Finding the inverse metric tensor from a given line element

[Sayak Das]

Defining dS² as g_ijdxⁱdx^j and
given dS² = (dx¹)² + (dx²)² + 4(dx¹)(dx²). Find g^ijNow here is my take on the solution: Since the cross terms are present in the line element equation, we can say that it's a non-orthogonal system. So what I did was express the metric tensor in form of a 2x2 matrix, and checked the corresponding coefficient in the equation. But I am having a problem getting to the cross terms, and how to find the corresponding coefficients to the metric tensor.

 

[phyzguy]

Isn't $g^{ij}$ just the inverse of the 2x2 matrix representing $g_{ij}$?

 

[vela]

Defining dS² as g_ijdxⁱdx^j and
given dS² = (dx¹)² + (dx²)² + 4(dx¹)(dx²). Find g^ijNow here is my take on the solution: Since the cross terms are present in the line element equation, we can say that it's a non-orthogonal system. So what I did was express the metric tensor in form of a 2x2 matrix, and checked the corresponding coefficient in the equation. But I am having a problem getting to the cross terms, and how to find the corresponding coefficients to the metric tensor.

So did you figure out the matrix or not? In the second sentence, you say you found it, but in the third, you imply that you did not.