- #1

- 212

- 4

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?