I recently derived a matrix which I believe to be the metric tensor in spherical polar coordinates in 3-D. Here were the components of the tensor that I derived. I will show my work afterwards:(adsbygoogle = window.adsbygoogle || []).push({});

g_{11}= sin^{2}(ø) + cos^{2}(θ)

g_{12}= -rsin(θ)cos(θ)

g_{13}= rsin(ø)cos(ø)

g_{21}= -rsin(θ)cos(θ)

g_{22}= r^{2}sin^{2}(ø) + r^{2}sin^{2}(θ)

g_{23}= 0

g_{31}= rsin(ø)cos(ø)

g_{32}= 0

g_{33}= r^{2}cos^{2}(ø)

The above is what I derived, but when I tried to verify to see if my answer was correct by checking various websites, I did not see any site have what I derived.

Here is my work:

The axes were:

x^{1}= r

x^{2}= θ

x^{3}= ø

The vector that I differentiated was:

<rcos(θ)sin(ø) , rsin(θ)sin(ø) , rcos(θ)>

I then differentiated the vector with respect to the various axes in order to derive my tangential basis vectors.

Here were my basis vectors:

e_{r}= <cos(θ)sin(ø) , sin(θ)sin(ø), cos(θ)>

e_{θ}= <-rsin(θ)sin(ø), rcos(θ)sin(ø) , -rsin(θ)>

e_{ø}= <rcos(θ)cos(ø), rsin(θ)cos(ø) , 0>

Finally, I did the dot product with these basis vectors to derive the components of my metric tensor.

That is how I got what I derived, but I don't see any confirmation of this online.

Can anyone please either verify if I am right with this metric tensor or tell me where I went wrong?

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# Metric Tensor in Spherical Coordinates

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