differential geometry – coordinate transformation of the Laplace Beltrami Operator

My Laplace-Beltrami operator isn’t transforming correctly under a change of coordinates. What am I doing wrong?


The Laplace-Betrami operator has the following expressing in local coordinates of a Riemannian manifold $(M, g)$,
begin{align}
Delta f(x) = g^{ij}(x) frac{partial^2 f(x)}{partial x_i partial x_j} + g^{jk}Gamma^i_{jk} frac{partial f(x)}{partial x_i},
end{align}

where $Gamma^i_{jk}$ are Christoffel symbols. Written in this way, it is clear that the Laplace-Beltrami operator is a second-order elliptic operator on the manifold.

Many books such as (Markov Processes by Dynkin)(1) (page 151) or (Functional Analysis by Yosida)(2) (page 426) state that the coefficient functions of elliptic operators must transform in a prescribed way in order to give a consistent result on the manifold. Namely, consider an elliptic operator of the form (written in local coordinates $x$)
begin{align}
Af(x) = b^i(x) frac{partial f(x)}{partial x_i} + a^{ij}(x) frac{partial^2 f(x)}{partial x_ipartial x_j}.
end{align}

Then in another coordinate system $tilde{x}$ the coefficient functions transform as
begin{align}
tilde{b}^i(x) &= b^k(x) frac{partial tilde{x}_i}{partial x_k} + a^{kl}(x)frac{partial^2 tilde{x}_i}{partial x_kpartial x_l} \\
tilde{a}^{ij}(x) &= a^{kl}(x) frac{partial tilde{x}_i}{partial x_k}frac{partial tilde{x}_j}{partial x_l}.
end{align}

The Laplace-Beltrami operator should then obey this transformation rule with $a^{ij} =g^{ij}$ and $b^i = g^{jk}Gamma^i_{ij}$. The fact that $g^{ij}$ obeys the correct transformation rule is apparent; however, I am having a hard time seeing that $g^{jk}Gamma^i_{ij}$ obeys the correct transformation.


What I want to show is that
begin{align}
tilde{g}^{jk}tilde{Gamma}^i_{jk} = g^{pq}Gamma^k_{pq} frac{partial tilde{x}_i}{partial x_k} + g^{pq}frac{partial^2 tilde{x}_p}{partial x_kpartial x_q}
end{align}

I know that the transformation rule for the Christoffel symbols is as follows:
begin{align}
tilde{Gamma}^i_{jk} &= frac{partial x_p}{partialtilde{x_j}}frac{partial x_q}{partialtilde{x}_k} Gamma_{pq}^r frac{tilde{x}_i}{partial x_r} + frac{partial^2 x_r}{partial tilde{x}_jpartial tilde{x}_k} frac{partial tilde{x}_i}{partial x_r} \
&= frac{partial x_p}{partialtilde{x_j}}frac{partial x_q}{partialtilde{x}_k} Gamma_{pq}^r frac{tilde{x}_i}{partial x_r} – frac{partial x_p}{partial tilde{x}_j} frac{partial^2 tilde{x}_i}{partial x_ppartial x_q} frac{partial x_q}{partial tilde{x}_k}
end{align}

If I multiply both sides by $tilde{g}^{jk}$ and use the transformation law of the inverse metric in coordinates, I obtain,
begin{align}
tilde{g}^{jk} tilde{Gamma}^i_{jk} &= tilde{g}^{jk} frac{partial x_p}{partialtilde{x_j}}frac{partial x_q}{partialtilde{x}_k} Gamma_{pq}^r frac{tilde{x}_i}{partial x_r} – tilde{g}^{jk}frac{partial x_p}{partial tilde{x}_j} frac{partial^2 tilde{x}_i}{partial x_ppartial x_q} frac{partial x_q}{partial tilde{x}_k} \
&= g^{pq} Gamma_{pq}^r frac{tilde{x}_i}{partial x_r} – g^{pq} frac{partial^2 tilde{x}_i}{partial x_ppartial x_q}.
end{align}

This is very nearly what I wanted to show, but differs from the expected result by a negative sign in the second term. What have I done wrong?

