logo

Gradient of a Scalar Function in Curvilinear Coordinates 📂Mathematical Physics

Gradient of a Scalar Function in Curvilinear Coordinates

Theorem

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

∇f=1h1∂f∂q1q^1+1h2∂f∂q2q^2+1h3∂f∂q3q^3=∑i=131hi∂f∂qiq^i \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}=\sum \limits _{i=1} ^{3}\frac{1}{h_{i}}\frac{ \partial f}{ \partial q_{i}}\hat{\mathbf{q}}_{i}

hih_{i} is the scale factor.

Formulas

  • Cartesian Coordinate System:

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

    ∇f=∂f∂xx^+∂f∂yy^+∂f∂zz^ \nabla f= \frac{\partial f}{\partial x}\mathbf{\hat{\mathbf{x}} }+ \frac{\partial f}{\partial y}\mathbf{\hat{\mathbf{y}}} + \frac{\partial f}{\partial z}\mathbf{\hat{\mathbf{z}}}

  • Cylindrical Coordinate System:

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

    ∇f=∂f∂ρρ^+1ρ∂f∂ϕϕ^+∂f∂zz^ \nabla f = \frac{\partial f}{\partial \rho}\boldsymbol{\hat \rho} + \frac{1}{\rho}\frac{\partial f}{\partial \phi}\boldsymbol{\hat \phi} + \frac{\partial f}{\partial z}\mathbf{\hat{\mathbf{z}}}

  • Spherical Coordinate System:

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

    ∇f=∂f∂rr^+1r∂f∂θθ^+1rsin⁡θ∂f∂ϕϕ^ \nabla f= \frac{\partial f}{\partial r} \mathbf{\hat r} + \frac{1}{r}\frac{\partial f}{\partial \theta} \boldsymbol{\hat \theta} + \frac{1}{r\sin\theta}\frac{\partial f}{\partial \phi}\boldsymbol{\hat \phi}

Derivation

In the three-dimensional Cartesian coordinate system, a\mathbf{a}, which satisfies the following equation, is defined as the gradient of ff, and is denoted as ∇f\nabla f.

df=a⋅dr d f =\mathbf{a} \cdot d\mathbf{r}

The same definition is applied to any curvilinear coordinate system. The total differential of ff is as follows.

df=∂f∂q1dq1+∂f∂q2dq2+∂f∂q3dq3 d f = \frac{ \partial f}{ \partial q_{1} }dq_{1}+\frac{ \partial f}{ \partial q_{2}}dq_{2}+\frac{ \partial f}{ \partial q_{3}}dq_{3}

In a curvilinear coordinate system, the infinitesimal change in the position vector r\mathbf{r} is as follows.

dr=h1dq1q^1+h2dq2q^2+h3dq3q^3 d\mathbf{r}=h_{1}dq_{1}\hat{\mathbf{q}}_{1}+h_{2}dq_{2}\hat{\mathbf{q}}_{2}+h_{3}dq_{3}\hat{\mathbf{q}}_{3}

Now, what we seek is a\mathbf{a} that satisfies the equation below.

df=a⋅dr \begin{equation} df=\mathbf{a} \cdot d\mathbf{r} \end{equation}

Given a=a1q^1+a2q^2+a3q^3\mathbf{a}=a_{1}\hat{\mathbf{q}}_{1}+a_{2}\hat{\mathbf{q}}_{2}+a_{3}\hat{\mathbf{q}}_{3}, then (1)(1) is as follows.

∂f∂q1dq1+∂f∂q2dq2+∂f∂q3dq3=a1h1dq1+a2h2dq2+a3h3dq3 \frac{ \partial f}{ \partial q_{1} }dq_{1}+\frac{ \partial f}{ \partial q_{2}}dq_{2}+\frac{ \partial f}{ \partial q_{3}}dq_{3} = a_{1}h_{1}dq_{1}+a_{2}h_{2}dq_{2}+a_{3}h_{3}dq_{3}

Therefore, given ai=1hi∂f∂qia_{i}=\dfrac{1 }{h_{i}}\dfrac{ \partial f}{ \partial q_{i} }, the following holds true.

a=1h1∂f∂q1q^1+1h2∂f∂q2q^2+1h3∂f∂q3q^3 \quad \mathbf{a}=\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}

Now, the vector a\mathbf{a} defined as the gradient of ff and denoted as ∇f\nabla f.

■

See Also