Change of variable in 2D integral












0












$begingroup$


If I have an integral of the form



$$
I = int_{-T}^T dxint_{-T}^T dy i(x - y)
tag1$$



Where $i$ is any function depending just on the relative variable, i.e., $x - y$. But, let's suppose that I need to perform the following change of variable:



$$
r = frac{x + y}{2}, quad R = x - y
tag2$$



Then, I think that $I$ will become



$$
I = int_{-sqrt{2}T/2}^{sqrt{2}T/2}dr int_{-sqrt{2}T}^{sqrt{2}T}dR i(R)
tag3$$



But I'm not sure. My reasoning to get this new limits is as follows: from Eq. (2), you can imagine $R$ as the module of a 2D vector in the cartesian plane, so



$$vec{R}_{maximum} = (T, 0) - (0, T) Rightarrow R_{maximum} = sqrt{2}T$$



Similarly,
$$R_{minimum} = -R_{maximum}$$



Moreover,
$$vec{r}_{maximum} = 0.5((T, 0) + (0, T)) Rightarrow r_{maximum} = frac{sqrt{2}T}{2}$$



and, as before,



$$r_{minimum} = -r_{maximum}$$



Is this correct or did I fail something?










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$endgroup$








  • 1




    $begingroup$
    Your change of variables maps a rectangle in the $(x,y)$-plane to a rhombus in the $(r,R)$-plane, not a rectangle. You need more elaborate limits on your new integral. I suggest drawing a picture.
    $endgroup$
    – Lord Shark the Unknown
    Dec 23 '18 at 6:05






  • 1




    $begingroup$
    The correct limits for an inner integral, after a change of variables, are not the largest and smallest possible values that variable can take ever; they are the largest and smallest possible values that variable can take given a particular value $r$ of the outer variable. In particular, the limits of the inner integral usually are functions of $r$ (or whatever the outer variable is).
    $endgroup$
    – Greg Martin
    Dec 23 '18 at 6:18
















0












$begingroup$


If I have an integral of the form



$$
I = int_{-T}^T dxint_{-T}^T dy i(x - y)
tag1$$



Where $i$ is any function depending just on the relative variable, i.e., $x - y$. But, let's suppose that I need to perform the following change of variable:



$$
r = frac{x + y}{2}, quad R = x - y
tag2$$



Then, I think that $I$ will become



$$
I = int_{-sqrt{2}T/2}^{sqrt{2}T/2}dr int_{-sqrt{2}T}^{sqrt{2}T}dR i(R)
tag3$$



But I'm not sure. My reasoning to get this new limits is as follows: from Eq. (2), you can imagine $R$ as the module of a 2D vector in the cartesian plane, so



$$vec{R}_{maximum} = (T, 0) - (0, T) Rightarrow R_{maximum} = sqrt{2}T$$



Similarly,
$$R_{minimum} = -R_{maximum}$$



Moreover,
$$vec{r}_{maximum} = 0.5((T, 0) + (0, T)) Rightarrow r_{maximum} = frac{sqrt{2}T}{2}$$



and, as before,



$$r_{minimum} = -r_{maximum}$$



Is this correct or did I fail something?










share|cite|improve this question











$endgroup$








  • 1




    $begingroup$
    Your change of variables maps a rectangle in the $(x,y)$-plane to a rhombus in the $(r,R)$-plane, not a rectangle. You need more elaborate limits on your new integral. I suggest drawing a picture.
    $endgroup$
    – Lord Shark the Unknown
    Dec 23 '18 at 6:05






  • 1




    $begingroup$
    The correct limits for an inner integral, after a change of variables, are not the largest and smallest possible values that variable can take ever; they are the largest and smallest possible values that variable can take given a particular value $r$ of the outer variable. In particular, the limits of the inner integral usually are functions of $r$ (or whatever the outer variable is).
    $endgroup$
    – Greg Martin
    Dec 23 '18 at 6:18














0












0








0





$begingroup$


If I have an integral of the form



$$
I = int_{-T}^T dxint_{-T}^T dy i(x - y)
tag1$$



Where $i$ is any function depending just on the relative variable, i.e., $x - y$. But, let's suppose that I need to perform the following change of variable:



$$
r = frac{x + y}{2}, quad R = x - y
tag2$$



Then, I think that $I$ will become



$$
I = int_{-sqrt{2}T/2}^{sqrt{2}T/2}dr int_{-sqrt{2}T}^{sqrt{2}T}dR i(R)
tag3$$



But I'm not sure. My reasoning to get this new limits is as follows: from Eq. (2), you can imagine $R$ as the module of a 2D vector in the cartesian plane, so



$$vec{R}_{maximum} = (T, 0) - (0, T) Rightarrow R_{maximum} = sqrt{2}T$$



Similarly,
$$R_{minimum} = -R_{maximum}$$



Moreover,
$$vec{r}_{maximum} = 0.5((T, 0) + (0, T)) Rightarrow r_{maximum} = frac{sqrt{2}T}{2}$$



and, as before,



$$r_{minimum} = -r_{maximum}$$



Is this correct or did I fail something?










