Understanding the cosine as a partial derivative.












0












$begingroup$


From the first chapter of Arfken's Mathematical Methods for physicists (rotation of the coordinate axis):




To go on to three and, later, four dimensions, we find it convenient to use a more compact notation. Let
begin{equation}
x → x_1,
textbf{ }
y → x_2
end{equation}



begin{equation}
a_{11} = cosphi,textbf{ } a_{12} = sinphi
end{equation}



begin{equation}
a_{21} = −sinphi, textbf{ } a_{22} = cosphi
end{equation}



Then Eqs. become



begin{equation}
x′_1 = a_{11}x_1 + a_{12}x_2
end{equation}



begin{equation}
x′_2 = a_{21}x_1 + a_{22}x_2.
end{equation}



The coefficient $a_{ij}$ may be interpreted as a direction cosine, the cosine of the angle between $x'_i$ and $x_j$ ; that is,




This is all good. Later, the book states:




From the definition of $a_{ij}$ as the cosine of the angle between the positive $x′_i$ direction and the positive $x_j$ direction we may write (Cartesian coordinates):



begin{equation}
a_{ij}=frac{partial x'_i}{partial x_j}
end{equation}




Where did that come from? I can't find anything about the cosine as a partial derivative, and I don't see how that works.






trigonometry partial-derivative rotations







share|cite|improve this question











$endgroup$












  • $begingroup$
    Take the derivatives of the $x'_i$s from your two equations at the end of your first box.
    $endgroup$
    – John Douma
    Dec 29 '18 at 17:01










  • $begingroup$
    @JohnDouma The derivatives with respect to what? I'm sorry, I can't see it. Thank you.
    $endgroup$
    – IchVerloren
    Dec 29 '18 at 17:15










  • $begingroup$
    Start with $frac{partial x'_1}{partial x_1}$.
    $endgroup$
    – John Douma
    Dec 29 '18 at 17:48
















0












$begingroup$


From the first chapter of Arfken's Mathematical Methods for physicists (rotation of the coordinate axis):




To go on to three and, later, four dimensions, we find it convenient to use a more compact notation. Let
begin{equation}
x → x_1,
textbf{ }
y → x_2
end{equation}



begin{equation}
a_{11} = cosphi,textbf{ } a_{12} = sinphi
end{equation}



begin{equation}
a_{21} = −sinphi, textbf{ } a_{22} = cosphi
end{equation}



Then Eqs. become



begin{equation}
x′_1 = a_{11}x_1 + a_{12}x_2
end{equation}



begin{equation}
x′_2 = a_{21}x_1 + a_{22}x_2.
end{equation}



The coefficient $a_{ij}$ may be interpreted as a direction cosine, the cosine of the angle between $x'_i$ and $x_j$ ; that is,




This is all good. Later, the book states:




From the definition of $a_{ij}$ as the cosine of the angle between the positive $x′_i$ direction and the positive $x_j$ direction we may write (Cartesian coordinates):



begin{equation}
a_{ij}=frac{partial x'_i}{partial x_j}
end{equation}




Where did that come from? I can't find anything about the cosine as a partial derivative, and I don't see how that works.






trigonometry partial-derivative rotations







share|cite|improve this question











$endgroup$












  • $begingroup$
    Take the derivatives of the $x'_i$s from your two equations at the end of your first box.
    $endgroup$
    – John Douma
    Dec 29 '18 at 17:01










  • $begingroup$
    @JohnDouma The derivatives with respect to what? I'm sorry, I can't see it. Thank you.
    $endgroup$
    – IchVerloren
    Dec 29 '18 at 17:15










  • $begingroup$
    Start with $frac{partial x'_1}{partial x_1}$.
    $endgroup$
    – John Douma
    Dec 29 '18 at 17:48














0












0








0





$begingroup$


From the first chapter of Arfken's Mathematical Methods for physicists (rotation of the coordinate axis):




