Gradient and Divergence in Orthonormal Curvilinear Coordinates

Gradient and Divergence in Orthonormal Curvilinear Coordinates Swapnil Sunil Jain Aug 7, 2006

Gradient and Divergence in Orthonormal Curvilinear Coordinates

Gradient in Curvilinear Coordinates

In rectangular coordinates (where f=f(x,y,z)), an infinitesimalMathworldPlanetmathPlanetmath length vector dl is given by


the gradient is given by


and the differentialMathworldPlanetmath change in the output is given by


Similarly in orthonormal curvilinear coordinates ( where f=f(q1,q2,q3)), the infinitesimal length vector is given by11See my article Unit Vectors in Curvilinear Coordinates for an insight into this expression.



hi=k(xkqi)2 and q^i=1hi(xkqi) for i1,2,3

So if


then since we know that




this implies that



= 1h1q1q^1+1h2q2q^2+1h3q3q^3
= i1hiqiq^i

Divergence in Curvilinear Coordinates

In the previous sectionMathworldPlanetmath we concluded that in curvilinear coordinates, the gradient operator is given by


Then for F=F1q^1+F2q^2+F3q^3, the divergence of F is given by


which is not equal to


as one would think! The real expression can be derived the following way,

=i[(1hiq^i)jq^jFjqi]call it A+i[(1hiq^i)jFjq^jqi]call it B
A = i[(1hiq^i)jq^jFjqi]
= i[1hij(q^iq^j)δijFjqi]
= i1hiFiqi
B = i[(1hiq^i)jFjq^jqi]

Using the following equality22The proof of this identity is left as an exercise for the reader.

q^jqi=q^ihjhiqj  ij

we can write B as

B = i[(1hiq^i)jFj(q^i1hjhiqj)]  ij
= i[1hijFj(q^iq^i)11hjhiqj]  ij
= i[1hijFj1hjhiqj]  ij
= ijFjhjhihiqj
= i1F1h1hihiq1+i2F2h2hihiq2+i3F3h3hihiq3
F = A+B
= i1hiFiqi+i1F1h1hihiq1+i2F2h2hihiq2+i3F3h3hihiq3
= [1h1F1q1+1h2F2q2+1h3F3q3]

Collecting similarMathworldPlanetmathPlanetmath terms together we get,

F = [1h1F1q1+F1h1h2h2q1+F1h1h3h3q1]

If we define ΩΠihi, we can further write the above expression as

F = [h2h3ΩF1q1+F1h3Ωh2q1+F1h2Ωh3q1]
= 1Ω([F1q1h2h3+F1h3h2q1+F1h2h3q1]
= 1Ω(q1(F1h2h3)+q2(h1F2h3)+q3(h1h2F3))


F = 1Ωiqi(ΩhiFi) where Ω=Πihi
Title Gradient and Divergence in Orthonormal Curvilinear Coordinates
Canonical name GradientAndDivergenceInOrthonormalCurvilinearCoordinates1
Date of creation 2013-03-11 19:26:23
Last modified on 2013-03-11 19:26:23
Owner swapnizzle (13346)
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Numerical id 1
Author swapnizzle (0)
Entry type Definition