Covariant derivative in cylindrical coordinates
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I am confused by the Wolfram article on cylindrical coordinates. Specifically, I do not understand how they go from equation (48) to equations (49)-(57).
Equation (48) shows that the covariant derivative is:
$$A_{j;k} = frac{1}{g_{kk}}frac{partial A_j}{partial x_k} - Gamma^i_{jk}A_i$$
The next few equations expand this for the case of cylindrical coordinates, equation (50) is:
$$A_{r;theta} = frac{1}{r}frac{partial A_r}{partial theta} - frac{A_theta}{r}$$
The contravariant metric tensor has non-zero elements:
$$g^{11} = 1$$
$$g^{22} = frac{1}{r^2}$$
$$g^{33} = 1$$
And the Christoffel symbols of the second kind have non-zero elements:
$$Gamma^1_{22} = -r$$
$$Gamma^2_{12} = frac{1}{r}$$
$$Gamma^2_{21} = frac{1}{r}$$
If I plug these values back into their definition of the covariant derivative I get for equation (50):
$$A_{r;theta} = frac{1}{r^2}frac{partial A_r}{partial theta} - frac{A_theta}{r}$$
Why does this not match up with their results?
differential-geometry
asked Jun 13 '14 at 18:44
OSEOSE
95110
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add a comment |
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I am confused by the Wolfram article on cylindrical coordinates. Specifically, I do not understand how they go from equation (48) to equations (49)-(57).
Equation (48) shows that the covariant derivative is:
$$A_{j;k} = frac{1}{g_{kk}}frac{partial A_j}{partial x_k} - Gamma^i_{jk}A_i$$
The next few equations expand this for the case of cylindrical coordinates, equation (50) is:
$$A_{r;theta} = frac{1}{r}frac{partial A_r}{partial theta} - frac{A_theta}{r}$$
The contravariant metric tensor has non-zero elements:
$$g^{11} = 1$$
$$g^{22} = frac{1}{r^2}$$
$$g^{33} = 1$$
And the Christoffel symbols of the second kind have non-zero elements:
$$Gamma^1_{22} = -r$$
$$Gamma^2_{12} = frac{1}{r}$$
$$Gamma^2_{21} = frac{1}{r}$$
If I plug these values back into their definition of the covariant derivative I get for equation (50):
$$A_{r;theta} = frac{1}{r^2}frac{partial A_r}{partial theta} - frac{A_theta}{r}$$
Why does this not match up with their results?
differential-geometry
asked Jun 13 '14 at 18:44
OSEOSE
95110
$endgroup$
add a comment |
$begingroup$
I am confused by the Wolfram article on cylindrical coordinates. Specifically, I do not understand how they go from equation (48) to equations (49)-(57).
Equation (48) shows that the covariant derivative is:
$$A_{j;k} = frac{1}{g_{kk}}frac{partial A_j}{partial x_k} - Gamma^i_{jk}A_i$$
The next few equations expand this for the case of cylindrical coordinates, equation (50) is:
$$A_{r;theta} = frac{1}{r}frac{partial A_r}{partial theta} - frac{A_theta}{r}$$
The contravariant metric tensor has non-zero elements:
$$g^{11} = 1$$
$$g^{22} = frac{1}{r^2}$$
$$g^{33} = 1$$
And the Christoffel symbols of the second kind have non-zero elements:
$$Gamma^1_{22} = -r$$
$$Gamma^2_{12} = frac{1}{r}$$
$$Gamma^2_{21} = frac{1}{r}$$
If I plug these values back into their definition of the covariant derivative I get for equation (50):
$$A_{r;theta} = frac{1}{r^2}frac{partial A_r}{partial theta} - frac{A_theta}{r}$$
Why does this not match up with their results?
differential-geometry
asked Jun 13 '14 at 18:44
OSEOSE
95110
$endgroup$
I am confused by the Wolfram article on cylindrical coordinates. Specifically, I do not understand how they go from equation (48) to equations (49)-(57).
