[ik,j]+[jk,i]=12(๐gij๐xk+๐gkj๐xiโ๐gik๐xj+๐gij๐xk+๐gki๐xjโ๐gjk๐xi)open bracket i k comma j close bracket plus open bracket j k comma i close bracket equals one-half open paren the fraction with numerator partial g sub i j end-sub and denominator partial x to the k-th power end-fraction plus the fraction with numerator partial g sub k j end-sub and denominator partial x to the i-th power end-fraction minus the fraction with numerator partial g sub i k end-sub and denominator partial x to the j-th power end-fraction plus the fraction with numerator partial g sub i j end-sub and denominator partial x to the k-th power end-fraction plus the fraction with numerator partial g sub k i end-sub and denominator partial x to the j-th power end-fraction minus the fraction with numerator partial g sub j k end-sub and denominator partial x to the i-th power end-fraction close paren Notice that
The line element in polar coordinates is .The non-zero metric tensor components are:
): Transform using the partial derivatives of the new coordinates with respect to the old ones. They typically represent positional vectors or velocities.
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Aฬj=๐xฬj๐xiAicap A bar to the j-th power equals the fraction with numerator partial x bar to the j-th power and denominator partial x to the i-th power end-fraction cap A to the i-th power
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Aฬ2=๐xฬ2๐x1A1+๐xฬ2๐x2A2cap A bar squared equals the fraction with numerator partial x bar squared and denominator partial x to the first power end-fraction cap A to the first power plus the fraction with numerator partial x bar squared and denominator partial x squared end-fraction cap A squared Calculate the partial derivatives: It is available on Scribd
By the chain rule of partial differentiation, this product simplifies directly to:
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Show that the inner product of a contravariant vector Aicap A to the i-th power and a covariant vector Bicap B sub i is a scalar (invariant under coordinate transformation). Solution: Write the transformation laws for Aicap A to the i-th power Bicap B sub i into a new coordinate system xโฒpx prime to the p-th power
into an expression. Alternatively, we can raise one of the indices of Tijcap T sub i j end-sub to contract it directly with Bkcap B to the k-th power
): Placed as superscripts. They transform like coordinate differentials ( dxid x to the i-th power Covariant Indices ( Aicap A sub i