Christoffel symbols are a measure of how a coordinate system's basis vectors change from point to point. In the linear coordinate system on the left, the basis vectors (v₁ and v₂) are the same everywhere. Since there is no change, their derivatives are zero, which means all Christoffel symbols are zero. This holds true even though the system is non-orthogonal. In the spherical (curvilinear) coordinate system on the right, the basis vectors (∂/∂r and ∂/∂θ) constantly change direction. This change is precisely what the non-zero Christoffel symbols account for.
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compares a simple linear coordinate system with zero Christoffel symbols to a curvilinear system with non-zero
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$\gg$The Metric Tensor Covariant Derivatives and Tensor Densities
$\ggg$Mathematical Structures Underlying Physical Laws
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