share|cite|improve this question











$endgroup$




If I have an integral of the form



$$
I = int_{-T}^T dxint_{-T}^T dy i(x - y)
tag1$$



Where $i$ is any function depending just on the relative variable, i.e., $x - y$. But, let's suppose that I need to perform the following change of variable:



$$
r = frac{x + y}{2}, quad R = x - y
tag2$$



Then, I think that $I$ will become



$$
I = int_{-sqrt{2}T/2}^{sqrt{2}T/2}dr int_{-sqrt{2}T}^{sqrt{2}T}dR i(R)
tag3$$



But I'm not sure. My reasoning to get this new limits is as follows: from Eq. (2), you can imagine $R$ as the module of a 2D vector in the cartesian plane, so



$$vec{R}_{maximum} = (T, 0) - (0, T) Rightarrow R_{maximum} = sqrt{2}T$$



Similarly,
$$R_{minimum} = -R_{maximum}$$



Moreover,
$$vec{r}_{maximum} = 0.5((T, 0) + (0, T)) Rightarrow r_{maximum} = frac{sqrt{2}T}{2}$$



and, as before,



$$r_{minimum} = -r_{maximum}$$



Is this correct or did I fail something?







calculus integration multivariable-calculus definite-integrals change-of-variable






share|cite|improve this question















share|cite|improve this question













share|cite|improve this question




share|cite|improve this question








edited Dec 23 '18 at 4:54







Vicky

















asked Dec 23 '18 at 4:29









VickyVicky

2627




2627








  • 1




    $begingroup$
    Your change of variables maps a rectangle in the $(x,y)$-plane to a rhombus in the $(r,R)$-plane, not a rectangle. You need more elaborate limits on your new integral. I suggest drawing a picture.
    $endgroup$
    – Lord Shark the Unknown
    Dec 23 '18 at 6:05






  • 1




    $begingroup$
    The correct limits for an inner integral, after a change of variables, are not the largest and smallest possible values that variable can take ever; they are the largest and smallest possible values that variable can take given a particular value $r$ of the outer variable. In particular, the limits of the inner integral usually are functions of $r$ (or whatever the outer variable is).
    $endgroup$
    – Greg Martin
    Dec 23 '18 at 6:18














  • 1




    $begingroup$
    Your change of variables maps a rectangle in the $(x,y)$-plane to a rhombus in the $(r,R)$-plane, not a rectangle. You need more elaborate limits on your new integral. I suggest drawing a picture.
    $endgroup$
    – Lord Shark the Unknown
    Dec 23 '18 at 6:05






  • 1




    $begingroup$
    The correct limits for an inner integral, after a change of variables, are not the largest and smallest possible values that variable can take ever; they are the largest and smallest possible values that variable can take given a particular value $r$ of the outer variable. In particular, the limits of the inner integral usually are functions of $r$ (or whatever the outer variable is).
    $endgroup$
    – Greg Martin
    Dec 23 '18 at 6:18








1




1




$begingroup$
Your change of variables maps a rectangle in the $(x,y)$-plane to a rhombus in the $(r,R)$-plane, not a rectangle. You need more elaborate limits on your new integral. I suggest drawing a picture.
$endgroup$
– Lord Shark the Unknown
Dec 23 '18 at 6:05




$begingroup$
Your change of variables maps a rectangle in the $(x,y)$-plane to a rhombus in the $(r,R)$-plane, not a rectangle. You need more elaborate limits on your new integral. I suggest drawing a picture.
$endgroup$
– Lord Shark the Unknown
Dec 23 '18 at 6:05




1




1




$begingroup$
The correct limits for an inner integral, after a change of variables, are not the largest and smallest possible values that variable can take ever; they are the largest and smallest possible values that variable can take given a particular value $r$ of the outer variable. In particular, the limits of the inner integral usually are functions of $r$ (or whatever the outer variable is).
$endgroup$
– Greg Martin
Dec 23 '18 at 6:18




$begingroup$
The correct limits for an inner integral, after a change of variables, are not the largest and smallest possible values that variable can take ever; they are the largest and smallest possible values that variable can take given a particular value $r$ of the outer variable. In particular, the limits of the inner integral usually are functions of $r$ (or whatever the outer variable is).
$endgroup$
– Greg Martin
Dec 23 '18 at 6:18










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