Volume of $M:=big(x,y,z)in mathbbR^3,:,ax^2+2bxy+cy^2leq z leq 3big$ [closed]

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Let beginpmatrix a & b \ b & c endpmatrix be a positive definite matrix. How can I calculate the volume of $$M:=big(x,y,z)in mathbb R^3,:,ax^2+2bxy+cy^2leq z leq 3big,?$$

I am not sure whe the property of the matrix comes into play...

real-analysis linear-algebra volume bilinear-form positive-definite

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edited Jul 30 at 16:17

asked Jul 30 at 15:53

Marc

676421

closed as off-topic by amWhy, Mostafa Ayaz, Isaac Browne, Rhys Steele, José Carlos Santos Jul 30 at 22:33

This question appears to be off-topic. The users who voted to close gave this specific reason:

• "This question is missing context or other details: Please improve the question by providing additional context, which ideally includes your thoughts on the problem and any attempts you have made to solve it. This information helps others identify where you have difficulties and helps them write answers appropriate to your experience level." – amWhy, Mostafa Ayaz, Isaac Browne, Rhys Steele, José Carlos Santos

If this question can be reworded to fit the rules in the help center, please edit the question.

• 
  
  
  

  
  1
  

  
  
  
  
  
  

  
  The PD property is telling you that $alpha^T A alpha > 0$, where $A$ is the matrix you wrote and $alpha = [x y ]^T$. Notice that $alpha^T A alpha = ax^2 + 2bxy + cy^2$
  – Ahmad Bazzi
  Jul 30 at 15:57
  
  

  
  
  
  


• 
  
  
  

  
  2
  

  
  
  
  
  
  

  
  Did you mean $beginbmatrixa&b\b&cendbmatrix$ instead? You have one unused variable $d$.
  – Batominovski
  Jul 30 at 16:13
  
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  Yes, I edited my post. Thank you!
  – Marc
  Jul 30 at 16:14
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  Thanks :) my bad...
  – Marc
  Jul 30 at 16:18
  

  
  
  
  

add a comment | 

up vote
1
down vote

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2

Let beginpmatrix a & b \ b & c endpmatrix be a positive definite matrix. How can I calculate the volume of $$M:=big(x,y,z)in mathbb R^3,:,ax^2+2bxy+cy^2leq z leq 3big,?$$

I am not sure whe the property of the matrix comes into play...

real-analysis linear-algebra volume bilinear-form positive-definite

share|cite|improve this question

edited Jul 30 at 16:17

asked Jul 30 at 15:53

Marc

676421

closed as off-topic by amWhy, Mostafa Ayaz, Isaac Browne, Rhys Steele, José Carlos Santos Jul 30 at 22:33

This question appears to be off-topic. The users who voted to close gave this specific reason:

• "This question is missing context or other details: Please improve the question by providing additional context, which ideally includes your thoughts on the problem and any attempts you have made to solve it. This information helps others identify where you have difficulties and helps them write answers appropriate to your experience level." – amWhy, Mostafa Ayaz, Isaac Browne, Rhys Steele, José Carlos Santos

If this question can be reworded to fit the rules in the help center, please edit the question.

• 
  
  
  

  
  1
  

  
  
  
  
  
  

  
  The PD property is telling you that $alpha^T A alpha > 0$, where $A$ is the matrix you wrote and $alpha = [x y ]^T$. Notice that $alpha^T A alpha = ax^2 + 2bxy + cy^2$
  – Ahmad Bazzi
  Jul 30 at 15:57
  
  

  
  
  
  


• 
  
  
  

  
  2
  

  
  
  
  
  
  

  
  Did you mean $beginbmatrixa&b\b&cendbmatrix$ instead? You have one unused variable $d$.
  – Batominovski
  Jul 30 at 16:13
  
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  Yes, I edited my post. Thank you!
  – Marc
  Jul 30 at 16:14
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  Thanks :) my bad...
  – Marc
  Jul 30 at 16:18
  

