18 at 14:35
Joel PereiraJoel Pereira
74519
$endgroup$
$begingroup$
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
add a comment |
$begingroup$
"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.
answered Dec 8 '18 at 14:35
Joel PereiraJoel Pereira
74519
$endgroup$
"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.
answered Dec 8 '18 at 14:35
Joel PereiraJoel Pereira
74519
answered Dec 8 '18 at 14:35
Joel PereiraJoel Pereira
74519
answered Dec 8 '18 at 14:35
Joel PereiraJoel Pereira
74519
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
add a comment |
$begingroup$
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
add a comment |
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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