A metric tensor is used to measure distances based on a given coordinate system. In terms of the Jacobian, the metric tensor can be found from where is the transpose of the Jacobian. Since is a symmetric matrix for any matrix , the metric tensor is always symmetric. (In fancy-pants math lingo this is called a symmetric bilinear form.) What is the real-life consequences of this? The distance from to is always the same as the distance from to , no matter what kind of crazy coordinate system you are living in!

If we want to calculate the length of a parameterized curve where is a parameter with respect to some coordinate system, then we can write an infintesmal displacement element as . The length of this displacement is and the length of the curve from to is .

So we need the metric tensor to define distance along a curve when we are in non-cartesian coordinate systems, such as spherical or toroidal. From the metric tensor one can then start to study the "curvature" of a coordinate system. More soon!

If we want to calculate the length of a parameterized curve where is a parameter with respect to some coordinate system, then we can write an infintesmal displacement element as . The length of this displacement is and the length of the curve from to is .

So we need the metric tensor to define distance along a curve when we are in non-cartesian coordinate systems, such as spherical or toroidal. From the metric tensor one can then start to study the "curvature" of a coordinate system. More soon!