  
  
  
  

add a comment | 

up vote
1
down vote

favorite

2

up vote
1
down vote

favorite

2

2

Let beginpmatrix a & b \ b & c endpmatrix be a positive definite matrix. How can I calculate the volume of $$M:=big(x,y,z)in mathbb R^3,:,ax^2+2bxy+cy^2leq z leq 3big,?$$

I am not sure whe the property of the matrix comes into play...

real-analysis linear-algebra volume bilinear-form positive-definite

share|cite|improve this question

edited Jul 30 at 16:17

asked Jul 30 at 15:53

Marc

676421

Let beginpmatrix a & b \ b & c endpmatrix be a positive definite matrix. How can I calculate the volume of $$M:=big(x,y,z)in mathbb R^3,:,ax^2+2bxy+cy^2leq z leq 3big,?$$

I am not sure whe the property of the matrix comes into play...

real-analysis linear-algebra volume bilinear-form positive-definite

share|cite|improve this question

edited Jul 30 at 16:17

asked Jul 30 at 15:53

Marc

676421

share|cite|improve this question

share|cite|improve this question

edited Jul 30 at 16:17

edited Jul 30 at 16:17

edited Jul 30 at 16:17

asked Jul 30 at 15:53

Marc

676421

asked Jul 30 at 15:53

Marc

676421

asked Jul 30 at 15:53

Marc

676421

676421

closed as off-topic by amWhy, Mostafa Ayaz, Isaac Browne, Rhys Steele, José Carlos Santos Jul 30 at 22:33

This question appears to be off-topic. The users who voted to close gave this specific reason:

• "This question is missing context or other details: Please improve the question by providing additional context, which ideally includes your thoughts on the problem and any attempts you have made to solve it. This information helps others identify where you have difficulties and helps them write answers appropriate to your experience level." – amWhy, Mostafa Ayaz, Isaac Browne, Rhys Steele, José Carlos Santos

If this question can be reworded to fit the rules in the help center, please edit the question.

closed as off-topic by amWhy, Mostafa Ayaz, Isaac Browne, Rhys Steele, José Carlos Santos Jul 30 at 22:33

This question appears to be off-topic. The users who voted to close gave this specific reason:

• "This question is missing context or other details: Please improve the question by providing additional context, which ideally includes your thoughts on the problem and any attempts you have made to solve it. This information helps others identify where you have difficulties and helps them write answers appropriate to your experience level." – amWhy, Mostafa Ayaz, Isaac Browne, Rhys Steele, José Carlos Santos

If this question can be reworded to fit the rules in the help center, please edit the question.

• 
  
  
  

  
  1
  

  
  
  
  
  
  

  
  The PD property is telling you that $alpha^T A alpha > 0$, where $A$ is the matrix you wrote and $alpha = [x y ]^T$. Notice that $alpha^T A alpha = ax^2 + 2bxy + cy^2$
  – Ahmad Bazzi
  Jul 30 at 15:57
  
  

  
  
  
  


• 
  
  
  

  
  2
  

  
  
  
  
  
  

  
  Did you mean $beginbmatrixa&b\b&cendbmatrix$ instead? You have one unused variable $d$.
  – Batominovski
  Jul 30 at 16:13
  
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  Yes, I edited my post. Thank you!
  – Marc
  Jul 30 at 16:14
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  Thanks :) my bad...
  – Marc
  Jul 30 at 16:18
  

  
  
  
  

add a comment | 

• 
  
  
  

  
  1
  

  
  
  
  
  
  

  
  The PD property is telling you that $alpha^T A alpha > 0$, where $A$ is the matrix you wrote and $alpha = [x y ]^T$. Notice that $alpha^T A alpha = ax^2 + 2bxy + cy^2$
  – Ahmad Bazzi
  Jul 30 at 15:57
  
  

