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    Hello all,

    As per the title, I'm a stuck on a question, that involves a change of coordinate system. I am keen to do through the question myself afterhopefully resolving my queries. Here is the question, and below it the queries:

    x and y are related to u and v by: x = u+v, y = uv
    A region, S, of the xy plane corresponds to the trianglein the uv plane with vertices (u,v) = (0,0) (0,1) (1,1)
    Find the area of S in the xy plane.

    Queries:
    1. Is it explicit that the Jacobian determinant method is onlyvalid for going from or to the orthogonal/Cartesian to new system (say(a,b)). Or does the Jacobian cope when both sets of variables arenon-orthogonal ie not involving Cartesian?

    2. From using the Jacobian method (sorry I am clueless about inserting equations on website fields so will type in long hand):

    dxdy = mod(del(x,y)/del(u,v))*dudv

    I get a new double integral, where f(x,y) must be set =1whenever one requires the 2D area of the planar domain/not the usual volumebeneath the surface. However it seems unsolvable:

    double integral, limits 0 to 1, 2y to y respectively of(dxdy)/(u-v)

    Basically I clearly don't really understand the double integralchange of variable method and would be very grateful for some assistance!

    Also, if someone could provide instruction on how to insertproperly formatted equations into a website field like a number of the regular contributors on this forum, I would be doubly grateful.
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    (Original post by Sm0key)
    Hello all,

    As per the title, I'm a stuck on a question, that involves a change of coordinate system. I am keen to do through the question myself afterhopefully resolving my queries. Here is the question, and below it the queries:

    x and y are related to u and v by: x = u+v, y = uv
    A region, S, of the xy plane corresponds to the trianglein the uv plane with vertices (u,v) = (0,0) (0,1) (1,1)
    Find the area of S in the xy plane.
    Unless I'm misunderstanding the question, you don't need to use the Jacobian here at all. You are being asked to compute two different integrals, namely:

    A_1 = \iint_{R1} du dv
    A_2 = \iint_{R2} dx dy

    where R1 is the region in the x-y plane that corresponds to the R2 triangular region in the u-v plane. So the problem is not to compute the same integral by transforming it between coord systems, but to compute two integrals with different values.

    To do this, you need to choose the limits in the x-y plane that map back to the triangular region in the u-v plane. So it's really a "how do the limits correspond" question, I think.

    If you were to include the Jacobian in the second integral above, you would calculate same answer in both cases. It's the 2D analogy of doing a change-of-variables in a 1D integral e.g. You can compute:

    I = \int_0^1 \frac{1}{1+x^2} dx

    or

    I = \int_0^{\pi/4} \frac{1}{1+\tan^2 \theta} \frac{dx}{d\theta} d\theta

    where x=\tan \theta, and you get the same answer. Here dx/d\theta is the 1D Jacobian.

    1. Is it explicit that the Jacobian determinant method is onlyvalid for going from or to the orthogonal/Cartesian to new system (say(a,b)). Or does the Jacobian cope when both sets of variables arenon-orthogonal ie not involving Cartesian?
    Jacobians are completely general; they transform between any two coordinate systems, as long as there is a bijection between the two.

    Also, if someone could provide instruction on how to insertproperly formatted equations into a website field like a number of the regular contributors on this forum, I would be doubly grateful.
    Look up latex. There is a guide on this site, and plenty of examples available by googling.
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    (Original post by Sm0key)
    Also, if someone could provide instruction on how to insertproperly formatted equations into a website field like a number of the regular contributors on this forum, I would be doubly grateful.
    Use this website: http://www.codecogs.com/latex/eqneditor.php

    But you need to format it like: [ tex ] insert code here [ /tex ] with the spaces removed.
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    (Original post by atsruser)
    Unless I'm misunderstanding the question, you don't need to use the Jacobian here at all. You are being asked to compute two different integrals, namely:

    A_1 = \iint_{R1} du dv
    A_2 = \iint_{R2} dx dy

    where R1 is the region in the x-y plane that corresponds to the R2 triangular region in the u-v plane. So the problem is not to compute the same integral by transforming it between coord systems, but to compute two integrals with different values.

    To do this, you need to choose the limits in the x-y plane that map back to the triangular region in the u-v plane. So it's really a "how do the limits correspond" question, I think.
    Thanks for your reply. Please could you elaborate on the above - I still don't know how to proceed!
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    (Original post by Sm0key)
    Thanks for your reply. Please could you elaborate on the above - I still don't know how to proceed!
    1. Draw out the region in the u-v plane, noting the lines which bound it.
    2. Consider the transformation equations.
    3. Figure out to which region in the x-y plane the boundary lines in 1. are mapped.