(1) https://www.springer.com/gp/book/9783662000335
(2) https://www.springer.com/gp/book/9783540586548

Google Dataflow Job is unable to perform transformation step

Reference: https://cloud.google.com/composer/docs/how-to/using/using-dataflow-template-operator

Code: https://github.com/kolban-google/composer-dataflow

Submitted job using following script but the Transformation step doesn’t generate any data and just shows following error.

message: “Warning: Nashorn engine is planned to be removed from a future JDK release” step: “JavascriptTextTransformer.TransformTextViaJavascript/ParDo(Anonymous)”

As a result ‘Insert to Bigquery’ step is failing.

–region us-east4
–subnetwork regions/us-east4/subnetworks/test-subnetwork
–worker-zone us-east4-b
–disable-public-ips
–gcs-location gs://dataflow-templates/latest/GCS_Text_to_BigQuery
–parameters
javascriptTextTransformFunctionName=transformCSVtoJSON,
JSONPath=gs://test-dataflow-temp-bucket/sample/jsonSchema.json,
javascriptTextTransformGcsPath=gs://test-dataflow-temp-bucket/sample/transformCSVtoJSON.js,
inputFilePattern=gs://test-dataflow-temp-bucket/sample/inputFile.txt,
outputTable=test-project:average_weather.average_weather,
bigQueryLoadingTemporaryDirectory=gs://test-dataflow-temp-bucket/sample/tmp/“`

warning messages – What should I make of “Simplify: Time spent on a transformation exceeded -4.03955*10^12 seconds”?

Running version “12.0.0 for Microsoft Windows (64-bit) (April 6, 2019)”, in case that matters. I fed Mathematica the command

Simplify(
  Integrate(
    e*Exp(-e*t)*F*Exp(-F *t)* Exp(-(Lambda) - b*t)*
    Integrate(
      Sum((b*s + (Lambda))^y/y! *
        Sum(2^(-y)*Binomial(y, z)*(b*(t - s))^(x - z)/(x - z)!, {z, 0, x}),
      {y, 0, Infinity}),
    {s, 0, t}),
  {t, 0, Infinity}),
e > 0 && F > 0 && (Lambda) > 0 && b > 0 && Element(x, Integers))

(line breaks added for readability, though I don’t think that matters). It output the warning message

Simplify::time: Time spent on a transformation exceeded -4.03955*10^12 seconds, and the transformation was aborted. Increasing the value of TimeConstraint option may improve the result of simplification.

In fact, it gave me that message three times, and then

General::stop: Further output of Simplify::time will be suppressed during this calculation.

Twelve hours after that, it gave me an answer. The warning bugs me for a couple of reasons. First, 4.03955*10^12 seconds is thousands of years. Second, I don’t know what to make of the negative number of seconds. Third, when I Googled I didn’t see any results about this. (The only results on this site that I found are all of the form “Time spent on a transformation exceeded 300 seconds”, which is rather different.)

The number -4.03955*10^12 makes me think there’s some type of overflow error going on, but I’ve got no idea other than that.

dynamical systems – Canonical transformation and generating function

I am stuck on an exercise of which I do not understand the solution. The exercise is the following:

Consider the Hamiltonian system
$$H(theta, p, t) = dfrac{(p-omega t)^2}{2} – kcos(theta)$$
Find a generating function for a canonical transformation that completes the relation
$$P = p-omega t$$
and calculate the new Hamiltonian and find $k$ such that an elliptic equilibrium point does exit.

Solution

A generating function of second type can be written as

$$F(theta, P) = theta(P + omega t) ~~~~~~~~~~~~~~~~~ textbf{why??}$$

and the hew Hamiltonian becomes

$$H(phi, P) = dfrac{P^2}{2} – kcosphi + omega phi ~~~~~~~~~~~~~ textbf{why??}$$

The fixed point can be found by solving the equations

$$P = 0 ~~~~~~~~~~~~~ -ksinphi = omega ~~~~~~~~~~~~~~~~~ textbf{why??}$$

hence the condition for an elliptic point is $-1 < omega / k < 1$.