  
  
  
  


• 
  
  
  

  
  2
  

  
  
  
  
  
  

  
  Did you mean $beginbmatrixa&b\b&cendbmatrix$ instead? You have one unused variable $d$.
  – Batominovski
  Jul 30 at 16:13
  
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  Yes, I edited my post. Thank you!
  – Marc
  Jul 30 at 16:14
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  Thanks :) my bad...
  – Marc
  Jul 30 at 16:18
  

  
  
  
  

1

1

The PD property is telling you that $alpha^T A alpha > 0$, where $A$ is the matrix you wrote and $alpha = [x y ]^T$. Notice that $alpha^T A alpha = ax^2 + 2bxy + cy^2$
– Ahmad Bazzi
Jul 30 at 15:57

The PD property is telling you that $alpha^T A alpha > 0$, where $A$ is the matrix you wrote and $alpha = [x y ]^T$. Notice that $alpha^T A alpha = ax^2 + 2bxy + cy^2$
– Ahmad Bazzi
Jul 30 at 15:57

2

2

Did you mean $beginbmatrixa&b\b&cendbmatrix$ instead? You have one unused variable $d$.
– Batominovski
Jul 30 at 16:13

Did you mean $beginbmatrixa&b\b&cendbmatrix$ instead? You have one unused variable $d$.
– Batominovski
Jul 30 at 16:13

Yes, I edited my post. Thank you!
– Marc
Jul 30 at 16:14

Yes, I edited my post. Thank you!
– Marc
Jul 30 at 16:14

Thanks :) my bad...
– Marc
Jul 30 at 16:18

Thanks :) my bad...
– Marc
Jul 30 at 16:18

add a comment | 

2 Answers
2

active

oldest

votes

up vote
4
down vote



accepted

$ax^2+2bxy + cy^2 = beginbmatrix x&yendbmatrixbeginbmatrix a&b\b&cendbmatrixbeginbmatrix x\yendbmatrix le z le 3 $

Compare this matrix with what you show above.

M is an elliptical paraboloid.

Since this matrix is symmetric, it can be daigonalized with ortho-normal basis.

$mathbf x^T P^T Lambda P mathbf x le z le 3$

and since P is ortho-normal it doesn't distort distances (or volumes), and we can say

$mathbf x^T Lambda mathbf x le z le 3$

or

$lambda_1 x^2 + lambda_2 y^2 le z$

Integrating in polar coordinates:

$int_0^2piint_0^3int_0^sqrt z frac rsqrtlambda_1lambda_2 dr dz dtheta\
int_0^2piint_0^3frac r^22sqrtlambda_1lambda_2 |_0^sqrt z dz dtheta\
int_0^2piint_0^3frac z2sqrtlambda_1lambda_2 dz dtheta\
int_0^2pifrac z^24sqrtlambda_1lambda_2 |_0^3 dtheta\
frac 9pi2sqrtlambda_1lambda_2
$

The product of eigenvalues?

$lambda_1lambda_2 = ac-b^2$

share|cite|improve this answer

edited Jul 30 at 16:25

answered Jul 30 at 16:02

Doug M

39k31749

• 
  
  
  

  
  

  
  
  
  
  
  

  
  +1 for completeness I would add: "The product of eigenvalues is positive since we have positive definite matrix..."
  – user190080
  Jul 30 at 17:55
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  @user190080 The positive definite assumption is required before we can describe the solid as an elliptic paraboloid, and say that the volume is, in fact, finite.
  – Doug M
  Jul 30 at 19:41
  

  
  
  
  

add a comment | 

up vote
2
down vote

You can write $ax^2+2bxy+cy^2=lambda,u^2+mu,v^2$ for some $lambda,mu>0$, where $u$ and $v$ are linear functions in terms of $x$ and $y$ (depending on, of course, the parameters $a,b,c$). We may assume that $lambdageqmu$. For a fixed $hinmathbbR_geq 0$ (in this problem, $h=3$), I shall write
$$M:=big(x,y,z)inmathbbR^3,big$$ instead.

