angular coordinates , where the angles range over radians (or over degrees) and ranges over radians (or over degrees). If are the Cartesian coordinates, then we may compute from with: Except in the special cases described below, the inverse transformation is unique: where if for some but all of are zero then when , and (180 degrees) when . There are some special cases where the inverse transform is not unique; for any will be ambiguous whenever all of are zero; in this case may be chosen to be zero. Spherical volume and area elements To express the volume element of -dimensional Euclidean space in terms of spherical coordinates, first observe that the Jacobian matrix of the transformation is: The determinant of this matrix can be calculated by induction. When , a straightforward computation shows that the determinant is . For larger , observe that can be constructed from as follows. Except in column , rows and of are the same as row of , but multiplied by an extra factor of in row and an extra factor of in row . In column , rows and of are the same as column of row of , but multiplied by extra factors of in row and in row , respectively. The determinant of can be calculated by Laplace expansion in the final column. By the recursive description of , the submatrix formed by deleting the entry at and its row and column almost equals , except that its last row is multiplied by . Similarly, the submatrix formed by deleting the entry at and its row and column almost equals , except that its last row is multiplied by . Therefore the determinant of is Induction then gives a closed-form expression for the volume element in spherical coordinates The formula for the volume of the -ball can be derived from this by integration. Similarly the surface area element of the -sphere of radius , which generalizes the area element of the 2-sphere, is given by The natural choice of an orthogonal basis over the angular coordinates is a product of ultraspherical polynomials, for , and the for the angle in concordance with the spherical harmonics. Polyspherical coordinates The standard spherical coordinate system arises from writing as the product . These two factors may be related using polar coordinates. For each point of , the standard Cartesian coordinates can be transformed