To go on to three and, later, four dimensions, we find it convenient to use a more compact notation. Let
begin{equation}
x → x_1,
textbf{ }
y → x_2
end{equation}



begin{equation}
a_{11} = cosphi,textbf{ } a_{12} = sinphi
end{equation}



begin{equation}
a_{21} = −sinphi, textbf{ } a_{22} = cosphi
end{equation}



Then Eqs. become



begin{equation}
x′_1 = a_{11}x_1 + a_{12}x_2
end{equation}



begin{equation}
x′_2 = a_{21}x_1 + a_{22}x_2.
end{equation}



The coefficient $a_{ij}$ may be interpreted as a direction cosine, the cosine of the angle between $x'_i$ and $x_j$ ; that is,




This is all good. Later, the book states:




From the definition of $a_{ij}$ as the cosine of the angle between the positive $x′_i$ direction and the positive $x_j$ direction we may write (Cartesian coordinates):



begin{equation}
a_{ij}=frac{partial x'_i}{partial x_j}
end{equation}




Where did that come from? I can't find anything about the cosine as a partial derivative, and I don't see how that works.






trigonometry partial-derivative rotations







share|cite|improve this question











$endgroup$




From the first chapter of Arfken's Mathematical Methods for physicists (rotation of the coordinate axis):




To go on to three and, later, four dimensions, we find it convenient to use a more compact notation. Let
begin{equation}
x → x_1,
textbf{ }
y → x_2
end{equation}



begin{equation}
a_{11} = cosphi,textbf{ } a_{12} = sinphi
end{equation}



begin{equation}
a_{21} = −sinphi, textbf{ } a_{22} = cosphi
end{equation}



Then Eqs. become



begin{equation}
x′_1 = a_{11}x_1 + a_{12}x_2
end{equation}



begin{equation}
x′_2 = a_{21}x_1 + a_{22}x_2.
end{equation}



The coefficient $a_{ij}$ may be interpreted as a direction cosine, the cosine of the angle between $x'_i$ and $x_j$ ; that is,




This is all good. Later, the book states:




From the definition of $a_{ij}$ as the cosine of the angle between the positive $x′_i$ direction and the positive $x_j$ direction we may write (Cartesian coordinates):



begin{equation}
a_{ij}=frac{partial x'_i}{partial x_j}
end{equation}




Where did that come from? I can't find anything about the cosine as a partial derivative, and I don't see how that works.






trigonometry partial-derivative rotations




trigonometry partial-derivative rotations






share|cite|improve this question















share|cite|improve this question













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edited Dec 29 '18 at 18:47









Christian Blatter

176k9115328




176k9115328










asked Dec 29 '18 at 16:51









IchVerlorenIchVerloren

21310




21310












  • $begingroup$
    Take the derivatives of the $x'_i$s from your two equations at the end of your first box.
    $endgroup$
    – John Douma
    Dec 29 '18 at 17:01










  • $begingroup$
    @JohnDouma The derivatives with respect to what? I'm sorry, I can't see it. Thank you.
    $endgroup$
    – IchVerloren
    Dec 29 '18 at 17:15










  • $begingroup$
    Start with $frac{partial x'_1}{partial x_1}$.
    $endgroup$
    – John Douma
    Dec 29 '18 at 17:48


















  • $begingroup$
    Take the derivatives of the $x'_i$s from your two equations at the end of your first box.
    $endgroup$
    – John Douma
    Dec 29 '18 at 17:01










  • $begingroup$
    @JohnDouma The derivatives with respect to what? I'm sorry, I can't see it. Thank you.
    $endgroup$
    – IchVerloren
    Dec 29 '18 at 17:15










  • $begingroup$
    Start with $frac{partial x'_1}{partial x_1}$.
    $endgroup$
    – John Douma
    Dec 29 '18 at 17:48
















$begingroup$
Take the derivatives of the $x'_i$s from your two equations at the end of your first box.
$endgroup$
– John Douma
Dec 29 '18 at 17:01