Now, in the $(u,v,z)$-coordinates, the slice $(u,v,z)$ of $M$ at a fixed $z=zetageq 0$ satisfies $$lambda,u^2+mu,v^2leq zeta,.$$ This is an ellipse $E_zeta$ with semiminor axis $sqrtdfraczetalambda$ and semimajor axis $sqrtdfraczetamu$. The area of $E_zeta$ equals $$pileft(sqrtdfraczetalambdaright)left(sqrtdfraczetamuright)=dfracpi,zetasqrtlambdamu,.$$
Hence, the volume of $M$ in the $(u,v,z)$-coordinate is thus
$$int_0^h,dfracpi,zsqrtlambdamu,textdz=fracpi,h^22,sqrtlambdamu,.$$

Now, assume that $T$ is the linear transformation sending $(x,y)$ to $(u,v)$; in other words,
$$beginbmatrixu\vendbmatrix=T,beginbmatrixx\yendbmatrix,.$$
By the Change-of-Variables Formula, we get that
$$textdu,textdv=big|det(T)big|,textdx,textdy,.$$
That is, the volume of $M$ in the $(x,y,z)$-coordinates is equal to
$$frac1,left(fracpi,h^22,sqrtlambdamuright),.$$

Here is the kick. Show that
$$big(det(T)big)^2,lambdamu=detleft(beginbmatrixa&b\b&cendbmatrixright),.$$
That is, the required volume is simply
$$fracpi,h^22,sqrtac-b^2,.$$

Hint: We have $T^top,beginbmatrixlambda&0\0&muendbmatrix,T=beginbmatrixa&b\b&cendbmatrix$.

share|cite|improve this answer

answered Jul 30 at 16:35

Batominovski

22.8k22777

add a comment | 

2 Answers
2

active

oldest

votes

2 Answers
2

active

oldest

votes

active

oldest

votes

active

oldest

votes

up vote
4
down vote



accepted

$ax^2+2bxy + cy^2 = beginbmatrix x&yendbmatrixbeginbmatrix a&b\b&cendbmatrixbeginbmatrix x\yendbmatrix le z le 3 $

Compare this matrix with what you show above.

M is an elliptical paraboloid.

Since this matrix is symmetric, it can be daigonalized with ortho-normal basis.

$mathbf x^T P^T Lambda P mathbf x le z le 3$

and since P is ortho-normal it doesn't distort distances (or volumes), and we can say

$mathbf x^T Lambda mathbf x le z le 3$

or

$lambda_1 x^2 + lambda_2 y^2 le z$

Integrating in polar coordinates:

$int_0^2piint_0^3int_0^sqrt z frac rsqrtlambda_1lambda_2 dr dz dtheta\
int_0^2piint_0^3frac r^22sqrtlambda_1lambda_2 |_0^sqrt z dz dtheta\
int_0^2piint_0^3frac z2sqrtlambda_1lambda_2 dz dtheta\
int_0^2pifrac z^24sqrtlambda_1lambda_2 |_0^3 dtheta\
frac 9pi2sqrtlambda_1lambda_2
$

The product of eigenvalues?

$lambda_1lambda_2 = ac-b^2$

share|cite|improve this answer

edited Jul 30 at 16:25

answered Jul 30 at 16:02

Doug M

39k31749

• 
  
  
  

  
  

  
  
  
  
  
  

  
  +1 for completeness I would add: "The product of eigenvalues is positive since we have positive definite matrix..."
  – user190080
  Jul 30 at 17:55
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  @user190080 The positive definite assumption is required before we can describe the solid as an elliptic paraboloid, and say that the volume is, in fact, finite.
  – Doug M
  Jul 30 at 19:41
  

  
  
  
  

add a comment | 

up vote
4
down vote



accepted

$ax^2+2bxy + cy^2 = beginbmatrix x&yendbmatrixbeginbmatrix a&b\b&cendbmatrixbeginbmatrix x\yendbmatrix le z le 3 $

Compare this matrix with what you show above.

