grid figure fixup
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@ -387,41 +387,43 @@ to make the illustration simpler; we need at least two cells in the x dimension
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demonstrate how nonuniform `dx` affects the various components.
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Place the E fore-vectors at integer indices \\( r = (m, n, p) \\) and the H back-vectors
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at fractional indices \\( r + \\frac{1}{2} = (m + \\frac{1}{2}, n + \\frac{1}{2}, p + \\frac{1}{2}.
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Remember that these are indices and not coordinates; they can coorespond to arbitrary
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(monotonically increasing) coordinates depending on the cell widths.
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at fractional indices \\( r + \\frac{1}{2} = (m + \\frac{1}{2}, n + \\frac{1}{2},
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p + \\frac{1}{2}) \\). Remember that these are indices and not coordinates; they can
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correspond to arbitrary (monotonically increasing) coordinates depending on the cell widths.
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Draw lines to denote the planes on which the H components and back-vectors are defined.
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For simplicity, don't draw the equivalent planes for the E components and fore-vectors,
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except as necessary to show their locations -- it's easiest to just connect them to their
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associated H-equivalents. The result looks something like this:
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associated H-equivalents.
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The result looks something like this:
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[figure: Component centers]
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p=
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[H]__________Hx___________[H]______Hx______[H] __ +1/2
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z y /: /: /: /: /| | |
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|/_x / : / : / : / : / | | |
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/ : / : / : / : / | | |
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Hy : Ez...........Hy : Ez......Hy | | |
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/: : : : /: : : : /| | | |
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/ : Hz : Ey....../.:..Hz : Ey./.|..Hz __ 0 | dz[0]
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/ : /: : / / : /: : / / | /| | |
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/_________________________/________________/ | / | | |
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| :/ : :/ | :/ : :/ | |/ | | |
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| Ex : [E].......|..Ex : [E]..|..Ex | | |
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| : | : | | | |
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| [H]..........Hx....|......[H].....Hx|......[H] __ --------- (m=+1/2, p=-1/2)
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| / | / | / / /
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| / | / | / / /
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Hz / Hz / Hz / / /
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| Hy | Hy | Hy __ 0 / dy[0]
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| / | / | / / /
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| / | / | / / /
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|/ |/ |/ / /
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[H]__________Hx___________[H]______Hx______[H] __ -1/2 /
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p=
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[H]__________Hx___________[H]_____Hx______[H] __ +1/2
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z y /: /: /: /: /| | |
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|/_x / : / : / : / : / | | |
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/ : / : / : / : / | | |
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Hy : Ez...........Hy : Ez......Hy | | |
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/: : : : /: : : : /| | | |
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/ : Hz : Ey....../.:..Hz : Ey./.|..Hz __ 0 | dz[0]
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/ : /: : / / : /: : / / | /| | |
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/_________________________/_______________/ | / | | |
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| :/ : :/ | :/ : :/ | |/ | | |
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| Ex : [E].......|..Ex : [E]..|..Ex | | |
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| : | : | | | |
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| [H]..........Hx....|......[H].....H|x.....[H] __ --------- (n=+1/2, p=-1/2)
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| / | / | / / /
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Hz / Hz / Hz / / /
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| / | / | / / /
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| Hy | Hy | Hy __ 0 / dy[0]
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| / | / | / / /
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| / | / | / / /
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|/ |/ |/ / /
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[H]__________Hx___________[H]_____Hx______[H] __ -1/2 /
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=n
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|------------|------------|--------|------|
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-1/2 0 +1/2 +1 +3/2 = m
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|------------|------------|-------|-------|
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-1/2 0 +1/2 +1 +3/2 = m
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------------------------- ----------------
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dx[0] dx[1]
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@ -438,36 +440,40 @@ what volumes those components are responsible for representing. Consider the Ex
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two of its nearest neighbors are E fore-vectors, labeled `[E]` in the figure.
