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:
$$A_j;k = frac1g_kkfracpartial A_jpartial x_k - Gamma^i_jkA_i$$
The next few equations expand this for the case of cylindrical coordinates, equation (50) is:
$$A_r;theta = frac1rfracpartial A_rpartial theta - fracA_thetar$$
The contravariant metric tensor has non-zero elements:
$$g^11 = 1$$
$$g^22 = frac1r^2$$
$$g^33 = 1$$
And the Christoffel symbols of the second kind have non-zero elements:
$$Gamma^1_22 = -r$$
$$Gamma^2_12 = frac1r$$
$$Gamma^2_21 = frac1r$$
If I plug these values back into their definition of the covariant derivative I get for equation (50):
$$A_r;theta = frac1r^2fracpartial A_rpartial theta - fracA_thetar$$
Why does this not match up with their results?
differential-geometry
asked Jun 13 '14 at 18:44
OSE
9218
add a comment |Â
up vote
1
down vote
favorite
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:
$$A_j;k = frac1g_kkfracpartial A_jpartial x_k - Gamma^i_jkA_i$$
The next few equations expand this for the case of cylindrical coordinates, equation (50) is:
$$A_r;theta = frac1rfracpartial A_rpartial theta - fracA_thetar$$
The contravariant metric tensor has non-zero elements:
$$g^11 = 1$$
$$g^22 = frac1r^2$$
$$g^33 = 1$$
And the Christoffel symbols of the second kind have non-zero elements:
$$Gamma^1_22 = -r$$
$$Gamma^2_12 = frac1r$$
$$Gamma^2_21 = frac1r$$
If I plug these values back into their definition of the covariant derivative I get for equation (50):
$$A_r;theta = frac1r^2fracpartial A_rpartial theta - fracA_thetar$$
Why does this not match up with their results?
differential-geometry
asked Jun 13 '14 at 18:44
OSE
9218
add a comment |Â
up vote
1
down vote
favorite
up vote
1
down vote
favorite
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:
$$A_j;k = frac1g_kkfracpartial A_jpartial x_k - Gamma^i_jkA_i$$
The next few equations expand this for the case of cylindrical coordinates, equation (50) is:
$$A_r;theta = frac1rfracpartial A_rpartial theta - fracA_thetar$$
The contravariant metric tensor has non-zero elements:
$$g^11 = 1$$
$$g^22 = frac1r^2$$
$$g^33 = 1$$
And the Christoffel symbols of the second kind have non-zero elements:
$$Gamma^1_22 = -r$$
$$Gamma^2_12 = frac1r$$
$$Gamma^2_21 = frac1r$$
If I plug these values back into their definition of the covariant derivative I get for equation (50):
$$A_r;theta = frac1r^2fracpartial A_rpartial theta - fracA_thetar$$
Why does this not match up with their results?
differential-geometry
asked Jun 13 '14 at 18:44
OSE
9218
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:
$$A_j;k = frac1g_kkfracpartial A_jpartial x_k - Gamma^i_jkA_i$$
The next few equations expand this for the case of cylindrical coordinates, equation (50) is:
$$A_r;theta = frac1rfracpartial A_rpartial theta - fracA_thetar$$
The contravariant metric tensor has non-zero elements:
$$g^11 = 1$$
$$g^22 = frac1r^2$$
$$g^33 = 1$$
And the Christoffel symbols of the second kind have non-zero elements:
$$Gamma^1_22 = -r$$
$$Gamma^2_12 = frac1r$$
$$Gamma^2_21 = frac1r$$
If I plug these values back into their definition of the covariant derivative I get for equation (50):
$$A_r;theta = frac1r^2fracpartial A_rpartial theta - fracA_thetar$$
Why does this not match up with their results?
differential-geometry
differential-geometry
asked Jun 13 '14 at 18:44
OSE
9218
asked Jun 13 '14 at 18:44
OSE
9218
edited Jun 16 '14 at 20:11
asked Jun 13 '14 at 18:44
OSE
9218
asked Jun 13 '14 at 18:44
OSE
9218
asked Jun 13 '14 at 18:44
OSE
9218
9218
add a comment |Â
add a comment |Â
1 Answer
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The thing you forgot is the scale factor $frac1r$ given in equation (14). See Scale Factor in Mathworld.
answered Jun 13 '14 at 20:10
Mark Fischler
32k12049
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...
– OSE
Jun 13 '14 at 21:05
add a comment |Â
1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
up vote
0
down vote
The thing you forgot is the scale factor $frac1r$ given in equation (14). See Scale Factor in Mathworld.
answered Jun 13 '14 at 20:10
Mark Fischler
32k12049
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...
– OSE
Jun 13 '14 at 21:05
add a comment |Â
up vote
0
down vote
The thing you forgot is the scale factor $frac1r$ given in equation (14). See Scale Factor in Mathworld.
answered Jun 13 '14 at 20:10
Mark Fischler
32k12049
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...
– OSE
Jun 13 '14 at 21:05
add a comment |Â
up vote
0
down vote
up vote
0
down vote
The thing you forgot is the scale factor $frac1r$ given in equation (14). See Scale Factor in Mathworld.
answered Jun 13 '14 at 20:10
Mark Fischler
32k12049
The thing you forgot is the scale factor $frac1r$ given in equation (14). See Scale Factor in Mathworld.
answered Jun 13 '14 at 20:10
Mark Fischler
32k12049
answered Jun 13 '14 at 20:10
Mark Fischler
32k12049
answered Jun 13 '14 at 20:10
Mark Fischler
32k12049
answered Jun 13 '14 at 20:10
Mark Fischler
32k12049
32k12049
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...
– OSE
Jun 13 '14 at 21:05
add a comment |Â
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...
– OSE
Jun 13 '14 at 21:05
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...
– OSE
Jun 13 '14 at 21:05
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...
– OSE
Jun 13 '14 at 21:05
add a comment |Â
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