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#### Line Elements and **Metric Tensor**s

- The
*line element***ds**for a curve or a space is the shortest distance between two neighboring points. Using calculus,**ds**can be used to calculate distances, lengths, angles, areas, volumes, and geodesics (the shortest line between two points). - The line element is determined by the
*Metric Tensor*and*coordinates*. - Thus:
**Metric Tensor + Coordinates ➞ Line Element****Line Element + Calculus ➞ Distances, Lengths, Angles, Areas, Volumes, Geodesics**

#### Examples

**Metric Tensor for Euclidean Plane in Cartesian Coordinates**

**Metric Tensor for Euclidean Space in ****Spherical Coordinate**

**Metric Tensor for Euclidean Space in**

**Spherical Coordinate****Metric Tensor for Four Dimensional Space in Cartesian Coordinates**

**Metric Tensor for Four Dimensional Space in Cartesian Coordinates**

**Metric Tensor for Flat Spacetime **in Cartesian Coordinates

**in Cartesian Coordinates****Metric Tensor for Flat Spacetime****Metric Tensor for Flat Spacetime **in Spherical Coordinates

**in Spherical Coordinates****Metric Tensor for Flat Spacetime**