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[figure: Ex volumes]
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<__________________________________________>
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z y << /: / /: >> |
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|/_x < < / : / / : > > |
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< < / : / / : > > |
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< < / : / / : > > |
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<: < / : : / : >: > |
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< : < / : : / : > : > | dz[0]
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< : < / : : / : > : > |
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<____________/_____________________________> : > |
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< : < | : :| : > : > |
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< Ex < | : Ex| : > Ex > |
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< : < | : :| : > : > |
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< : <....|.......:........:|.......:...>...:...>
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< : < | / :| / / > : > /
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< : < | / :| / / > : > /
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< :< | / :|/ / > :> /
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< < | / :| / > > /
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< < | / | / > > / dy[0]
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< < | / | / > > /
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<< |/ |/ >> /
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<____________|_________________|___________> /
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p=
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<_________________________________________> __ +1/2
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z y << /: / /: >> | |
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|/_x < < / : / / : > > | |
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< < / : / / : > > | |
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< < / : / / : > > | |
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<: < / : : / : >: > | |
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< : < / : : / : > : > __ 0 | dz[0]
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< : < / : : / :> : > | |
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<____________/____________________/_______> : > | |
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< : < | : : | > : > | |
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< Ex < | : Ex | > Ex > | |
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< : < | : : | > : > | |
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< : <....|.......:........:...|.......>...:...> __ --------- (n=+1/2, p=-1/2)
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< : < | / : /| /> : > / /
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< : < | / : / | / > : > / /
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< :< | / :/ | / > :> / /
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< < | / : | / > > _ 0 / dy[0]
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< < | / | / > > / /
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< < | / | / > > / /
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<< |/ |/ >> / /
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<____________|____________________|_______> __ -1/2 /
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=n
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|------------|------------|-------|-------|
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-1/2 0 +1/2 +1 +3/2 = m
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~------------ ----------------- -----------~
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dx'[-1] dx'[0] dx'[1]
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~------------ -------------------- -------~
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dx'[-1] dx'[0] dx'[1]
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The Ex values are positioned on the x-faces of the base
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grid. They represent the Ex field in volumes shifted by
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a half-cell in the x-dimension, as shown here. Only the
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center cell is fully shown; the other two are truncated
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(shown using >< markers).
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center cell (with width dx'[0]) is fully shown; the
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other two are truncated (shown using >< markers).
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Note that the Ex positions are the in the same positions
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as the previous figure; only the cell boundaries have moved.
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@ -477,28 +483,34 @@ two of its nearest neighbors are E fore-vectors, labeled `[E]` in the figure.
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center cell here.
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[figure: Hy volumes]
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z y mmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmm s
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|/_x << m: m: >> |
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< < m : m : > > | dz'[1]
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Hy............m...........Hy........m......Hy > |
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< < m : m : > > |
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< < m : m : > > |
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< _______m_____:_______________m_____:_>______
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mmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmm > |
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< < | / : | / :> > |
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< < | / : | / :> > | dz'[0]
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< < | / : | / :> > |
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< wwwwww|w/wwwwwwwwwwwwwwwwwww|w/wwwww>wwwwwww s
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< < |/ w |/ w > > /
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_____________|_____________________|________ > /
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< Hy........|...w...........Hy....|...w...>..Hy /
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< < | w | w > > / dy[0]
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< < | w | w > > /
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<< |w |w >> /
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wwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwww
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p=
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z y mmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmm __ +1/2 s
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|/_x << m: m: >> | |
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< < m : m : > > | | dz'[1]
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< < m : m : > > | |
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Hy........... m........Hy...........m......Hy > | |
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< < m : m : > > | |
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< ______ m_____:_______________m_____:_>______ __ 0
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< < m /: m / > > | |
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mmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmm > | |
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< < | / : | / > > | | dz'[0]
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< < | / : | / > > | |
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< < | / : | / > > | |
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< wwwww|w/wwwwwwwwwwwwwwwwwww|w/wwwww>wwwwwwww __ s
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< < |/ w |/ w> > / /
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_____________|_____________________|________ > / /
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< < | w | w > > / /
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< Hy........|...w........Hy.......|...w...>..Hy _ 0 / dy[0]
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< < | w | w > > / /
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<< | w | w > > / /
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< |w |w >> / /
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wwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwww __ -1/2 /
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|------------|------------|--------|-------|
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-1/2 0 +1/2 +1 +3/2 = m
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~------------ --------------------- -------~
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dx'[-1] dx'[0] dx'[1]
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dx'[-1] dx'[0] dx'[1]
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The Hy values are positioned on the y-edges of the base
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grid. Again here, the 'Hy' labels represent the same points
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