M is an elliptical paraboloid.

Since this matrix is symmetric, it can be daigonalized with ortho-normal basis.

$mathbf x^T P^T Lambda P mathbf x le z le 3$

and since P is ortho-normal it doesn't distort distances (or volumes), and we can say

$mathbf x^T Lambda mathbf x le z le 3$

or

$lambda_1 x^2 + lambda_2 y^2 le z$

Integrating in polar coordinates:

$int_0^2piint_0^3int_0^sqrt z frac rsqrtlambda_1lambda_2 dr dz dtheta\
int_0^2piint_0^3frac r^22sqrtlambda_1lambda_2 |_0^sqrt z dz dtheta\
int_0^2piint_0^3frac z2sqrtlambda_1lambda_2 dz dtheta\
int_0^2pifrac z^24sqrtlambda_1lambda_2 |_0^3 dtheta\
frac 9pi2sqrtlambda_1lambda_2
$

The product of eigenvalues?

$lambda_1lambda_2 = ac-b^2$

share|cite|improve this answer

edited Jul 30 at 16:25

answered Jul 30 at 16:02

Doug M

39k31749

• 
  
  
  

  
  

  
  
  
  
  
  

  
  +1 for completeness I would add: "The product of eigenvalues is positive since we have positive definite matrix..."
  – user190080
  Jul 30 at 17:55
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  @user190080 The positive definite assumption is required before we can describe the solid as an elliptic paraboloid, and say that the volume is, in fact, finite.
  – Doug M
  Jul 30 at 19:41
  

  
  
  
  

add a comment | 

up vote
4
down vote



accepted

up vote
4
down vote



accepted

$ax^2+2bxy + cy^2 = beginbmatrix x&yendbmatrixbeginbmatrix a&b\b&cendbmatrixbeginbmatrix x\yendbmatrix le z le 3 $

Compare this matrix with what you show above.

M is an elliptical paraboloid.

Since this matrix is symmetric, it can be daigonalized with ortho-normal basis.

$mathbf x^T P^T Lambda P mathbf x le z le 3$

and since P is ortho-normal it doesn't distort distances (or volumes), and we can say

$mathbf x^T Lambda mathbf x le z le 3$

or

$lambda_1 x^2 + lambda_2 y^2 le z$

Integrating in polar coordinates:

$int_0^2piint_0^3int_0^sqrt z frac rsqrtlambda_1lambda_2 dr dz dtheta\
int_0^2piint_0^3frac r^22sqrtlambda_1lambda_2 |_0^sqrt z dz dtheta\
int_0^2piint_0^3frac z2sqrtlambda_1lambda_2 dz dtheta\
int_0^2pifrac z^24sqrtlambda_1lambda_2 |_0^3 dtheta\
frac 9pi2sqrtlambda_1lambda_2
$

The product of eigenvalues?

$lambda_1lambda_2 = ac-b^2$

share|cite|improve this answer

edited Jul 30 at 16:25

answered Jul 30 at 16:02

Doug M

39k31749

$ax^2+2bxy + cy^2 = beginbmatrix x&yendbmatrixbeginbmatrix a&b\b&cendbmatrixbeginbmatrix x\yendbmatrix le z le 3 $

Compare this matrix with what you show above.

M is an elliptical paraboloid.

Since this matrix is symmetric, it can be daigonalized with ortho-normal basis.