I really need some calrifications to understand those points…

Thank you!

ordinary differential equations – Solving ODE using spatial transformation

Could you please help me with how to solve ODE’s using linear spatial transformation.

Theory

Consider the differential equation of the form: $$ y’=X(t, y) $$

Then my textbook says I can transform this ODE with a linear spatial
transformation of the form: $$ Psi_{t}(mathrm{y})=Phi(t)
mathrm{y}+g(t) $$
such that the transformed ODE becomes $$
mathrm{y}’=X_{Psi}(t, mathrm{y}) $$
with $$ X_{Psi}(t,
mathrm{y})=Phi(t)^{-1}(-Phi'(t) mathrm{y}-g'(t)+X(t, Psi_{t}(mathrm{y}))) $$

And likewise I can use this transformation to transform between the solutions to each ODE (I know I have to be careful about the region, when using transformations, but that’s not relevant for my question.)

Problem

I have already found solutions to
$$
y’=frac{1}{2 y}
$$

I’m now being asked to solve the following ODE
$$y’=frac{y}{t+1}+frac{t+1}{2 y}$$

As a hint, I’m being asked to use the transformation $Psi_{t}(y)=(t+1) y$.

Solution attempt

My issue is that I don’t seem to arrive at the correct equation using this transformation. Looking at the hint, gives me $Phi(t)=t+1, quad g(t)=0$. And using the above formula, would then give me:

$$ X_{Psi}(t,
mathrm{y})=Phi(t)^{-1}(-Phi'(t) mathrm{y}-g'(t)+X(t, Psi_{t}(mathrm{y}))) =frac{1}{t+1}left(-y+frac{1}{2(t+1) y}right)$$

$$=frac{1}{2(t+1)^{2} y}-frac{y}{t+1}$$
Which isn’t quite equal to $frac{y}{t+1}+frac{t+1}{2 y}$ as far as I can tell.
So can you see what I’m doing wrong here?
I believe I know how to find the correct solutions If I manage to find a suitable transformation that transforms the original ODE into the new one.

coordinate transformation – How to troubleshoot TransformedField functionality in Mathematica?

I list up this method to transform a complex function to Cartestian form, which can be used on virtually any function:

such as:

u0(r_, phi_) := Sum(I^(-n) BesselJ(n, r) Exp(I n phi), {n, -5, 5});

TransformedField("Polar" -> "Cartesian",  u0(r, phi), {r, phi} -> {x, y})

which yields:

  u0(x_, y_) := 
 BesselJ(0, Sqrt(x^2 + y^2)) + 
  1/(x^2 + y^2)^(5/2)
    2 (-I x (x^2 + y^2)^2 BesselJ(1, Sqrt(x^2 + y^2)) + 
      Sqrt(x^2 + y^2) (-x^4 + y^4) BesselJ(2, Sqrt(x^2 + y^2)) + 
      I x^5 BesselJ(3, Sqrt(x^2 + y^2)) - 
      2 I x^3 y^2 BesselJ(3, Sqrt(x^2 + y^2)) - 
      3 I x y^4 BesselJ(3, Sqrt(x^2 + y^2)) + 
      x^4 Sqrt(x^2 + y^2) BesselJ(4, Sqrt(x^2 + y^2)) - 
      6 x^2 y^2 Sqrt(x^2 + y^2) BesselJ(4, Sqrt(x^2 + y^2)) + 
      y^4 Sqrt(x^2 + y^2) BesselJ(4, Sqrt(x^2 + y^2)) - 
      I x^5 BesselJ(5, Sqrt(x^2 + y^2)) + 
      10 I x^3 y^2 BesselJ(5, Sqrt(x^2 + y^2)) - 
      5 I x y^4 BesselJ(5, Sqrt(x^2 + y^2)))

or

a Hankel and Bessel function together:

u(r_, phi_) :=  Piecewise({{BesselJ(1.5 r, 5)*Exp(I 5 phi), 
0 < r < 1/2}, {(BesselJ(3 r, 5) + BesselY(3 r, 5))*Exp(I 5 phi), 
1/2 < r < 1}, {HankelH1(r, 5)*Exp(I 5 phi), r > 1}})

which yields:

