Covariant Differentiation

Relevant parts to questions...

• Vectors: $A_{i,k}=\frac{\partial A_i}{\partial x^k}-{\Gamma }_{ik}^j{A}_j$, $A_{\cdot k}^i=\frac{\partial A^i}{\partial x^k}+{\Gamma }_{jk}^i{A}^j$.
• Sign rule: lower index → $-\Gamma$, upper index → $+\Gamma$.
• Extension rule: each index of a tensor contributes its own $\Gamma$ correction.
• In a fixed basis, all $\Gamma =0$ and covariant derivatives become ordinary partials.

Covariant Differentiation is the correct way to differentiate tensors in a Local Basis. It modifies the ordinary partial derivative with Christoffel Symbols so that the result is itself a tensor - even when the basis vectors vary with position.

Vectors

For the covariant and contravariant components of a vector $\mathbf{A}$:

$$\boxed{A_{i,k}=\frac{\partial A_i}{\partial x^k}-{\Gamma }_{ik}^j{A}_j}$$ $$\boxed{A_{\cdot k}^i=\frac{\partial A^i}{\partial x^k}+{\Gamma }_{jk}^i{A}^j}$$

The first term differentiates the components; the second term - the Christoffel correction - accounts for how the basis itself changes.

Sign and index placement

The covariant component picks up a minus; the contravariant component picks up a plus. The free index on $\Gamma$ matches the component’s index position (up or down). Swap them and the whole calculation collapses.

Higher-Rank Tensors

Each index of the tensor contributes one Christoffel correction:

• Rank-2 covariant ($T_{ik}$):

$$T_{ik,\ell }=\frac{\partial T_{ik}}{\partial x^{\ell }}-{\Gamma }_{i\ell }^m{T}_{mk}-{\Gamma }_{k\ell }^m{T}_{im}$$

• Rank-2 contravariant ($T^{ik}$):

$$T_{\cdot \cdot ,\ell }^{ik}=\frac{\partial T^{ik}}{\partial x^{\ell }}+{\Gamma }_{m\ell }^i{T}^{mk}+{\Gamma }_{m\ell }^k{T}^{im}$$

• Rank-2 mixed ($T_{\cdot k}^i$):

$$T_{\cdot k,\ell }^i=\frac{\partial T_{\cdot k}^i}{\partial x^{\ell }}+{\Gamma }_{m\ell }^i{T}_{\cdot k}^m-{\Gamma }_{k\ell }^m{T}_{\cdot m}^i$$

Rule of thumb: partial derivative + one $\Gamma$ per index ($+$ for up, $-$ for down).

Properties

• Always produces a tensor: that is the entire reason for the correction. Partial derivatives of tensor components, by themselves, do not transform as tensors (see Christoffel Symbols - they transform with a non-tensorial extra term, and those extra terms are exactly what the covariant derivative cancels).
• Reduces to ordinary partials in Cartesian coordinates, since $\Gamma =0$ there.
• Rank-raising::$T_{ik,\ell }$ is rank 3 - the covariant derivative adds one lower index.
• Does not commute with itself in curved space: ${\nabla }_j{\nabla }_k{A}_i-{\nabla }_k{\nabla }_j{A}_i=R_{ijk}^r{A}_r$ introduces the Riemann-Christoffel Tensor (see Ricci’s Theorem for the metric case).

Applications

1. Writing vector calculus identities that work in any coordinate system. For example, $\text{div}(\mathbf{A})=A_{\cdot i}^i$ is valid in spherical, cylindrical, and curved spaces.
2. Generalising ODEs/PDEs to manifolds, by replacing $\frac{\partial }{\partial x^i}$ with ${\nabla }_i$ throughout.
3. Detecting curvature, via the non-commutativity of second covariant derivatives (see Riemann-Christoffel Tensor).

Covariant derivative collapses in Cartesian coordinates

In Cartesian coordinates, ${\Gamma }_{jk}^i=0$. So:

$A_{i,k}=\frac{\partial A_i}{\partial x^k}-0=\boxed{\frac{\partial A_i}{\partial x^k}}$ - the covariant derivative is just the ordinary partial.

This matches what we have been doing all along in standard vector calculus: it was secretly a special case of covariant differentiation.

Partial derivative + one $\Gamma$ per index, $+$ for up, $-$ for down.