Equation (48) shows that the covariant derivative is:
$$A_{j;k} = frac{1}{g_{kk}}frac{partial A_j}{partial x_k} - Gamma^i_{jk}A_i$$
The next few equations expand this for the case of cylindrical coordinates, equation (50) is:
$$A_{r;theta} = frac{1}{r}frac{partial A_r}{partial theta} - frac{A_theta}{r}$$
The contravariant metric tensor has non-zero elements:
$$g^{11} = 1$$
$$g^{22} = frac{1}{r^2}$$
$$g^{33} = 1$$
And the Christoffel symbols of the second kind have non-zero elements:
$$Gamma^1_{22} = -r$$
$$Gamma^2_{12} = frac{1}{r}$$
$$Gamma^2_{21} = frac{1}{r}$$
If I plug these values back into their definition of the covariant derivative I get for equation (50):
$$A_{r;theta} = frac{1}{r^2}frac{partial A_r}{partial theta} - frac{A_theta}{r}$$
Why does this not match up with their results?
differential-geometry
differential-geometry
asked Jun 13 '14 at 18:44
OSEOSE
95110
asked Jun 13 '14 at 18:44
OSEOSE
95110
edited Jun 16 '14 at 20:11
OSE
asked Jun 13 '14 at 18:44
OSEOSE
95110
asked Jun 13 '14 at 18:44
OSEOSE
95110
asked Jun 13 '14 at 18:44
OSEOSE
95110
95110
add a comment |
add a comment |
1 Answer
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The thing you forgot is the scale factor $frac{1}{r}$ given in equation (14). See Scale Factor in Mathworld.
answered Jun 13 '14 at 20:10
Mark FischlerMark Fischler
34k12552
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I don't see how the scale factor is involved with the first term. Could you explain it a little more explicitly? My understanding of this is a little shaky...
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– OSE
Jun 13 '14 at 21:05
add a comment |
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1 Answer
1
active
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1 Answer
1
active
oldest
votes
active
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active
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votes
$begingroup$
The thing you forgot is the scale factor $frac{1}{r}$ given in equation (14). See Scale Factor in Mathworld.
answered Jun 13 '14 at 20:10
Mark FischlerMark Fischler
34k12552
$endgroup$
$begingroup$
I don't see how the scale factor is involved with the first term. Could you explain it a little more explicitly? My understanding of this is a little shaky...
$endgroup$
– OSE
Jun 13 '14 at 21:05
add a comment |
$begingroup$
The thing you forgot is the scale factor $frac{1}{r}$ given in equation (14). See Scale Factor in Mathworld.
answered Jun 13 '14 at 20:10
Mark FischlerMark Fischler
34k12552
$endgroup$
$begingroup$
I don't see how the scale factor is involved with the first term. Could you explain it a little more explicitly? My understanding of this is a little shaky...
$endgroup$
– OSE
Jun 13 '14 at 21:05
add a comment |
$begingroup$
The thing you forgot is the scale factor $frac{1}{r}$ given in equation (14). See Scale Factor in Mathworld.
answered Jun 13 '14 at 20:10
Mark FischlerMark Fischler
34k12552
$endgroup$
The thing you forgot is the scale factor $frac{1}{r}$ given in equation (14). See Scale Factor in Mathworld.
answered Jun 13 '14 at 20:10
Mark FischlerMark Fischler
34k12552
answered Jun 13 '14 at 20:10
Mark FischlerMark Fischler
34k12552
answered Jun 13 '14 at 20:10
Mark FischlerMark Fischler
34k12552
answered Jun 13 '14 at 20:10
Mark FischlerMark Fischler
34k12552
34k12552
$begingroup$
I don't see how the scale factor is involved with the first term. Could you explain it a little more explicitly? My understanding of this is a little shaky...
$endgroup$
– OSE
Jun 13 '14 at 21:05
add a comment |
$begingroup$
I don't see how the scale factor is involved with the first term. Could you explain it a little more explicitly? My understanding of this is a little shaky...
$endgroup$
– OSE
Jun 13 '14 at 21:05
$begingroup$
I don't see how the scale factor is involved with the first term. Could you explain it a little more explicitly? My understanding of this is a little shaky...
$endgroup$
– OSE
Jun 13 '14 at 21:05
$begingroup$
I don't see how the scale factor is involved with the first term. Could you explain it a little more explicitly? My understanding of this is a little shaky...
$endgroup$
– OSE
Jun 13 '14 at 21:05
add a comment |
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