$mathbf x^T P^T Lambda P mathbf x le z le 3$

and since P is ortho-normal it doesn't distort distances (or volumes), and we can say

$mathbf x^T Lambda mathbf x le z le 3$

or

$lambda_1 x^2 + lambda_2 y^2 le z$

Integrating in polar coordinates:

$int_0^2piint_0^3int_0^sqrt z frac rsqrtlambda_1lambda_2 dr dz dtheta\
int_0^2piint_0^3frac r^22sqrtlambda_1lambda_2 |_0^sqrt z dz dtheta\
int_0^2piint_0^3frac z2sqrtlambda_1lambda_2 dz dtheta\
int_0^2pifrac z^24sqrtlambda_1lambda_2 |_0^3 dtheta\
frac 9pi2sqrtlambda_1lambda_2
$

The product of eigenvalues?

$lambda_1lambda_2 = ac-b^2$

share|cite|improve this answer

edited Jul 30 at 16:25

answered Jul 30 at 16:02

Doug M

39k31749

share|cite|improve this answer

share|cite|improve this answer

edited Jul 30 at 16:25

edited Jul 30 at 16:25

edited Jul 30 at 16:25

answered Jul 30 at 16:02

Doug M

39k31749

answered Jul 30 at 16:02

Doug M

39k31749

answered Jul 30 at 16:02

Doug M

39k31749

39k31749

• 
  
  
  

  
  

  
  
  
  
  
  

  
  +1 for completeness I would add: "The product of eigenvalues is positive since we have positive definite matrix..."
  – user190080
  Jul 30 at 17:55
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  @user190080 The positive definite assumption is required before we can describe the solid as an elliptic paraboloid, and say that the volume is, in fact, finite.
  – Doug M
  Jul 30 at 19:41
  

  
  
  
  

add a comment | 

• 
  
  
  

  
  

  
  
  
  
  
  

  
  +1 for completeness I would add: "The product of eigenvalues is positive since we have positive definite matrix..."
  – user190080
  Jul 30 at 17:55
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  @user190080 The positive definite assumption is required before we can describe the solid as an elliptic paraboloid, and say that the volume is, in fact, finite.
  – Doug M
  Jul 30 at 19:41
  

  
  
  
  

+1 for completeness I would add: "The product of eigenvalues is positive since we have positive definite matrix..."
– user190080
Jul 30 at 17:55

+1 for completeness I would add: "The product of eigenvalues is positive since we have positive definite matrix..."
– user190080
Jul 30 at 17:55

@user190080 The positive definite assumption is required before we can describe the solid as an elliptic paraboloid, and say that the volume is, in fact, finite.
– Doug M
Jul 30 at 19:41

@user190080 The positive definite assumption is required before we can describe the solid as an elliptic paraboloid, and say that the volume is, in fact, finite.
– Doug M
Jul 30 at 19:41

add a comment | 

up vote
2
down vote

You can write $ax^2+2bxy+cy^2=lambda,u^2+mu,v^2$ for some $lambda,mu>0$, where $u$ and $v$ are linear functions in terms of $x$ and $y$ (depending on, of course, the parameters $a,b,c$). We may assume that $lambdageqmu$. For a fixed $hinmathbbR_geq 0$ (in this problem, $h=3$), I shall write
$$M:=big(x,y,z)inmathbbR^3,big$$ instead.

Now, in the $(u,v,z)$-coordinates, the slice $(u,v,z)$ of $M$ at a fixed $z=zetageq 0$ satisfies $$lambda,u^2+mu,v^2leq zeta,.$$ This is an ellipse $E_zeta$ with semiminor axis $sqrtdfraczetalambda$ and semimajor axis $sqrtdfraczetamu$. The area of $E_zeta$ equals $$pileft(sqrtdfraczetalambdaright)left(sqrtdfraczetamuright)=dfracpi,zetasqrtlambdamu,.$$
Hence, the volume of $M$ in the $(u,v,z)$-coordinate is thus
$$int_0^h,dfracpi,zsqrtlambdamu,textdz=fracpi,h^22,sqrtlambdamu,.$$