$begingroup$
Take the derivatives of the $x'_i$s from your two equations at the end of your first box.
$endgroup$
– John Douma
Dec 29 '18 at 17:01












$begingroup$
@JohnDouma The derivatives with respect to what? I'm sorry, I can't see it. Thank you.
$endgroup$
– IchVerloren
Dec 29 '18 at 17:15




$begingroup$
@JohnDouma The derivatives with respect to what? I'm sorry, I can't see it. Thank you.
$endgroup$
– IchVerloren
Dec 29 '18 at 17:15












$begingroup$
Start with $frac{partial x'_1}{partial x_1}$.
$endgroup$
– John Douma
Dec 29 '18 at 17:48




$begingroup$
Start with $frac{partial x'_1}{partial x_1}$.
$endgroup$
– John Douma
Dec 29 '18 at 17:48










1 Answer
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$begingroup$

You can verify-



$frac{partial x_1'}{partial x_1}=a_{11},frac{partial x_1'}{partial x_2}=a_{12},frac{partial x_2'}{partial x_1}=a_{21} and frac{partial x_2'}{partial x_2}=a_{22}$






share|cite|improve this answer









$endgroup$













  • $begingroup$
    these are from the equations for $x_1' and x_2'$ which you have given
    $endgroup$
    – Mustang
    Dec 29 '18 at 18:40












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1 Answer
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active

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1 Answer
1






active

oldest

votes









active

oldest

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active

oldest

votes









0












$begingroup$

You can verify-



$frac{partial x_1'}{partial x_1}=a_{11},frac{partial x_1'}{partial x_2}=a_{12},frac{partial x_2'}{partial x_1}=a_{21} and frac{partial x_2'}{partial x_2}=a_{22}$






share|cite|improve this answer









$endgroup$













  • $begingroup$
    these are from the equations for $x_1' and x_2'$ which you have given
    $endgroup$
    – Mustang
    Dec 29 '18 at 18:40
















0












$begingroup$

You can verify-



$frac{partial x_1'}{partial x_1}=a_{11},frac{partial x_1'}{partial x_2}=a_{12},frac{partial x_2'}{partial x_1}=a_{21} and frac{partial x_2'}{partial x_2}=a_{22}$






share|cite|improve this answer









$endgroup$













  • $begingroup$
    these are from the equations for $x_1' and x_2'$ which you have given
    $endgroup$
    – Mustang
    Dec 29 '18 at 18:40














0












0








0





$begingroup$

You can verify-



$frac{partial x_1'}{partial x_1}=a_{11},frac{partial x_1'}{partial x_2}=a_{12},frac{partial x_2'}{partial x_1}=a_{21} and frac{partial x_2'}{partial x_2}=a_{22}$






share|cite|improve this answer









$endgroup$



You can verify-



$frac{partial x_1'}{partial x_1}=a_{11},frac{partial x_1'}{partial x_2}=a_{12},frac{partial x_2'}{partial x_1}=a_{21} and frac{partial x_2'}{partial x_2}=a_{22}$







share|cite|improve this answer












share|cite|improve this answer



share|cite|improve this answer










answered Dec 29 '18 at 18:37









MustangMustang

3367




3367












  • $begingroup$
    these are from the equations for $x_1' and x_2'$ which you have given
    $endgroup$
    – Mustang
    Dec 29 '18 at 18:40


















  • $begingroup$
    these are from the equations for $x_1' and x_2'$ which you have given
    $endgroup$
    – Mustang
    Dec 29 '18 at 18:40
















$begingroup$
these are from the equations for $x_1' and x_2'$ which you have given
$endgroup$
– Mustang
Dec 29 '18 at 18:40




$begingroup$
these are from the equations for $x_1' and x_2'$ which you have given
$endgroup$
– Mustang
Dec 29 '18 at 18:40


















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