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How do I properly squish an n-dimensional Scalar Field down to 3D?
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Let:
$mathcal P_n$ and $mathcal P_3$ be two manifolds isomorphic to the Euclidian Space, where $mathcal P_n$ is $n$-dimensional and, equivalentl,y, $mathcal P_3$ is $3$-dimensional.
$phi$ be a mapping such that $phi:mathcalP_n rightarrow mathcalP_3$. $phi$ is the Squishification mapping.
$p in mathcalP_n$ and $p' in mathcalP_3$.
$f$ be a real scalar field defined in $mathcalP_n$ such that $f:mathcalP_n rightarrow mathbbR$
Let $nabla f(p) in T_p mathcalP_n$ where $T_p mathcalP_n$ is the tangent space of $mathcalP_n$ at point $p$ and $nabla f(p)$ is the gradient of $f$ at $p$.
I don't have extensive training in math, so I am going to do a quick introduction to the problem which might contain mistakes and inconsistencies. Feel free to point it out.
Anyway, I am faced with a number of doubts that involves the tranformation of $f$ under $phi$. For example:
$mathcalP_n$ has a higher dimensionality than $mathcalP_3$, it is likely that more than one point in $mathcalP_n$ will be mapped to a point in $mathcalP_3$ under $phi$. Suppose that for $p,q in mathcalP_n$ we have that $phi(p)=p'$ and $phi(q)=p'$. How are the values $f(p)$ and $f(q)$ mapped into $mathcalP_3$?
Also, I am assuming that if $phi(p)=p'$ then $T_phi(p) mathcalP_3 = T_p' mathcalP_3$. Given these conditions, is it true that there will be a vector $u in T_p' mathcalP_3$ that will be "projection" of $nabla f(p)$ in the new tangent space? If so, how can I obtain this projection vector by knowing $phi$? in other words, how are the gradients of $f$ mapped into $mathcalP_3$? after it is squished?
The same problem involves Hessians. A Hessian will be a $ n times n$ matrix, where $n$ is the dimensionality of the space in which it is defined. How can I map a Hessian of $f$ defined at $p$ to a Hessian defined at $p'$ once $phi$ is applied?
Ideally I would like to generalize this to differential structures that includes derivatives of any order, and to spaces $mathcalP_n$ that are not necessarily Euclidian (which is currently the case). The question is:
How does a scalar field and its differential structures, defined on $mathcalP_n$, are mapped into $mathcalP_3$ under a specific mapping $phi$?
Even if there is no known body of knowledge about this particular topic, I would be happy to be provided with references I could consult, videos I could watch, lectures, notes... Anything really that would, in the least, point me towards the particular direction of research in which the answer might be found, if it exists.
I want to know, more than a specific solution, how do I go about dealing with this kind of problem. What branch of mathematics does this? How do I describe these things? What is the formalism and the key ideas behind the correct tackling of this particular question?
Additionally, the idea is to have generalized solutions to the transformations, in terms of an unspecified $phi$ but in the actual application, $phi$ will be provided.
P.S. I realize that the text is a bit vague and potentially unclear. But in my head the idea is very clear, I just might need help putting it into paper in a way that we can math it up. So there might be a few edits going on as you people give feedback about what are your interpretations of the text before we can really zoom in on the actual issue.
Anyway, let the math begin. I'm going to grab a cup of coffee.
differential-geometry differential-topology smooth-manifolds
asked Jul 24 at 20:57
urquiza
1338
add a comment |Â
up vote
1
down vote
favorite
Let:
$mathcal P_n$ and $mathcal P_3$ be two manifolds isomorphic to the Euclidian Space, where $mathcal P_n$ is $n$-dimensional and, equivalentl,y, $mathcal P_3$ is $3$-dimensional.
$phi$ be a mapping such that $phi:mathcalP_n rightarrow mathcalP_3$. $phi$ is the Squishification mapping.
$p in mathcalP_n$ and $p' in mathcalP_3$.
$f$ be a real scalar field defined in $mathcalP_n$ such that $f:mathcalP_n rightarrow mathbbR$
Let $nabla f(p) in T_p mathcalP_n$ where $T_p mathcalP_n$ is the tangent space of $mathcalP_n$ at point $p$ and $nabla f(p)$ is the gradient of $f$ at $p$.