Now, assume that $T$ is the linear transformation sending $(x,y)$ to $(u,v)$; in other words,
$$beginbmatrixu\vendbmatrix=T,beginbmatrixx\yendbmatrix,.$$
By the Change-of-Variables Formula, we get that
$$textdu,textdv=big|det(T)big|,textdx,textdy,.$$
That is, the volume of $M$ in the $(x,y,z)$-coordinates is equal to
$$frac1,left(fracpi,h^22,sqrtlambdamuright),.$$

Here is the kick. Show that
$$big(det(T)big)^2,lambdamu=detleft(beginbmatrixa&b\b&cendbmatrixright),.$$
That is, the required volume is simply
$$fracpi,h^22,sqrtac-b^2,.$$

Hint: We have $T^top,beginbmatrixlambda&0\0&muendbmatrix,T=beginbmatrixa&b\b&cendbmatrix$.

share|cite|improve this answer

answered Jul 30 at 16:35

Batominovski

22.8k22777

add a comment | 

up vote
2
down vote

You can write $ax^2+2bxy+cy^2=lambda,u^2+mu,v^2$ for some $lambda,mu>0$, where $u$ and $v$ are linear functions in terms of $x$ and $y$ (depending on, of course, the parameters $a,b,c$). We may assume that $lambdageqmu$. For a fixed $hinmathbbR_geq 0$ (in this problem, $h=3$), I shall write
$$M:=big(x,y,z)inmathbbR^3,big$$ instead.

Now, in the $(u,v,z)$-coordinates, the slice $(u,v,z)$ of $M$ at a fixed $z=zetageq 0$ satisfies $$lambda,u^2+mu,v^2leq zeta,.$$ This is an ellipse $E_zeta$ with semiminor axis $sqrtdfraczetalambda$ and semimajor axis $sqrtdfraczetamu$. The area of $E_zeta$ equals $$pileft(sqrtdfraczetalambdaright)left(sqrtdfraczetamuright)=dfracpi,zetasqrtlambdamu,.$$
Hence, the volume of $M$ in the $(u,v,z)$-coordinate is thus
$$int_0^h,dfracpi,zsqrtlambdamu,textdz=fracpi,h^22,sqrtlambdamu,.$$

Now, assume that $T$ is the linear transformation sending $(x,y)$ to $(u,v)$; in other words,
$$beginbmatrixu\vendbmatrix=T,beginbmatrixx\yendbmatrix,.$$
By the Change-of-Variables Formula, we get that
$$textdu,textdv=big|det(T)big|,textdx,textdy,.$$
That is, the volume of $M$ in the $(x,y,z)$-coordinates is equal to
$$frac1,left(fracpi,h^22,sqrtlambdamuright),.$$

Here is the kick. Show that
$$big(det(T)big)^2,lambdamu=detleft(beginbmatrixa&b\b&cendbmatrixright),.$$
That is, the required volume is simply
$$fracpi,h^22,sqrtac-b^2,.$$

Hint: We have $T^top,beginbmatrixlambda&0\0&muendbmatrix,T=beginbmatrixa&b\b&cendbmatrix$.

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answered Jul 30 at 16:35

Batominovski

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You can write $ax^2+2bxy+cy^2=lambda,u^2+mu,v^2$ for some $lambda,mu>0$, where $u$ and $v$ are linear functions in terms of $x$ and $y$ (depending on, of course, the parameters $a,b,c$). We may assume that $lambdageqmu$. For a fixed $hinmathbbR_geq 0$ (in this problem, $h=3$), I shall write
$$M:=big(x,y,z)inmathbbR^3,big$$ instead.

