logo

Divergence of Vector Functions in Curvilinear Coordinates 📂Mathematical Physics

Divergence of Vector Functions in Curvilinear Coordinates

Theorem

The divergence of the vector function F=F(q1,q2,q3)=F1q^1+F2q^2+F3q^3\mathbf{F}=\mathbf{F}(q_{1},q_{2},q_{3})=F_{1}\hat{\mathbf{q}}_{1}+F_{2}\hat{\mathbf{q}}_{2}+F_{3}\hat{\mathbf{q}}_{3} in curvilinear coordinates is as follows.

∇⋅F=1h1h2h3[∂∂q1(h2h3F1)+∂∂q2(h1h3F2)+∂∂q3(h1h2F3)] \nabla \cdot \mathbf{F}=\frac{1}{h_{1}h_{2}h_{3}}\left[ \frac{ \partial }{ \partial q_{1} }(h_{2}h_{3}F_{1})+\frac{ \partial }{ \partial q_{2} }(h_{1}h_{3}F_{2})+\frac{ \partial }{ \partial q_{3} }(h_{1}h_{2}F_{3}) \right]

hih_{i} is the scale factor.

Formulas

  • Cartesian coordinates:

    h1=h2=h3=1 h_{1}=h_{2}=h_{3}=1

    ∇⋅F=∂Fx∂x+∂Fy∂y+∂Fz∂z \begin{align*} \nabla \cdot \mathbf{F} =\frac{\partial F_{x}}{\partial x}+\frac{\partial F_{y}}{\partial y}+\frac{\partial F_{z}}{\partial z} \end{align*}

  • Cylindrical coordinates:

    h1=1,h2=ρ,h3=1 h_{1}=1,\quad h_{2}=\rho,\quad h_{3}=1

    ∇⋅F=1ρ(∂(ρFρ)∂ρ+∂(Fϕ)∂ϕ+∂(ρFz)∂z)=1ρ∂(ρFρ)∂ρ+1ρ∂Fϕ∂ϕ+∂Fz∂z \begin{align*} \nabla \cdot \mathbf{F} &= \frac{1}{\rho} \left( \frac{\partial (\rho F_{\rho})}{\partial \rho} + \frac{\partial (F_{\phi})}{\partial \phi} + \frac{\partial (\rho F_{z})}{\partial z} \right) \\ &= \frac{1}{\rho} \frac{\partial (\rho F_{\rho})}{\partial \rho} + \frac{1}{\rho}\frac{\partial F_{\phi}}{\partial \phi} + \frac{\partial F_{z}}{\partial z} \end{align*}

    • Spherical coordinates:

    h1=1,h2=r,h3=rsin⁡θ h_{1}=1,\quad h_{2}=r\quad, h_{3}=r\sin\theta

    ∇⋅F=1r2sin⁡θ(∂(r2sin⁡θFr)∂r+∂(rsin⁡θFθ)∂θ+∂(rFϕ)∂ϕ)=1r2∂(r2Fr)∂r+1rsin⁡θ∂(sin⁡θFθ)∂θ+1rsin⁡θ∂Fϕ∂ϕ \begin{align*} \nabla \cdot \mathbf{F} &= \frac{1}{r^{2}\sin\theta}\left( \frac{\partial (r^{2}\sin\theta F_{r})}{\partial r}+\frac{\partial (r\sin\theta F_{\theta})}{\partial \theta}+\frac{\partial (rF_{\phi})}{\partial \phi} \right) \\ &= \frac{1}{r^{2}}\frac{\partial (r^{2} F_{r})}{\partial r}+\frac{1}{r\sin\theta}\frac{\partial (\sin\theta F_{\theta})}{\partial \theta}+\frac{1}{r\sin\theta}\frac{\partial F_{\phi}}{\partial \phi} \end{align*}

Derivation

In the 3D Cartesian coordinate system, the divergence ∇⋅F\nabla \cdot \mathbf{F} of the vector function F\mathbf{F} told us how F\mathbf{F} flows. We can get the divergence in the curvilinear coordinates in the same way. Let’s first compute it only in the direction of q1q_{1} axis. The amount passing through da1d\mathbf{a}_{1} and da2d\mathbf{a}_{2} can be calculated by the inner product of the two vectors. The computation is the same as in the Cartesian coordinates, so some parts are omitted.