    To do the last step, you need, for each line, to find a relation y=f(x) in the x-y plane. To do that, you need to eliminate u,v from the transformation equations, taking into account their values and ranges on each boundary line in the u-v plane. Note that for two of these lines, either u or v is constant, and for the third, it's a linear relationship.

    I get an answer of 1/6 for the area in the x-y plane, BTW (both by double integration, or by evaluating a standard 1D area problem).
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    (Original post by atsruser)
    1. Draw out the region in the u-v plane, noting the lines which bound it.
    2. Consider the transformation equations.
    3. Figure out to which region in the x-y plane the boundary lines in 1. are mapped.

    To do the last step, you need, for each line, to find a relation y=f(x) in the x-y plane. To do that, you need to eliminate u,v from the transformation equations, taking into account their values and ranges on each boundary line in the u-v plane. Note that for two of these lines, either u or v is constant, and for the third, it's a linear relationship.

    I get an answer of 1/6 for the area in the x-y plane, BTW (both by double integration, or by evaluating a standard 1D area problem).
    Edit: I forgot to say that someone told me that the answer is indeed 1/6. So you definitely understood the question!

    I want to use the double integral method, and am still struggling . This is what I get to so far:

    Let S' denote the required area in the xy plane, and S denote the starting area in the uv plane.

    Using the Jacobian definition, in going from a 2D corodinate system (a,b) to (e,f):

    da.db =\left | \frac{\partial (a,b)}{\partial (e,f)} \right |de.df

    in this question I'll require:

    du.dv = \left | \frac{\partial u}{\partial x}\frac{\partial v}{\partial y}-\frac{\partial u}{\partial y}\frac{\partial v}{\partial x} \right | dx.dy

    but this is where the problems start - I cannot explicitly write u=... and v=... from the given 2 transformations. Hence cannot calculate the Jacobian...

    Ditto for changing my limits on each integral to (x,y)


    Thanks for your help so far. I appreciate it.
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    (Original post by Sm0key)
    Edit: I forgot to say that someone told me that the answer is indeed 1/6. So you definitely understood the question!
    Good. At least we're not wasting our time solving the wrong problem.

    I want to use the double internal method,
    Perhaps you're looking for a more *cough* specialist website?...

    Let S' denote the required area in the xy plane, and S denote the starting area in the uv plane.

    Using the Jacobian definition, in going from a 2D corodinate system (a,b) to (e,f):

    da.db =\left | \frac{\partial (a,b)}{\partial (e,f)} \right |de.df
    So I say that you don't need to use the Jacobian, and you use the Jacobian... Anyway, I commend you on your use of nice Latex.

    You don't seem to have followed my prescription above. Try that and post up your working here.

    You don't need the Jacobian as you are not being asked to transform an integral from the u-v plane to the x-y plane. You have to compute an area in the x-y plane, and that area is bounded by some lines that must be mapped from the u-v plane. It is the transformation equations that do that; the Jacobian tells you the ratio of an infinitesimal area in the u-v plane to that in the x-y plane.
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    I managed to get it sorted in the end. Thanks for your help.
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    (Original post by Sm0key)
    I managed to get it sorted in the end. Thanks for your help.
    Good. Did you do it the way that I suggested?
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    (Original post by atsruser)
    Good. Did you do it the way that I suggested?
    Yes. I ended up with boundary mappings (in xy plane) of:
    y=0, y=x-1, y=x^2 which reduced the problem to a simple 1D integral yielding 1/6.

    Thanks again.
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    (Original post by Sm0key)
    Yes. I ended up with boundary mappings (in xy plane) of:
    y=0, y=x-1, y=x^2 which reduced the problem to a simple 1D integral yielding 1/6.

    Thanks again.
    I got y=\frac{x^2}{4} - is it a typo above?

    Also, it would probably help you if you were to evaluate the area via a double integral approach. The main difficulty is finding the correct limits, but it's quite straightforward.
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    (Original post by atsruser)
    I got y=\frac{x^2}{4} - is it a typo above?

    Also, it would probably help you if you were to evaluate the area via a double integral approach. The main difficulty is finding the correct limits, but it's quite straightforward.
    Yes a typo from me. I will try the double integral approach during my consolidation.
 
 
 
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