Now, in the $(u,v,z)$-coordinates, the slice $(u,v,z)$ of $M$ at a fixed $z=zetageq 0$ satisfies $$lambda,u^2+mu,v^2leq zeta,.$$ This is an ellipse $E_zeta$ with semiminor axis $sqrtdfraczetalambda$ and semimajor axis $sqrtdfraczetamu$. The area of $E_zeta$ equals $$pileft(sqrtdfraczetalambdaright)left(sqrtdfraczetamuright)=dfracpi,zetasqrtlambdamu,.$$
Hence, the volume of $M$ in the $(u,v,z)$-coordinate is thus
$$int_0^h,dfracpi,zsqrtlambdamu,textdz=fracpi,h^22,sqrtlambdamu,.$$

Now, assume that $T$ is the linear transformation sending $(x,y)$ to $(u,v)$; in other words,
$$beginbmatrixu\vendbmatrix=T,beginbmatrixx\yendbmatrix,.$$
By the Change-of-Variables Formula, we get that
$$textdu,textdv=big|det(T)big|,textdx,textdy,.$$
That is, the volume of $M$ in the $(x,y,z)$-coordinates is equal to
$$frac1,left(fracpi,h^22,sqrtlambdamuright),.$$

Here is the kick. Show that
$$big(det(T)big)^2,lambdamu=detleft(beginbmatrixa&b\b&cendbmatrixright),.$$
That is, the required volume is simply
$$fracpi,h^22,sqrtac-b^2,.$$

Hint: We have $T^top,beginbmatrixlambda&0\0&muendbmatrix,T=beginbmatrixa&b\b&cendbmatrix$.

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answered Jul 30 at 16:35

Batominovski

22.8k22777

You can write $ax^2+2bxy+cy^2=lambda,u^2+mu,v^2$ for some $lambda,mu>0$, where $u$ and $v$ are linear functions in terms of $x$ and $y$ (depending on, of course, the parameters $a,b,c$). We may assume that $lambdageqmu$. For a fixed $hinmathbbR_geq 0$ (in this problem, $h=3$), I shall write
$$M:=big(x,y,z)inmathbbR^3,big$$ instead.

Now, in the $(u,v,z)$-coordinates, the slice $(u,v,z)$ of $M$ at a fixed $z=zetageq 0$ satisfies $$lambda,u^2+mu,v^2leq zeta,.$$ This is an ellipse $E_zeta$ with semiminor axis $sqrtdfraczetalambda$ and semimajor axis $sqrtdfraczetamu$. The area of $E_zeta$ equals $$pileft(sqrtdfraczetalambdaright)left(sqrtdfraczetamuright)=dfracpi,zetasqrtlambdamu,.$$
Hence, the volume of $M$ in the $(u,v,z)$-coordinate is thus
$$int_0^h,dfracpi,zsqrtlambdamu,textdz=fracpi,h^22,sqrtlambdamu,.$$

Now, assume that $T$ is the linear transformation sending $(x,y)$ to $(u,v)$; in other words,
$$beginbmatrixu\vendbmatrix=T,beginbmatrixx\yendbmatrix,.$$
By the Change-of-Variables Formula, we get that
$$textdu,textdv=big|det(T)big|,textdx,textdy,.$$
That is, the volume of $M$ in the $(x,y,z)$-coordinates is equal to
$$frac1,left(fracpi,h^22,sqrtlambdamuright),.$$

Here is the kick. Show that
$$big(det(T)big)^2,lambdamu=detleft(beginbmatrixa&b\b&cendbmatrixright),.$$
That is, the required volume is simply
$$fracpi,h^22,sqrtac-b^2,.$$

Hint: We have $T^top,beginbmatrixlambda&0\0&muendbmatrix,T=beginbmatrixa&b\b&cendbmatrix$.

share|cite|improve this answer

answered Jul 30 at 16:35

Batominovski

22.8k22777

share|cite|improve this answer

share|cite|improve this answer

answered Jul 30 at 16:35

Batominovski

22.8k22777

answered Jul 30 at 16:35

Batominovski

22.8k22777

answered Jul 30 at 16:35

Batominovski

22.8k22777

22.8k22777

add a comment | 

add a comment |