I don't have extensive training in math, so I am going to do a quick introduction to the problem which might contain mistakes and inconsistencies. Feel free to point it out.
Anyway, I am faced with a number of doubts that involves the tranformation of $f$ under $phi$. For example:
$mathcalP_n$ has a higher dimensionality than $mathcalP_3$, it is likely that more than one point in $mathcalP_n$ will be mapped to a point in $mathcalP_3$ under $phi$. Suppose that for $p,q in mathcalP_n$ we have that $phi(p)=p'$ and $phi(q)=p'$. How are the values $f(p)$ and $f(q)$ mapped into $mathcalP_3$?
Also, I am assuming that if $phi(p)=p'$ then $T_phi(p) mathcalP_3 = T_p' mathcalP_3$. Given these conditions, is it true that there will be a vector $u in T_p' mathcalP_3$ that will be "projection" of $nabla f(p)$ in the new tangent space? If so, how can I obtain this projection vector by knowing $phi$? in other words, how are the gradients of $f$ mapped into $mathcalP_3$? after it is squished?
The same problem involves Hessians. A Hessian will be a $ n times n$ matrix, where $n$ is the dimensionality of the space in which it is defined. How can I map a Hessian of $f$ defined at $p$ to a Hessian defined at $p'$ once $phi$ is applied?
Ideally I would like to generalize this to differential structures that includes derivatives of any order, and to spaces $mathcalP_n$ that are not necessarily Euclidian (which is currently the case). The question is:
How does a scalar field and its differential structures, defined on $mathcalP_n$, are mapped into $mathcalP_3$ under a specific mapping $phi$?
Even if there is no known body of knowledge about this particular topic, I would be happy to be provided with references I could consult, videos I could watch, lectures, notes... Anything really that would, in the least, point me towards the particular direction of research in which the answer might be found, if it exists.
I want to know, more than a specific solution, how do I go about dealing with this kind of problem. What branch of mathematics does this? How do I describe these things? What is the formalism and the key ideas behind the correct tackling of this particular question?
Additionally, the idea is to have generalized solutions to the transformations, in terms of an unspecified $phi$ but in the actual application, $phi$ will be provided.
P.S. I realize that the text is a bit vague and potentially unclear. But in my head the idea is very clear, I just might need help putting it into paper in a way that we can math it up. So there might be a few edits going on as you people give feedback about what are your interpretations of the text before we can really zoom in on the actual issue.
Anyway, let the math begin. I'm going to grab a cup of coffee.
differential-geometry differential-topology smooth-manifolds
asked Jul 24 at 20:57
urquiza
1338
1
There is no good way to map a scalar field on $mathcal P_n$ to a scalar field on $mathcal P_3$ unless your transformation function goes the other way, $phi : mathcal P_3 to mathcal P_n$, in which case you can define the new scalar field as the pullback $f circ phi$.
– Rahul
Jul 25 at 13:39
What would be "a good way"? Are there "not so good ways" of doing it if the transformation function doesn't go the other way? Also, does the transformation has to be bijective or just invertible?
– urquiza
Jul 25 at 15:10
add a comment |Â
up vote
1
down vote
favorite
up vote
1
down vote
favorite
Let:
$mathcal P_n$ and $mathcal P_3$ be two manifolds isomorphic to the Euclidian Space, where $mathcal P_n$ is $n$-dimensional and, equivalentl,y, $mathcal P_3$ is $3$-dimensional.
$phi$ be a mapping such that $phi:mathcalP_n rightarrow mathcalP_3$. $phi$ is the Squishification mapping.
$p in mathcalP_n$ and $p' in mathcalP_3$.
$f$ be a real scalar field defined in $mathcalP_n$ such that $f:mathcalP_n rightarrow mathbbR$
Let $nabla f(p) in T_p mathcalP_n$ where $T_p mathcalP_n$ is the tangent space of $mathcalP_n$ at point $p$ and $nabla f(p)$ is the gradient of $f$ at $p$.
I don't have extensive training in math, so I am going to do a quick introduction to the problem which might contain mistakes and inconsistencies. Feel free to point it out.
