Divergence in Spherical Coordinates Derivation and Verification

The derivation of the divergence $\nabla \cdot v$ in spherical coordinates begins with the general tensor calculus formula, $\nabla \cdot v=\frac{1}{\sqrt{g}} \partial_a\left(\sqrt{g} v^a\right)$. The crucial geometric factor for this coordinate system is the square root of the metric determinant, $\sqrt{ g }= r ^{ 2 } \sin (\theta)$. Substituting this into the formula and simplifying yields the divergence in terms of the contravariant components $\left(v^a\right): \nabla \cdot v= \frac{1}{r^2} \partial_r\left(r^2 v^r\right)+\frac{1}{\sin (\theta)} \partial_\theta\left(\sin (\theta) v^\theta\right)+\partial_{\varphi} v^{\varphi}$. To verify this result against the standard physics expression, the contravariant components were converted to the physical components ( $\tilde{v}_a$ ) using the relationship $\tilde{v}a=\sqrt{\text { g }{a a}} v^a$, which introduces specific scaling factors like $1 / r$ and $1 /(r \sin (\theta))$ for the $\theta$ and $\varphi$ components, confirming the tensor-based derivation is consistent with the traditional vector analysis formula.

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✍️Mathematical Proof

$\complement\cdots$Counselor

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Universal Divergence Formula: The general expression for the divergence of a vector $v$ in any curvilinear coordinate system is fundamental and was the starting point:

$$ \nabla \cdot v=\frac{1}{\sqrt{g}} \partial_a\left(\sqrt{g} v^a\right) $$

Metric Determinant ( $\sqrt{ g }$ ): The specific geometry of spherical coordinates is encoded entirely in $\sqrt{g}$. For $(r, \theta, \varphi)$, the determinant is $g=r^4 \sin ^2(\theta)$, yielding the essential factor:

$$ \sqrt{g}= r ^2 \sin (\theta) $$

Contravariant Component Result: Substituting $\sqrt{g}$ and simplifying the derivatives yields the divergence in terms of the contravariant components $\left(v^r, v^\theta, v^{\varphi}\right)$ :

$$ \nabla \cdot v=\frac{1}{r^2} \partial_r\left(r^2 v^r\right)+\frac{1}{\sin (\theta)} \partial_\theta\left(\sin (\theta) v^\theta\right)+\partial_{\varphi} v^{\varphi} $$

Physical vs. Contravariant Components: To compare the derived result with the standard physics formula (1.211b), it was necessary to relate the contravariant components ( $v^a$ ) to the physical components ( $\tilde{v}a$ ) using the metric's diagonal elements: $\tilde{v}a=\sqrt{g{a a}} v^a$.
Scaling Factors: The conversion from contravariant to physical components introduced the necessary scaling factors:

$$ v^\theta=\frac{1}{ r } \tilde{v}\theta \quad \text { and } \quad v^{\varphi}=\frac{1}{ r \sin (\theta)} \tilde{v}{\varphi} $$

Verification: Substituting these scaled components into the contravariant formula successfully reproduced the standard expression in terms of physical components, thus verifying the general tensor formula for divergence in spherical coordinates.

✍️Mathematical Proof

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🧄Proof and Derivation-1

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