F(q1+dq1,q2,q3)⋅da1=F1(q1+dq1,q2,q3)h2h3dq2dq3F(q1,q2,q3)⋅da2=−F1(q1,q2,q3)h2h3dq2dq3 \begin{align*} \mathbf{F}(q_{1}+dq_{1},q_{2},q_{3})\cdot d\mathbf{a}_{1} &= F_{1}(q_{1}+dq_{1},q_{2},q_{3})h_{2}h_{3}dq_{2}dq_{3} \\ \mathbf{F}(q_{1},q_{2},q_{3})\cdot d\mathbf{a}_{2} &=- F_{1}(q_{1},q_{2},q_{3})h_{2}h_{3}dq_{2}dq_{3} \end{align*}

And the sum of these two is the influx (outflux).

F1(q1+dq1,q2,q3)h2h3dq2dq3−F1(q1,q2,q3)h2h3dq2dq3=F1(q1+dq1,q2,q3)h2h3−F1(q1,q2,q3)h2h3dq1dq1dq2dq3≈∂(F1h2h3)∂q1dq1dq2dq3 \begin{align*} &F_{1}(q_{1}+dq_{1},q_{2},q_{3})h_{2}h_{3}dq_{2}dq_{3}- F_{1}(q_{1},q_{2},q_{3})h_{2}h_{3}dq_{2}dq_{3} \\ =& \frac{F_{1}(q_{1}+dq_{1},q_{2},q_{3})h_{2}h_{3}- F_{1}(q_{1},q_{2},q_{3})h_{2}h_{3} }{dq_{1}}dq_{1}dq_{2}dq_{3} \\ \approx &\frac{ \partial (F_{1}h_{2}h_{3})}{ \partial q_{1}}dq_{1}dq_{2}dq_{3} \end{align*}

In the same way, if calculated for q2q_{2} and q3q_{3}, it is as follows.

∂(F2h1h3)∂q2dq1dq2dq3and∂(F3h1h2)∂q3dq1dq2dq3 \frac{ \partial (F_{2}h_{1}h_{3})}{ \partial q_{2}}dq_{1}dq_{2}dq_{3}\quad \text{and} \quad \frac{ \partial (F_{3}h_{1}h_{2})}{ \partial q_{3}}dq_{1}dq_{2}dq_{3}

Adding these gives the amount of F\mathbf{F} entering or leaving, and dividing it by the volume dV=h1h2h3dq1dq2dq3dV=h_{1}h_{2}h_{3}dq_{1}dq_{2}dq_{3} gives the influx (outflux) per unit volume.

∂(F1h2h3)∂q1dq1dq2dq3+∂(F2h1h3)∂q2dq1dq2dq3+∂(F3h1h2)∂q3dq1dq2dq3= (∂(F1h2h3)∂q1+∂(F2h1h3)∂q2+∂(F3h1h2)∂q3)dq1dq2dq31dV×  ⟹  1h1h2h3(∂(F1h2h3)∂q1+∂(F2h1h3)∂q2+∂(F3h1h2)∂q3) \begin{align*} & \frac{ \partial (F_{1}h_{2}h_{3})}{ \partial q_{1}}dq_{1}dq_{2}dq_{3}+\frac{ \partial (F_{2}h_{1}h_{3})}{ \partial q_{2}}dq_{1}dq_{2}dq_{3}+\frac{ \partial (F_{3}h_{1}h_{2})}{ \partial q_{3}}dq_{1}dq_{2}dq_{3} \\ =&\ \left( \frac{ \partial (F_{1}h_{2}h_{3})}{ \partial q_{1}}+\frac{ \partial (F_{2}h_{1}h_{3})}{ \partial q_{2}}+\frac{ \partial (F_{3}h_{1}h_{2})}{ \partial q_{3}} \right)dq_{1}dq_{2}dq_{3} \\ \frac{1}{dV}\times \implies &\frac{1}{h_{1}h_{2}h_{3}}\left( \frac{ \partial (F_{1}h_{2}h_{3})}{ \partial q_{1}}+\frac{ \partial (F_{2}h_{1}h_{3})}{ \partial q_{2}}+\frac{ \partial (F_{3}h_{1}h_{2})}{ \partial q_{3}} \right) \end{align*}

■

See also