Hankel+Bessel

which yields the respective given plots, when plotted:

Plot_Bessel_Cartesian

and

Plot_Hankel+Bessel_Cartesian

However, my supervisor thinks these plots look “strange”.

How can I verify that the TransformedField command did the right job – for such extended functions?

In other words, how to trobleshoot TransformedField?

performance – Parallel 3D Discrete Cosine Transformation Implementation in Matlab

I am trying to implement 3D Discrete Cosine Transformation calculation in Matlab with parallel computing parfor. The formula of 3D Discrete Cosine Transformation is as follows.

The experimental implementation

The experimental implementation of 3D Discrete Cosine Transformation function is DCT3D.

function X=DCT3D(x)

N1=size(x,1);
N2=size(x,2);
N3=size(x,3);
X=zeros(N1,N2,N3);

for k1=0:N1-1
    for k2=0:N2-1
        for k3=0:N3-1
            sumResult=0;            
            parfor n1=0:N1-1
                for n2=0:N2-1
                    for n3=0:N3-1
                        sumResult=sumResult+...
                            x(n1+1,n2+1,n3+1)*...
                            cos(pi/(2*N1)*(2*n1+1)*k1)*...
                            cos(pi/(2*N2)*(2*n2+1)*k2)*...
                            cos(pi/(2*N3)*(2*n3+1)*k3);
                    end
                end
            end
            X(k1+1,k2+1,k3+1)=8*sumResult*CalculateK(k1)*CalculateK(k2)*CalculateK(k3)/(N1*N2*N3);            
        end
    end
end

Moreover, the used function CalculateK:

function output = CalculateK(x)
  output = ones(size(x));
  output(x==0) = 1 / sqrt(2);

Test cases

%% Create test cells
testCellsSize = 10;

test = zeros(testCellsSize, testCellsSize, testCellsSize);
for x = 1:size(test, 1)
    for y = 1:size(test, 2)
        for z = 1:size(test, 3)
            test(x, y, z) = x * 100 + y * 10 + z;
        end
    end
end

%% Perform test
result = DCT3D(test);

%% Print output
for z = 1:size(result, 3)
    fprintf('3D DCT result: %d planen' , z);
    for x = 1:size(result, 1)
        for y = 1:size(result, 2)
            fprintf('%ft' , result(x, y, z));
        end
        fprintf('n');
    end
    fprintf('n');
end

The output of the test code above:

3D DCT result: 1 plane
1726.754760 -80.720722  -0.000000   -8.646042   -0.000000   -2.828427   0.000000    -1.143708   -0.000000   -0.320717   
-807.207224 0.000000    0.000000    -0.000000   0.000000    -0.000000   -0.000000   -0.000000   0.000000    -0.000000   
-0.000000   -0.000000   0.000000    0.000000    0.000000    -0.000000   0.000000    -0.000000   0.000000    -0.000000   
-86.460422  -0.000000   0.000000    -0.000000   -0.000000   -0.000000   0.000000    -0.000000   0.000000    -0.000000   
-0.000000   -0.000000   0.000000    0.000000    -0.000000   0.000000    -0.000000   -0.000000   -0.000000   -0.000000   
-28.284271  -0.000000   0.000000    -0.000000   -0.000000   0.000000    0.000000    -0.000000   -0.000000   0.000000    
0.000000    0.000000    -0.000000   0.000000    -0.000000   0.000000    0.000000    -0.000000   0.000000    0.000000    
-11.437076  -0.000000   0.000000    0.000000    -0.000000   0.000000    -0.000000   -0.000000   0.000000    -0.000000   
-0.000000   0.000000    0.000000    -0.000000   -0.000000   0.000000    0.000000    -0.000000   -0.000000   0.000000    
-3.207174   -0.000000   0.000000    -0.000000   0.000000    0.000000    -0.000000   -0.000000   0.000000    0.000000    

