- #1

- 11

- 1

- Homework Statement
- For a function f( x, y ) = x^2 + y^2 find the metric tensor at

(0, 0), (0, 1), (1, 0), (1, 1)

- Relevant Equations
- ( DirectionalDerivitive( v, f( x, y ) ) )^2

I have been teaching myself general relativity and wanted to see if I got these metric tensors right, I have a feeling I didn't.

For the first one I get all my directional derivatives

(0, 0): (0)i + (0)j

(0, 1): (0)i + 2j

(1, 0): 2i + (0)j

(1, 1): 2i + 2j

Then I square them (FOIL):

(0, 0): (0)i + (0)j + (0)ij + (0)ij

(0, 1): (0)i

(1, 0): 4i

(1, 1): 4i

Then I put the products into a matrix

(0, 0):

g = [ [ 0, 0 ]

[ 0, 0 ] ]

(0, 1):

g = [ [ 0, 0 ]

[ 0, 4 ] ]

(1, 0):

g = [ [ 4, 0 ]

[ 0, 0 ] ]

(1, 1):

g = [ [ 4, 16 ]

[ 16, 4 ] ]

I tried searching for "metric tensor calculator" but couldn't find anything to verify my results.

I have been following these tutorials they are really good and exactly the format I want, I have also done a little looking into non - euclidean geometry.

For the first one I get all my directional derivatives

(0, 0): (0)i + (0)j

(0, 1): (0)i + 2j

(1, 0): 2i + (0)j

(1, 1): 2i + 2j

Then I square them (FOIL):

(0, 0): (0)i + (0)j + (0)ij + (0)ij

(0, 1): (0)i

^{2}+ (0)ij + (0)ij+ 4j^{2}= (0)i + (0)ij + (0)ij + 4j(1, 0): 4i

^{2}+ (0)ij + (0)j^{2}= 4i + (0)ij + (0)ij + (0)j(1, 1): 4i

^{2}+ 4j^{2}+ 16(i^{2})(j^{2}) + 16(i^{2})(j^{2}) = 4i + 4j + 16ij + 16ijThen I put the products into a matrix

(0, 0):

g = [ [ 0, 0 ]

[ 0, 0 ] ]

(0, 1):

g = [ [ 0, 0 ]

[ 0, 4 ] ]

(1, 0):

g = [ [ 4, 0 ]

[ 0, 0 ] ]

(1, 1):

g = [ [ 4, 16 ]

[ 16, 4 ] ]

I tried searching for "metric tensor calculator" but couldn't find anything to verify my results.

I have been following these tutorials they are really good and exactly the format I want, I have also done a little looking into non - euclidean geometry.