Throughout the question, keep in mind that I know very little about differential geometry. That is, only the intrinsic definitions of differentiable / Riemannian manifolds and the metric tensor, etc. I'm trying to understand the definition of the cross product given by Wikipedia here:

https://en.wikipedia.org/wiki/Cross_product#Index_notation_for_tensors

The article says that we can define the cross product. $ c $ of two vectors $ u $,$ v $ given an adequate "knit product" $ eta ^ {mi} $ as follows

$ c ^ m: = sum_ {i = 1} ^ 3 sum_ {j = 1} ^ 3 sum_ {k = 1} ^ 3 eta ^ {mi} epsilon_ {ijk} u ^ jv ^ k $

To demonstrate my current understanding of this definition, I will introduce some notation and terminology. Then I will show where my confusion arises with an example. I apologize in advance for the length of this post.

Leave $ M $ be a mild Riemannian variety in $ mathbb {R} ^ 3 $ with the metric tensor $ g $. Choose a coordinate table $ (U, phi) $ with $ phi $ A diffeomorphism. We define a collection. $ beta = {b_i: U to TM | i in {1,2,3 } } $ of vector fields, called coordinate vectors, as follows

$ b_i (x): = Big (x, big ( delta_x circ frac { partial { phi ^ {- 1}}} { partial {q_i}} circ phi big) (x ) Big) $

where $ delta_x: mathbb {R} ^ 3 to T_xM $ Denotes the canonical bijection. The coordinate vectors induce a natural base. $ gamma_x $ at each point $ x in U $ through the tangent space $ T_xM $. Leave $[g_x]_S $ denotes the matrix representation of the metric tensor at the point $ x $ in the standard basis for $ T_xM $ and let $[g_x]_ { gamma_x} $ denote the matrix representation in the base $ gamma_x $.

My understanding of the previous definition of the cross product now follows. Leave $ u, v in T_xM $ be tangent vectors and let

$[u]_ { gamma_x} = begin {bmatrix}

u_1 \

u_2 \

u_3

end {bmatrix} $ $ space space space space space space [v]_ { gamma_x} = begin {bmatrix}

v_1 \

v_2 \

v_3

end {bmatrix} $

denotes the coordinates of $ u, v $ at the base $ gamma_x $. Then we define the $ m $The coordinate of the cross product. $ u times v in T_xM $ at the base $ gamma_x $ as

$ big ([u times v]_ { gamma_x} big) _m: = sum_ {i = 1} ^ 3 sum_ {j = 1} ^ 3 sum_ {k = 1} ^ 3 big ([g_x]_ { gamma_x} big) _ {mi} epsilon_ {ijk} u_jv_k $

Now I will demonstrate my apparent misunderstanding with an example. Leave the multiple $ M $ be the usual Riemannian collector in $ mathbb {R} ^ 3 $ and let $ phi $ be given by

$ phi (x_1, x_2, x_3) = (x_1, x_2, x_3-x_1 ^ 2-x_2 ^ 2) $

$ phi ^ {- 1} (q_1, q_2, q_3) = (q_1, q_2, q_3 + q_1 ^ 2 + q_2 ^ 2) $

The Jacobian matrix $ J $ of $ phi ^ {- 1} $ is

$ J = begin {bmatrix}

1 & 0 & 0 \

0 & 1 & 0 \

2q_1 and 2q_2 and 1

end {bmatrix} $ $ space space space space space space J ^ {{1} = begin {bmatrix}

1 & 0 & 0 \

0 & 1 & 0 \

-2q_1 & -2q_2 & 1

end {bmatrix} $

And the matrix representation of the metric tensor in the base. $ gamma_x $ is

$[g_x]_ { gamma_x} = J ^ T[g_x]_SJ = begin {bmatrix}

1 + 4q_1 ^ 2 and 4q_1q_2 and 2q_1 \

4q_1q_2 & 1 + 4q_2 ^ 2 & 2q_2 \

2q_1 and 2q_2 and 1

end {bmatrix} $

Now choose $ x = (1,1, -1) $. The coordinates of $ x $ they are obviously $ phi (x) = (1,1,1) $ and the three above matrices are converted

$ J = begin {bmatrix}

1 & 0 & 0 \

0 & 1 & 0 \

2 and 2 and 1

end {bmatrix} $ $ space space space space space space J ^ {{1} = begin {bmatrix}

1 & 0 & 0 \

0 & 1 & 0 \

-2 & -2 & 1

end {bmatrix} $ $ space space space space space space [g_x]_ { gamma_x} = begin {bmatrix}

5 & ββ4 & 2 \

4 & 5 & 2 \

2 and 2 and 1

end {bmatrix} $

Now we calculate the cross product in the base. $ gamma_x $. Using my understanding of the definition as described above, I get

$[u times v]_ { gamma_x} = begin {bmatrix}

36 \

35 \

sixteen

end {bmatrix} $

If, on the other hand, we calculate the cross product on the standard basis, then, using my understanding of the definition, I get

$[u times v]_S = begin {bmatrix}

0 \

-one \

two

end {bmatrix} $

Naturally, these results should coincide if we make a base change in $[u times v]_ { gamma_x} $. Doing just that, I get

$[u times v]_S = J[u times v]_ { gamma_x} = begin {bmatrix}

1 & 0 & 0 \

0 & 1 & 0 \

2 and 2 and 1

end {bmatrix} begin {bmatrix}

36 \

35 \

sixteen

end {bmatrix} = begin {bmatrix}

36 \

35 \

158

end {bmatrix} $

Clearly, these do not agree. I can think of several reasons for this. Perhaps the definition given in Wikipedia is erroneous or only works for orthogonal coordinates. Maybe I am misinterpreting the definition given in Wikipedia. Or maybe I made a mistake somewhere in my calculation. My question then is as follows. How should I interpret the definition given in Wikipedia, and how should that definition be expressed using the notation provided here?