3D DCT result: 2 plane
-8.072072   0.000000    -0.000000   0.000000    -0.000000   0.000000    -0.000000   0.000000    0.000000    -0.000000   
-0.000000   -0.000000   -0.000000   -0.000000   -0.000000   -0.000000   -0.000000   -0.000000   0.000000    -0.000000   
-0.000000   0.000000    -0.000000   0.000000    -0.000000   0.000000    0.000000    0.000000    -0.000000   0.000000    
-0.000000   -0.000000   -0.000000   0.000000    -0.000000   -0.000000   -0.000000   0.000000    0.000000    -0.000000   
-0.000000   -0.000000   0.000000    -0.000000   -0.000000   0.000000    -0.000000   0.000000    0.000000    -0.000000   
-0.000000   0.000000    -0.000000   0.000000    0.000000    0.000000    0.000000    -0.000000   -0.000000   -0.000000   
0.000000    0.000000    -0.000000   -0.000000   0.000000    -0.000000   0.000000    -0.000000   -0.000000   0.000000    
-0.000000   0.000000    0.000000    -0.000000   0.000000    0.000000    0.000000    -0.000000   -0.000000   -0.000000   
0.000000    0.000000    0.000000    0.000000    0.000000    0.000000    0.000000    -0.000000   -0.000000   0.000000    
0.000000    -0.000000   -0.000000   0.000000    0.000000    0.000000    -0.000000   0.000000    -0.000000   0.000000    

3D DCT result: 3 plane
-0.000000   0.000000    0.000000    -0.000000   -0.000000   0.000000    -0.000000   -0.000000   0.000000    0.000000    
0.000000    -0.000000   -0.000000   -0.000000   0.000000    0.000000    -0.000000   -0.000000   0.000000    0.000000    
-0.000000   -0.000000   0.000000    -0.000000   0.000000    0.000000    0.000000    0.000000    -0.000000   0.000000    
0.000000    -0.000000   0.000000    -0.000000   0.000000    0.000000    0.000000    0.000000    -0.000000   0.000000    
0.000000    0.000000    0.000000    -0.000000   -0.000000   -0.000000   0.000000    0.000000    -0.000000   -0.000000   
-0.000000   0.000000    0.000000    -0.000000   0.000000    0.000000    -0.000000   -0.000000   -0.000000   -0.000000   
-0.000000   -0.000000   -0.000000   0.000000    0.000000    0.000000    0.000000    -0.000000   -0.000000   0.000000    
0.000000    0.000000    -0.000000   -0.000000   0.000000    -0.000000   -0.000000   0.000000    0.000000    0.000000    
-0.000000   0.000000    0.000000    -0.000000   -0.000000   0.000000    -0.000000   -0.000000   0.000000    0.000000    
0.000000    0.000000    -0.000000   -0.000000   -0.000000   0.000000    0.000000    0.000000    0.000000    0.000000    

