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Thread: First-order equations by introducing new coordinates.

  1. #1
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    First-order equations by introducing new coordinates.

    $\displaystyle 3u_x+4u_y-2u=1\Rightarrow\omega_{\xi}+k\omega=\varphi(\xi,\e ta)$

    The book states, "If the $\displaystyle (\xi,\eta)$-axes are obtained from $\displaystyle (x,y)$-axes by rotating through an angle $\displaystyle \alpha$, then $\displaystyle (\xi,\eta)$ and $\displaystyle (x,y)$ are related by either of the pair of equations:

    $\displaystyle \xi=x\cos{\alpha}+y\sin{\alpha}, \ \ x=\xi\cos{\alpha}-\eta\sin{\alpha},$

    $\displaystyle \eta=-x\sin{\alpha}+y\cos{\alpha}, \ \ y=\xi\sin{\alpha}+\eta\sin{\alpha}.\mbox{"}$

    How did the book obtain those equations?
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  2. #2
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    Draw the orthogonal axes for $\displaystyle x,y$, and the rotated axes of $\displaystyle \xi, \eta$ by $\displaystyle \alpha$.
    The equations are obtained by simple trig and vector addition.
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  3. #3
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    Quote Originally Posted by snowtea View Post
    Draw the orthogonal axes for $\displaystyle x,y$, and the rotated axes of $\displaystyle \xi, \eta$ by $\displaystyle \alpha$.
    The equations are obtained by simple trig and vector addition.
    I did that, but unfortunately, I still don't see how it works.
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    A drawing normally explains this very easily, but since I suck at drawing (and cannot seem to find a picture online for this common transformation), I will try to explain it analytically.

    Any axis can be defined by a unit vector parallel to it in the postive direction.

    So let $\displaystyle e_x,e_y,e_\xi,e_\eta$ be the unit vectors parallel to the corresponding axes.

    A point in the $\displaystyle x,y$ coordinate system can be represented by the vector $\displaystyle xe_x + ye_y$.

    For any vector $\displaystyle \vec{v}$, its representation in the $\displaystyle \xi, \eta$ (orthogonal) coordinate system is:
    $\displaystyle (\vec{v}\cdot e_\xi)e_\xi + (\vec{v}\cdot e_\eta)e_\eta$.

    So the $\displaystyle \xi$ coordinate is given by $\displaystyle \vec{v}\cdot e_\xi$.

    So for $\displaystyle xe_x + ye_y$, this is $\displaystyle x(e_x\cdot e_\xi) + y(e_y \cdot e_\xi) = x\cos{\alpha}+y\sin{\alpha}$.

    Similarly for $\displaystyle \eta$...

    [Edit: Found an image here: http://www.tutornext.com/system/file...Fig.1.36_0.GIF,
    perhaps this will help with intuition using simple trigonometry]
    Last edited by snowtea; Jan 7th 2011 at 02:23 PM.
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  5. #5
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    Quote Originally Posted by snowtea View Post

    $\displaystyle (e_x\cdot e_\xi)= \cos{\alpha}$.
    Why is that equal to cosine alpha?
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  6. #6
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    $\displaystyle e_x, e_\xi$ are unit vectors parallel to the axes. Their dot product is cosine of the angle between them (they have unit length).
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  7. #7
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    Ok that makes since since $\displaystyle \displaystyle\cos{\alpha}=\frac{u\cdot v}{||u|| \ ||v||}$ but I have never seen sine defined in this manner so how does sine come into play?
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  8. #8
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    $\displaystyle \cos(\frac{\pi}{2} - \alpha) = \sin(\alpha)$
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