logo

Laplacian of a Scalar Function in Curvilinear Coordinates 📂Mathematical Physics

Laplacian of a Scalar Function in Curvilinear Coordinates

Theorem

In the curvilinear coordinate system, the Laplacian of a scalar function f=f(q1,q2,q3)f=f(q_{1},q_{2},q_{3}) is as follows.

∇2f=1h1h2h3[∂∂q1(h2h3h1∂f∂q1)+∂∂q2(h1h3h2∂f∂q2)+∂∂q3(h1h2h3∂f∂q3)] \nabla ^{2}f= \frac{1}{h_{1}h_{2}h_{3}}\left[\frac{ \partial }{ \partial q_{1} } \left( \frac{h_{2}h_{3}}{h_{1}} \frac{ \partial f}{ \partial q_{1}}\right)+\frac{ \partial }{ \partial q_{2} } \left( \frac{h_{1}h_{3}}{h_{2}} \frac{ \partial f}{ \partial q_{2}}\right)+\frac{ \partial }{ \partial q_{3} } \left( \frac{h_{1}h_{2}}{h_{3}} \frac{ \partial f}{ \partial q_{3}}\right) \right]

Formulas

  • Cartesian coordinates:

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

    ∇2f=∂2f∂x2+∂2f∂y2+∂2f∂z2 \nabla ^2 f= \frac{ \partial^2 f}{ \partial x^2 }+\frac{ \partial^2 f}{ \partial y^2}+\frac{ \partial^2 f}{ \partial z^2}

  • Cylindrical coordinates:

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

    ∇2f=1ρ∂∂ρ(ρ∂f∂ρ)+1ρ2∂2f∂ϕ2+∂2f∂z2 \nabla ^{2}f= \frac{1}{\rho} \frac{ \partial }{ \partial \rho }\left( \rho\frac{ \partial f}{ \partial \rho} \right) + \frac{1}{\rho^{2}} \frac{\partial ^{2} f}{\partial \phi^{2} }+ \frac{\partial ^{2} f}{\partial z^{2} }

  • Spherical coordinates:

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

    ∇2f=1r2∂∂r(r2∂f∂r)+1r2sin⁡θ∂∂θ(sin⁡θ∂f∂θ)+1r2sin⁡2θ∂2f∂ϕ2 \nabla ^{2}f= \frac{1}{r^{2}} \frac{ \partial }{ \partial r }\left(r^{2} \frac{ \partial f}{ \partial r} \right) + \frac{1}{r^{2}\sin\theta}\frac{ \partial }{ \partial \theta }\left( \sin \theta \frac{ \partial f}{ \partial \theta} \right)+\frac{1}{r^{2}\sin^{2}\theta} \frac{\partial ^{2} f}{\partial \phi^{2} }

Derivation

By applying the gradient and divergence in the curvilinear coordinate system in sequence, it can be obtained. The gradient of a certain scalar function ff is as follows

∇f=1h1∂f∂q1q^1+1h2∂f∂q2q2^+1h3∂f∂q3q^3 \nabla f= \frac{1}{h_{1}}\frac{ \partial f }{ \partial q_{1} } \hat{\mathbf{q}}_{1} + \frac{1}{h_{2}}\frac{ \partial f }{ \partial q }_{2} \hat{\mathbf{q}_{2}}+\frac{1}{h_{3}}\frac{ \partial f }{ \partial q_{3} } \hat{\mathbf{q}}_{3}

and the divergence of a certain vector function F\mathbf{F} 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]

Therefore, the Laplacian of ff is as follows.

∇⋅(∇f)=∇⋅(1h1∂f∂q1q^1+1h2∂f∂q2q2^+1h3∂f∂q3q^3)=1h1h2h3[∂∂q1(h2h3h1∂f∂q1)+∂∂q2(h1h3h2∂f∂q2)+∂∂q3(h1h2h3∂f∂q3)] \begin{align*} \nabla \cdot (\nabla f) &=\nabla \cdot \left( \frac{1}{h_{1}}\frac{ \partial f }{ \partial q_{1} } \hat{\mathbf{q}}_{1} + \frac{1}{h_{2}}\frac{ \partial f }{ \partial q }_{2} \hat{\mathbf{q}_{2}}+\frac{1}{h_{3}}\frac{ \partial f }{ \partial q_{3} } \hat{\mathbf{q}}_{3} \right) \\ &=\frac{1}{h_{1}h_{2}h_{3}}\left[\frac{ \partial }{ \partial q_{1} } \left( \frac{h_{2}h_{3}}{h_{1}} \frac{ \partial f}{ \partial q_{1}}\right)+\frac{ \partial }{ \partial q_{2} } \left( \frac{h_{1}h_{3}}{h_{2}} \frac{ \partial f}{ \partial q_{2}}\right)+\frac{ \partial }{ \partial q_{3} } \left( \frac{h_{1}h_{2}}{h_{3}} \frac{ \partial f}{ \partial q_{3}}\right) \right] \end{align*}

■

See Also