3D DCT result: 4 plane
-0.864604   -0.000000   -0.000000   -0.000000   -0.000000   -0.000000   0.000000    0.000000    0.000000    -0.000000   
-0.000000   -0.000000   -0.000000   0.000000    0.000000    -0.000000   -0.000000   -0.000000   0.000000    -0.000000   
-0.000000   -0.000000   -0.000000   -0.000000   0.000000    0.000000    0.000000    0.000000    -0.000000   0.000000    
-0.000000   0.000000    0.000000    -0.000000   -0.000000   0.000000    0.000000    -0.000000   -0.000000   -0.000000   
-0.000000   0.000000    -0.000000   0.000000    0.000000    -0.000000   0.000000    0.000000    -0.000000   0.000000    
-0.000000   0.000000    -0.000000   0.000000    0.000000    -0.000000   0.000000    -0.000000   0.000000    -0.000000   
-0.000000   0.000000    -0.000000   0.000000    -0.000000   0.000000    -0.000000   0.000000    -0.000000   0.000000    
0.000000    -0.000000   0.000000    0.000000    -0.000000   -0.000000   -0.000000   0.000000    0.000000    0.000000    
-0.000000   -0.000000   -0.000000   -0.000000   -0.000000   0.000000    -0.000000   -0.000000   0.000000    0.000000    
0.000000    -0.000000   0.000000    -0.000000   -0.000000   -0.000000   -0.000000   0.000000    0.000000    -0.000000   

3D DCT result: 5 plane
-0.000000   0.000000    0.000000    0.000000    0.000000    0.000000    0.000000    -0.000000   -0.000000   0.000000    
0.000000    -0.000000   -0.000000   -0.000000   -0.000000   0.000000    0.000000    0.000000    0.000000    0.000000    
-0.000000   0.000000    -0.000000   -0.000000   -0.000000   -0.000000   -0.000000   -0.000000   0.000000    0.000000    
0.000000    -0.000000   -0.000000   -0.000000   0.000000    -0.000000   0.000000    0.000000    -0.000000   0.000000    
0.000000    0.000000    0.000000    -0.000000   -0.000000   0.000000    0.000000    0.000000    0.000000    -0.000000   
0.000000    -0.000000   -0.000000   -0.000000   0.000000    -0.000000   -0.000000   -0.000000   0.000000    0.000000    
0.000000    0.000000    -0.000000   -0.000000   0.000000    -0.000000   -0.000000   0.000000    0.000000    -0.000000   
-0.000000   -0.000000   0.000000    0.000000    -0.000000   0.000000    0.000000    0.000000    -0.000000   0.000000    
0.000000    0.000000    -0.000000   0.000000    0.000000    -0.000000   -0.000000   -0.000000   -0.000000   -0.000000   
-0.000000   0.000000    0.000000    0.000000    -0.000000   -0.000000   0.000000    -0.000000   0.000000    -0.000000   

3D DCT result: 6 plane
-0.282843   -0.000000   -0.000000   -0.000000   0.000000    -0.000000   -0.000000   -0.000000   0.000000    -0.000000   
-0.000000   0.000000    0.000000    0.000000    -0.000000   -0.000000   -0.000000   0.000000    0.000000    0.000000    
-0.000000   0.000000    0.000000    -0.000000   0.000000    -0.000000   -0.000000   -0.000000   0.000000    -0.000000   
-0.000000   -0.000000   0.000000    -0.000000   -0.000000   -0.000000   -0.000000   0.000000    0.000000    0.000000    
-0.000000   0.000000    0.000000    0.000000    -0.000000   0.000000    -0.000000   -0.000000   -0.000000   0.000000    
-0.000000   0.000000    -0.000000   -0.000000   -0.000000   0.000000    0.000000    0.000000    -0.000000   0.000000    
0.000000    -0.000000   -0.000000   0.000000    -0.000000   0.000000    -0.000000   0.000000    -0.000000   0.000000    
0.000000    -0.000000   -0.000000   0.000000    -0.000000   0.000000    -0.000000   0.000000    0.000000    -0.000000   
0.000000    -0.000000   0.000000    0.000000    0.000000    0.000000    -0.000000   -0.000000   0.000000    0.000000    
0.000000    -0.000000   0.000000    -0.000000   0.000000    -0.000000   -0.000000   -0.000000   0.000000    -0.000000   

