The Jacobian of $(x,y)mapsto (x+y^2,y+x^2)$ under the substitution $u=x+y^2$ and $v=y+x^2$.
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I am given the map $(x,y)mapsto (x+y^2,y+x^2)$. I am unable to find the Jacobian by making the substitution $u=x+y^2$ and $v=y+x^2$. Any hints would be appreciated.
(I am trying to find whether the map is area preserving? I know "the map $f:mathbb{R}^ntomathbb{R}^n$ is area and orientation preserving iff the determinant of the Jacobian is $pm1$".)
multivariable-calculus transformation area
edited Dec 9 '18 at 2:32
Shaun
9,570113684
asked Dec 9 '18 at 1:56
Yadati KiranYadati Kiran
2,1061621
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add a comment |
$begingroup$
I am given the map $(x,y)mapsto (x+y^2,y+x^2)$. I am unable to find the Jacobian by making the substitution $u=x+y^2$ and $v=y+x^2$. Any hints would be appreciated.
(I am trying to find whether the map is area preserving? I know "the map $f:mathbb{R}^ntomathbb{R}^n$ is area and orientation preserving iff the determinant of the Jacobian is $pm1$".)
multivariable-calculus transformation area
edited Dec 9 '18 at 2:32
Shaun
9,570113684
asked Dec 9 '18 at 1:56
Yadati KiranYadati Kiran
2,1061621
$endgroup$
1
$begingroup$
What is the definition of Jacobian?
$endgroup$
– zoidberg
Dec 9 '18 at 2:39
add a comment |
$begingroup$
I am given the map $(x,y)mapsto (x+y^2,y+x^2)$. I am unable to find the Jacobian by making the substitution $u=x+y^2$ and $v=y+x^2$. Any hints would be appreciated.
(I am trying to find whether the map is area preserving? I know "the map $f:mathbb{R}^ntomathbb{R}^n$ is area and orientation preserving iff the determinant of the Jacobian is $pm1$".)
multivariable-calculus transformation area
edited Dec 9 '18 at 2:32
Shaun
9,570113684
asked Dec 9 '18 at 1:56
Yadati KiranYadati Kiran
2,1061621
$endgroup$
I am given the map $(x,y)mapsto (x+y^2,y+x^2)$. I am unable to find the Jacobian by making the substitution $u=x+y^2$ and $v=y+x^2$. Any hints would be appreciated.
(I am trying to find whether the map is area preserving? I know "the map $f:mathbb{R}^ntomathbb{R}^n$ is area and orientation preserving iff the determinant of the Jacobian is $pm1$".)
multivariable-calculus transformation area
multivariable-calculus transformation area
edited Dec 9 '18 at 2:32
Shaun
9,570113684
asked Dec 9 '18 at 1:56
Yadati KiranYadati Kiran
2,1061621
edited Dec 9 '18 at 2:32
Shaun
9,570113684
asked Dec 9 '18 at 1:56
Yadati KiranYadati Kiran
2,1061621
edited Dec 9 '18 at 2:32
Shaun
9,570113684
edited Dec 9 '18 at 2:32
Shaun
9,570113684
edited Dec 9 '18 at 2:32
Shaun
9,570113684
9,570113684
asked Dec 9 '18 at 1:56
Yadati KiranYadati Kiran
2,1061621
asked Dec 9 '18 at 1:56
Yadati KiranYadati Kiran
2,1061621
asked Dec 9 '18 at 1:56
Yadati KiranYadati Kiran
2,1061621
2,1061621
1
$begingroup$
What is the definition of Jacobian?
$endgroup$
– zoidberg
Dec 9 '18 at 2:39
add a comment |
1
$begingroup$
What is the definition of Jacobian?
$endgroup$
– zoidberg
Dec 9 '18 at 2:39
1
1
$begingroup$
What is the definition of Jacobian?
$endgroup$
– zoidberg
Dec 9 '18 at 2:39
$begingroup$
What is the definition of Jacobian?
$endgroup$
– zoidberg
Dec 9 '18 at 2:39
add a comment |
1 Answer
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The Jacobian of a multivariable function $f:mathbb{R}^2 rightarrow mathbb{R}^2$ such that $f(x,y)=(u(x,y),v(x,y))$ is:
$$textbf{J}(f(x,y)) = detbegin{pmatrix}
frac{partial u}{partial x} & frac{partial u}{partial y} \
frac{partial v}{partial x} & frac{partial v}{partial y} \
end{pmatrix}$$
where $det$ denotes the determinant of the matrix.
