Metric Tensor in Spherical Coordinates

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:



g₁₁ = sin²(ø) + cos²(θ)



g₁₂ = -rsin(θ)cos(θ)



g₁₃ = rsin(ø)cos(ø)



g₂₁ = -rsin(θ)cos(θ)



g₂₂ = r²sin²(ø) + r²sin²(θ)



g₂₃ = 0



g₃₁ = rsin(ø)cos(ø)



g₃₂ = 0



g₃₃ = r²cos²(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:



x¹ = r



x² = θ



x³ = 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?