3D DCT result: 7 plane
0.000000    -0.000000   0.000000    -0.000000   -0.000000   -0.000000   -0.000000   -0.000000   0.000000    -0.000000   
-0.000000   -0.000000   0.000000    0.000000    0.000000    0.000000    0.000000    -0.000000   0.000000    0.000000    
0.000000    -0.000000   -0.000000   0.000000    -0.000000   -0.000000   0.000000    0.000000    -0.000000   -0.000000   
-0.000000   -0.000000   -0.000000   0.000000    0.000000    -0.000000   -0.000000   -0.000000   -0.000000   -0.000000   
0.000000    0.000000    -0.000000   0.000000    0.000000    0.000000    0.000000    0.000000    -0.000000   -0.000000   
-0.000000   0.000000    0.000000    -0.000000   0.000000    0.000000    0.000000    -0.000000   -0.000000   -0.000000   
-0.000000   -0.000000   0.000000    0.000000    0.000000    0.000000    -0.000000   0.000000    -0.000000   -0.000000   
0.000000    -0.000000   0.000000    0.000000    0.000000    0.000000    -0.000000   -0.000000   0.000000    0.000000    
0.000000    -0.000000   -0.000000   0.000000    0.000000    0.000000    0.000000    -0.000000   0.000000    -0.000000   
-0.000000   -0.000000   -0.000000   -0.000000   0.000000    0.000000    0.000000    -0.000000   0.000000    0.000000    

3D DCT result: 8 plane
-0.114371   -0.000000   -0.000000   0.000000    -0.000000   -0.000000   -0.000000   -0.000000   -0.000000   -0.000000   
-0.000000   -0.000000   -0.000000   0.000000    -0.000000   -0.000000   0.000000    0.000000    0.000000    -0.000000   
-0.000000   -0.000000   0.000000    0.000000    0.000000    -0.000000   0.000000    -0.000000   0.000000    0.000000    
-0.000000   0.000000    0.000000    -0.000000   -0.000000   0.000000    -0.000000   -0.000000   -0.000000   0.000000    
0.000000    -0.000000   -0.000000   -0.000000   0.000000    0.000000    -0.000000   -0.000000   -0.000000   -0.000000   
-0.000000   0.000000    0.000000    0.000000    -0.000000   -0.000000   0.000000    -0.000000   -0.000000   0.000000    
-0.000000   0.000000    0.000000    -0.000000   0.000000    -0.000000   0.000000    -0.000000   0.000000    0.000000    
-0.000000   0.000000    -0.000000   0.000000    -0.000000   -0.000000   -0.000000   0.000000    -0.000000   -0.000000   
-0.000000   0.000000    0.000000    -0.000000   -0.000000   -0.000000   -0.000000   -0.000000   -0.000000   0.000000    
0.000000    -0.000000   0.000000    0.000000    0.000000    -0.000000   -0.000000   0.000000    0.000000    0.000000    

3D DCT result: 9 plane
-0.000000   0.000000    0.000000    0.000000    -0.000000   0.000000    0.000000    0.000000    -0.000000   -0.000000   
0.000000    0.000000    0.000000    0.000000    -0.000000   0.000000    -0.000000   0.000000    0.000000    0.000000    
0.000000    -0.000000   0.000000    -0.000000   0.000000    -0.000000   0.000000    0.000000    -0.000000   -0.000000   
0.000000    0.000000    0.000000    -0.000000   -0.000000   0.000000    -0.000000   -0.000000   0.000000    -0.000000   
-0.000000   0.000000    0.000000    0.000000    -0.000000   -0.000000   0.000000    -0.000000   0.000000    -0.000000   
0.000000    -0.000000   -0.000000   0.000000    -0.000000   -0.000000   -0.000000   0.000000    0.000000    0.000000    
-0.000000   0.000000    0.000000    -0.000000   -0.000000   -0.000000   0.000000    -0.000000   0.000000    -0.000000   
0.000000    0.000000    0.000000    0.000000    0.000000    -0.000000   0.000000    -0.000000   0.000000    -0.000000   
-0.000000   -0.000000   0.000000    0.000000    -0.000000   0.000000    -0.000000   0.000000    -0.000000   -0.000000   
0.000000    -0.000000   -0.000000   -0.000000   -0.000000   -0.000000   0.000000    0.000000    0.000000    0.000000    

