Proving $f(x,y)=left ( frac{1}{4}sin(x+y),1+frac{2}{3}arctan(x-y) right )$ is a contraction
$begingroup$
$f(x,y)=left ( frac{1}{4}sin(x+y),1+frac{2}{3}arctan(x-y) right )$
Prove that $f$ is a contraction
$mathbb{R}^2$ is equipped with $d((x,y),(x',y'))=|x-x'|+|y-y'|$
My problem is that I'm not able to find an upper bound of $|sin(x+y)-sin(x'+y')|$
general-topology
edited Jan 16 at 3:29
Andrews
1,2812422
asked Jan 16 at 0:29
Pedro AlvarèsPedro Alvarès
946
$endgroup$
add a comment |
$begingroup$
$f(x,y)=left ( frac{1}{4}sin(x+y),1+frac{2}{3}arctan(x-y) right )$
Prove that $f$ is a contraction
$mathbb{R}^2$ is equipped with $d((x,y),(x',y'))=|x-x'|+|y-y'|$
My problem is that I'm not able to find an upper bound of $|sin(x+y)-sin(x'+y')|$
general-topology
edited Jan 16 at 3:29
Andrews
1,2812422
asked Jan 16 at 0:29
Pedro AlvarèsPedro Alvarès
946
$endgroup$
add a comment |
$begingroup$
$f(x,y)=left ( frac{1}{4}sin(x+y),1+frac{2}{3}arctan(x-y) right )$
Prove that $f$ is a contraction
$mathbb{R}^2$ is equipped with $d((x,y),(x',y'))=|x-x'|+|y-y'|$
My problem is that I'm not able to find an upper bound of $|sin(x+y)-sin(x'+y')|$
general-topology
edited Jan 16 at 3:29
Andrews
1,2812422
asked Jan 16 at 0:29
Pedro AlvarèsPedro Alvarès
946
$endgroup$
$f(x,y)=left ( frac{1}{4}sin(x+y),1+frac{2}{3}arctan(x-y) right )$
Prove that $f$ is a contraction
$mathbb{R}^2$ is equipped with $d((x,y),(x',y'))=|x-x'|+|y-y'|$
My problem is that I'm not able to find an upper bound of $|sin(x+y)-sin(x'+y')|$
general-topology
general-topology
edited Jan 16 at 3:29
Andrews
1,2812422
asked Jan 16 at 0:29
Pedro AlvarèsPedro Alvarès
946
edited Jan 16 at 3:29
Andrews
1,2812422
asked Jan 16 at 0:29
Pedro AlvarèsPedro Alvarès
946
edited Jan 16 at 3:29
Andrews
1,2812422
edited Jan 16 at 3:29
Andrews
1,2812422
edited Jan 16 at 3:29
Andrews
1,2812422
1,2812422
asked Jan 16 at 0:29
Pedro AlvarèsPedro Alvarès
946
asked Jan 16 at 0:29
Pedro AlvarèsPedro Alvarès
946
asked Jan 16 at 0:29
Pedro AlvarèsPedro Alvarès
946
946
add a comment |
add a comment |
1 Answer
1
active
oldest
votes
$begingroup$
By MVT $|sin(A)-sin(B)|leq |A-B|$.
answered Jan 16 at 0:32
Kavi Rama MurthyKavi Rama Murthy
71.5k53170
$endgroup$
$begingroup$
But A is a function of x and y
$endgroup$
– Pedro Alvarès
Jan 16 at 0:37
$begingroup$
Then what is ther derivative if sin(A)
$endgroup$
– Pedro Alvarès
Jan 16 at 0:37
add a comment |
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1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
$begingroup$
By MVT $|sin(A)-sin(B)|leq |A-B|$.
answered Jan 16 at 0:32
Kavi Rama MurthyKavi Rama Murthy
71.5k53170
$endgroup$
$begingroup$
But A is a function of x and y
$endgroup$
– Pedro Alvarès
Jan 16 at 0:37
$begingroup$
Then what is ther derivative if sin(A)
$endgroup$
– Pedro Alvarès
Jan 16 at 0:37
add a comment |
$begingroup$
By MVT $|sin(A)-sin(B)|leq |A-B|$.
answered Jan 16 at 0:32
Kavi Rama MurthyKavi Rama Murthy
71.5k53170
$endgroup$
$begingroup$
But A is a function of x and y
$endgroup$
– Pedro Alvarès
Jan 16 at 0:37
$begingroup$
Then what is ther derivative if sin(A)
$endgroup$
– Pedro Alvarès
Jan 16 at 0:37
add a comment |
$begingroup$
By MVT $|sin(A)-sin(B)|leq |A-B|$.
answered Jan 16 at 0:32
Kavi Rama MurthyKavi Rama Murthy
71.5k53170
$endgroup$
By MVT $|sin(A)-sin(B)|leq |A-B|$.
answered Jan 16 at 0:32
Kavi Rama MurthyKavi Rama Murthy
71.5k53170
answered Jan 16 at 0:32
Kavi Rama MurthyKavi Rama Murthy
71.5k53170
answered Jan 16 at 0:32
Kavi Rama MurthyKavi Rama Murthy
71.5k53170
answered Jan 16 at 0:32
Kavi Rama MurthyKavi Rama Murthy
71.5k53170
71.5k53170
$begingroup$
But A is a function of x and y
$endgroup$
– Pedro Alvarès
Jan 16 at 0:37
$begingroup$
Then what is ther derivative if sin(A)
$endgroup$
– Pedro Alvarès
Jan 16 at 0:37
add a comment |
$begingroup$
But A is a function of x and y
$endgroup$
– Pedro Alvarès
Jan 16 at 0:37
$begingroup$
Then what is ther derivative if sin(A)
$endgroup$
– Pedro Alvarès
Jan 16 at 0:37
$begingroup$
But A is a function of x and y
$endgroup$
– Pedro Alvarès
Jan 16 at 0:37
$begingroup$
But A is a function of x and y
$endgroup$
– Pedro Alvarès
Jan 16 at 0:37
$begingroup$
Then what is ther derivative if sin(A)
$endgroup$
– Pedro Alvarès
Jan 16 at 0:37
$begingroup$
Then what is ther derivative if sin(A)
$endgroup$
– Pedro Alvarès
Jan 16 at 0:37
add a comment |
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