rectangular coordinate systems with orthonormal bases together. For the tensor relating a vector to a vector, the vectors and tensors throughout the equation all belong to the same coordinate system and basis. Derivatives and Jacobian matrix elements The entries of L are partial derivatives of the new or old coordinates with respect to the old or new coordinates, respectively. Differentiating i with respect to xk: so is an element of the Jacobian matrix. There is a (partially mnemonical) correspondence between index positions attached to L and in the partial derivative: i at the top and j at the bottom, in each case, although for Cartesian tensors the indices can be lowered. Conversely, differentiating xj with respect to i: so is an element of the inverse Jacobian matrix, with a similar index correspondence. Many sources state transformations in terms of the partial derivatives: and the explicit matrix equations in 3d are: similarly for Projections along coordinate axes As with all linear transformations, L depends on the basis chosen. For two orthonormal bases projecting x to the axes: projecting x to the x axes: Hence the components reduce to direction cosines between the i and xj axes: where θij and θji are the angles between the i and xj axes. In general, θij is not equal to θji, because for example θ12 and θ21 are two different angles. The transformation of coordinates can be written: and the explicit matrix equations in 3d are: similarly for The geometric interpretation is the i components equal to the sum of projecting the xj components onto the j axes. The numbers ei⋅ej arranged into a matrix would form a symmetric matrix (a matrix equal to its own transpose) due to the symmetry in the dot products, in fact it is the metric tensor g. By contrast ei⋅j or i⋅ej do not form symmetric matrices in general, as displayed above. Therefore, while the L matrices are still orthogonal, they are not symmetric. Apart from a rotation about any one axis, in which the xi and i for some i coincide, the angles are not the same as Euler angles, and so the L matrices are not the same as the rotation matrices. Transformation of the dot and cross products (three dimensions only) The dot product and cross product occur very frequently, in applications of vector analysis to physics and engineering, examples include: power transferred P by an