Christoffel第一类符号变换律相关困惑求解

Hey there! I totally get your frustration—Schaum’s textbooks are great for practice, but they often skip over the nitty-gritty derivations that leave you scratching your head. Let’s break down this "p and q are symmetric" explanation for the Christoffel first kind symbol transformation law step by step.

First, let’s recap the key definitions to set the stage:

• The Christoffel symbol of the first kind is defined as:
  [ij,k] = ½(∂g_ik/∂x^j + ∂g_jk/∂x^i - ∂g_ij/∂x^k)
  where g_ij is the metric tensor, which is symmetric by definition: g_ij = g_ji for all i,j.
• The metric tensor transforms between coordinate systems (say, from x^p to x'^m) as:
  g'_ij = g_pq ∂x^p/∂x'^i ∂x^q/∂x'^j

The "p and q Symmetry" in the Transformation Law #

When deriving the transformation law for [ij,k], we start by taking partial derivatives of the transformed metric tensor and plugging into the Christoffel definition. The part that’s confusing is likely the mixed partial derivative terms that pop up. Let’s look at those:

When you expand the partial derivative of g'_ij with respect to x'^k, you get two types of terms:

1. Terms involving derivatives of the original metric tensor g_pq (these end up linking back to the original Christoffel symbols).
2. Mixed partial derivative terms like:
   g_pq ∂x^p/∂x'^i ∂²x^q/∂x'^j ∂x'^k
   and
   g_pq ∂x^q/∂x'^i ∂²x^p/∂x'^j ∂x'^k

Here’s where the symmetry comes in:

• Since the metric tensor is symmetric (g_pq = g_qp), swapping the indices p and q in either term doesn’t change the value of the term. For example, g_pq ∂x^p/∂x'^i ∂²x^q/∂x'^j ∂x'^k is identical to g_qp ∂x^q/∂x'^i ∂²x^p/∂x'^j ∂x'^k—and since g_qp = g_pq, these two terms are the same.
• Additionally, mixed partial derivatives are commutative (∂²x^p/∂x'^i ∂x'^j = ∂²x^p/∂x'^j ∂x'^i), which reinforces that swapping related indices doesn’t alter the term’s value.

Why This Matters #

This symmetry lets us combine or simplify these terms in the transformation law. Instead of writing out both separate terms, we can express them as a single term multiplied by 2 (or just keep one and recognize the other is equivalent). Schaum’s books often skip this derivation step because they assume familiarity with metric tensor symmetry, but it’s totally reasonable to want the full breakdown!

If you’re working through a specific line of the transformation law and want to dive deeper, feel free to share the exact equation you’re stuck on—I can walk through it with you.

内容的提问来源于stack exchange，提问作者Rima