3D DCT result: 10 plane
-0.032072   -0.000000   0.000000    -0.000000   -0.000000   0.000000    0.000000    0.000000    -0.000000   0.000000    
-0.000000   -0.000000   -0.000000   0.000000    -0.000000   -0.000000   0.000000    -0.000000   -0.000000   0.000000    
0.000000    0.000000    0.000000    -0.000000   -0.000000   0.000000    0.000000    0.000000    0.000000    0.000000    
-0.000000   0.000000    0.000000    -0.000000   -0.000000   0.000000    0.000000    -0.000000   0.000000    0.000000    
-0.000000   -0.000000   -0.000000   0.000000    0.000000    0.000000    -0.000000   -0.000000   -0.000000   0.000000    
-0.000000   -0.000000   0.000000    -0.000000   -0.000000   -0.000000   -0.000000   -0.000000   -0.000000   -0.000000   
0.000000    0.000000    -0.000000   -0.000000   0.000000    -0.000000   0.000000    0.000000    -0.000000   -0.000000   
-0.000000   0.000000    -0.000000   -0.000000   0.000000    -0.000000   0.000000    -0.000000   -0.000000   0.000000    
0.000000    -0.000000   0.000000    -0.000000   0.000000    0.000000    0.000000    -0.000000   -0.000000   0.000000    
-0.000000   0.000000    -0.000000   0.000000    0.000000    0.000000    0.000000    0.000000    0.000000    0.000000    

If there is any possible improvement, please let me know.

riemannian geometry – The stereographic projection is a conformal transformation.

The stereographic projection $sigma :mathbb S^n(R) setminus{N} to mathbb R^n$ is a conformal transformation .


To prove this theorem, we should show that for all $vin T_qmathbb R^n$ ($qin mathbb R^n $), we have $$langle T_qsigma^{-1}(v),T_qsigma^{-1}(v) rangle=flangle v,vrangle_0,$$
with $langle,rangle$ the canonical riemannian metric in $mathbb S_R^n$, $langle,rangle_0$ the Euclidean metric in $mathbb R^n$ and $sigma^{-1}$ is defined by $$sigma^{-1}(u,v)=(xi_1,cdotsxi_n,tau)=left(frac{2R^2u}{|u|^2+R^2},Rfrac{|u|^2-R^2}{|u|^2+R^2}right)$$

Let $displaystyle v=sum_{i=1}^n v_i frac{partial }{partial u_i}$, so $$T_qsigma^{-1}(v)=sum_{i=1}^nv(xi_i) frac{partial}{ partial xi_i}+v(tau) frac{partial}{partial tau}$$
I don’t know where this relation comes from !! I can prove the result directly by using the properties of pullback but my teacher use this one in the course.

Any help is highly appreciated !

real analysis – Proving Polar Transformation is Exists and is Unique

I am attempting a problem in the book Analysis on Manifolds that wants me to prove a form is closed but not unique. I have done the closed part, but for uniqueness following the steps in the problem, I am being asked to show for any $(x,y)in mathbb{R}-{z,0}$ there $zleq 0$, there exists a unqiue $tin (0,2pi)$ such that:

$$x=sqrt{x^2+y^2}cos(t)$$

$$y=sqrt{x^2+y^2}sin(t)$$

This is clearly polar coordinates, but how exactly would I show that such a $t$ value both exists and is unique?

real analysis – Values that make the transformation one to one

Find the preimage $Omega^{‘}$ of the set $Omega$ given by the canonical spherical coordinates (R$Phi$$Theta$) and find the values of $r, phi$ and $theta$ such that the transformation is one to one.

$$Omega = Big { (x,y,z): x^2+y^2 + z^2 le 2z space ; , ; z ge 3 sqrt{x^2+y^2} Big }$$

I tried to find the values for $phi$ and $r$ but got $phi$ in terms of $tan^{-1}(frac{1}{3})$ and im not really sure this is right.