Anyway, I am faced with a number of doubts that involves the tranformation of $f$ under $phi$. For example:
$mathcalP_n$ has a higher dimensionality than $mathcalP_3$, it is likely that more than one point in $mathcalP_n$ will be mapped to a point in $mathcalP_3$ under $phi$. Suppose that for $p,q in mathcalP_n$ we have that $phi(p)=p'$ and $phi(q)=p'$. How are the values $f(p)$ and $f(q)$ mapped into $mathcalP_3$?
Also, I am assuming that if $phi(p)=p'$ then $T_phi(p) mathcalP_3 = T_p' mathcalP_3$. Given these conditions, is it true that there will be a vector $u in T_p' mathcalP_3$ that will be "projection" of $nabla f(p)$ in the new tangent space? If so, how can I obtain this projection vector by knowing $phi$? in other words, how are the gradients of $f$ mapped into $mathcalP_3$? after it is squished?
The same problem involves Hessians. A Hessian will be a $ n times n$ matrix, where $n$ is the dimensionality of the space in which it is defined. How can I map a Hessian of $f$ defined at $p$ to a Hessian defined at $p'$ once $phi$ is applied?
Ideally I would like to generalize this to differential structures that includes derivatives of any order, and to spaces $mathcalP_n$ that are not necessarily Euclidian (which is currently the case). The question is:
How does a scalar field and its differential structures, defined on $mathcalP_n$, are mapped into $mathcalP_3$ under a specific mapping $phi$?
Even if there is no known body of knowledge about this particular topic, I would be happy to be provided with references I could consult, videos I could watch, lectures, notes... Anything really that would, in the least, point me towards the particular direction of research in which the answer might be found, if it exists.
I want to know, more than a specific solution, how do I go about dealing with this kind of problem. What branch of mathematics does this? How do I describe these things? What is the formalism and the key ideas behind the correct tackling of this particular question?
Additionally, the idea is to have generalized solutions to the transformations, in terms of an unspecified $phi$ but in the actual application, $phi$ will be provided.
P.S. I realize that the text is a bit vague and potentially unclear. But in my head the idea is very clear, I just might need help putting it into paper in a way that we can math it up. So there might be a few edits going on as you people give feedback about what are your interpretations of the text before we can really zoom in on the actual issue.
Anyway, let the math begin. I'm going to grab a cup of coffee.
differential-geometry differential-topology smooth-manifolds
asked Jul 24 at 20:57
urquiza
1338
Let:
$mathcal P_n$ and $mathcal P_3$ be two manifolds isomorphic to the Euclidian Space, where $mathcal P_n$ is $n$-dimensional and, equivalentl,y, $mathcal P_3$ is $3$-dimensional.
$phi$ be a mapping such that $phi:mathcalP_n rightarrow mathcalP_3$. $phi$ is the Squishification mapping.
$p in mathcalP_n$ and $p' in mathcalP_3$.
$f$ be a real scalar field defined in $mathcalP_n$ such that $f:mathcalP_n rightarrow mathbbR$
Let $nabla f(p) in T_p mathcalP_n$ where $T_p mathcalP_n$ is the tangent space of $mathcalP_n$ at point $p$ and $nabla f(p)$ is the gradient of $f$ at $p$.
I don't have extensive training in math, so I am going to do a quick introduction to the problem which might contain mistakes and inconsistencies. Feel free to point it out.
Anyway, I am faced with a number of doubts that involves the tranformation of $f$ under $phi$. For example:
$mathcalP_n$ has a higher dimensionality than $mathcalP_3$, it is likely that more than one point in $mathcalP_n$ will be mapped to a point in $mathcalP_3$ under $phi$. Suppose that for $p,q in mathcalP_n$ we have that $phi(p)=p'$ and $phi(q)=p'$. How are the values $f(p)$ and $f(q)$ mapped into $mathcalP_3$?
Also, I am assuming that if $phi(p)=p'$ then $T_phi(p) mathcalP_3 = T_p' mathcalP_3$. Given these conditions, is it true that there will be a vector $u in T_p' mathcalP_3$ that will be "projection" of $nabla f(p)$ in the new tangent space? If so, how can I obtain this projection vector by knowing $phi$? in other words, how are the gradients of $f$ mapped into $mathcalP_3$? after it is squished?