Calculating the partial derivatives of $u(x,y)=x+y^2$ and $v(x,y)=y+x^2$ we have that the Jacobian of your given function is:
$$detbegin{pmatrix}
1 & 2y \
2x & 1 \
end{pmatrix}=1-4xy$$
answered Dec 9 '18 at 3:20
MAXMAX
19218
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1 Answer
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1 Answer
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$begingroup$
The Jacobian of a multivariable function $f:mathbb{R}^2 rightarrow mathbb{R}^2$ such that $f(x,y)=(u(x,y),v(x,y))$ is:
$$textbf{J}(f(x,y)) = detbegin{pmatrix}
frac{partial u}{partial x} & frac{partial u}{partial y} \
frac{partial v}{partial x} & frac{partial v}{partial y} \
end{pmatrix}$$
where $det$ denotes the determinant of the matrix.
Calculating the partial derivatives of $u(x,y)=x+y^2$ and $v(x,y)=y+x^2$ we have that the Jacobian of your given function is:
$$detbegin{pmatrix}
1 & 2y \
2x & 1 \
end{pmatrix}=1-4xy$$
answered Dec 9 '18 at 3:20
MAXMAX
19218
$endgroup$
add a comment |
$begingroup$
The Jacobian of a multivariable function $f:mathbb{R}^2 rightarrow mathbb{R}^2$ such that $f(x,y)=(u(x,y),v(x,y))$ is:
$$textbf{J}(f(x,y)) = detbegin{pmatrix}
frac{partial u}{partial x} & frac{partial u}{partial y} \
frac{partial v}{partial x} & frac{partial v}{partial y} \
end{pmatrix}$$
where $det$ denotes the determinant of the matrix.
Calculating the partial derivatives of $u(x,y)=x+y^2$ and $v(x,y)=y+x^2$ we have that the Jacobian of your given function is:
$$detbegin{pmatrix}
1 & 2y \
2x & 1 \
end{pmatrix}=1-4xy$$
answered Dec 9 '18 at 3:20
MAXMAX
19218
$endgroup$
add a comment |
$begingroup$
The Jacobian of a multivariable function $f:mathbb{R}^2 rightarrow mathbb{R}^2$ such that $f(x,y)=(u(x,y),v(x,y))$ is:
$$textbf{J}(f(x,y)) = detbegin{pmatrix}
frac{partial u}{partial x} & frac{partial u}{partial y} \
frac{partial v}{partial x} & frac{partial v}{partial y} \
end{pmatrix}$$
where $det$ denotes the determinant of the matrix.
Calculating the partial derivatives of $u(x,y)=x+y^2$ and $v(x,y)=y+x^2$ we have that the Jacobian of your given function is:
$$detbegin{pmatrix}
1 & 2y \
2x & 1 \
end{pmatrix}=1-4xy$$
answered Dec 9 '18 at 3:20
MAXMAX
19218
$endgroup$
The Jacobian of a multivariable function $f:mathbb{R}^2 rightarrow mathbb{R}^2$ such that $f(x,y)=(u(x,y),v(x,y))$ is:
$$textbf{J}(f(x,y)) = detbegin{pmatrix}
frac{partial u}{partial x} & frac{partial u}{partial y} \
frac{partial v}{partial x} & frac{partial v}{partial y} \
end{pmatrix}$$
where $det$ denotes the determinant of the matrix.
Calculating the partial derivatives of $u(x,y)=x+y^2$ and $v(x,y)=y+x^2$ we have that the Jacobian of your given function is:
$$detbegin{pmatrix}
1 & 2y \
2x & 1 \
end{pmatrix}=1-4xy$$
answered Dec 9 '18 at 3:20
MAXMAX
19218
edited Dec 9 '18 at 3:45
answered Dec 9 '18 at 3:20
MAXMAX
19218
answered Dec 9 '18 at 3:20
MAXMAX
19218
answered Dec 9 '18 at 3:20
MAXMAX
19218
19218
add a comment |
add a comment |
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$begingroup$
What is the definition of Jacobian?
$endgroup$
– zoidberg
Dec 9 '18 at 2:39