The same problem involves Hessians. A Hessian will be a $ n times n$ matrix, where $n$ is the dimensionality of the space in which it is defined. How can I map a Hessian of $f$ defined at $p$ to a Hessian defined at $p'$ once $phi$ is applied?
Ideally I would like to generalize this to differential structures that includes derivatives of any order, and to spaces $mathcalP_n$ that are not necessarily Euclidian (which is currently the case). The question is:
How does a scalar field and its differential structures, defined on $mathcalP_n$, are mapped into $mathcalP_3$ under a specific mapping $phi$?
Even if there is no known body of knowledge about this particular topic, I would be happy to be provided with references I could consult, videos I could watch, lectures, notes... Anything really that would, in the least, point me towards the particular direction of research in which the answer might be found, if it exists.
I want to know, more than a specific solution, how do I go about dealing with this kind of problem. What branch of mathematics does this? How do I describe these things? What is the formalism and the key ideas behind the correct tackling of this particular question?
Additionally, the idea is to have generalized solutions to the transformations, in terms of an unspecified $phi$ but in the actual application, $phi$ will be provided.
P.S. I realize that the text is a bit vague and potentially unclear. But in my head the idea is very clear, I just might need help putting it into paper in a way that we can math it up. So there might be a few edits going on as you people give feedback about what are your interpretations of the text before we can really zoom in on the actual issue.
Anyway, let the math begin. I'm going to grab a cup of coffee.
differential-geometry differential-topology smooth-manifolds
asked Jul 24 at 20:57
urquiza
1338
edited Jul 25 at 13:21
asked Jul 24 at 20:57
urquiza
1338
asked Jul 24 at 20:57
urquiza
1338
asked Jul 24 at 20:57
urquiza
1338
1338
1
There is no good way to map a scalar field on $mathcal P_n$ to a scalar field on $mathcal P_3$ unless your transformation function goes the other way, $phi : mathcal P_3 to mathcal P_n$, in which case you can define the new scalar field as the pullback $f circ phi$.
– Rahul
Jul 25 at 13:39
What would be "a good way"? Are there "not so good ways" of doing it if the transformation function doesn't go the other way? Also, does the transformation has to be bijective or just invertible?
– urquiza
Jul 25 at 15:10
add a comment |Â
1
There is no good way to map a scalar field on $mathcal P_n$ to a scalar field on $mathcal P_3$ unless your transformation function goes the other way, $phi : mathcal P_3 to mathcal P_n$, in which case you can define the new scalar field as the pullback $f circ phi$.
– Rahul
Jul 25 at 13:39
What would be "a good way"? Are there "not so good ways" of doing it if the transformation function doesn't go the other way? Also, does the transformation has to be bijective or just invertible?
– urquiza
Jul 25 at 15:10
1
1
There is no good way to map a scalar field on $mathcal P_n$ to a scalar field on $mathcal P_3$ unless your transformation function goes the other way, $phi : mathcal P_3 to mathcal P_n$, in which case you can define the new scalar field as the pullback $f circ phi$.
– Rahul
Jul 25 at 13:39
There is no good way to map a scalar field on $mathcal P_n$ to a scalar field on $mathcal P_3$ unless your transformation function goes the other way, $phi : mathcal P_3 to mathcal P_n$, in which case you can define the new scalar field as the pullback $f circ phi$.
– Rahul
Jul 25 at 13:39
What would be "a good way"? Are there "not so good ways" of doing it if the transformation function doesn't go the other way? Also, does the transformation has to be bijective or just invertible?
– urquiza
Jul 25 at 15:10
What would be "a good way"? Are there "not so good ways" of doing it if the transformation function doesn't go the other way? Also, does the transformation has to be bijective or just invertible?
– urquiza
Jul 25 at 15:10
add a comment |Â
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1
There is no good way to map a scalar field on $mathcal P_n$ to a scalar field on $mathcal P_3$ unless your transformation function goes the other way, $phi : mathcal P_3 to mathcal P_n$, in which case you can define the new scalar field as the pullback $f circ phi$.
– Rahul
Jul 25 at 13:39
What would be "a good way"? Are there "not so good ways" of doing it if the transformation function doesn't go the other way? Also, does the transformation has to be bijective or just invertible?
– urquiza
Jul 25 at 15:10