In mathematical invariant theory, a transvectant is an invariant formed from n invariants in n variables using Cayley's é process.
If Q<sub>1</sub>,...,Q<sub>n</sub> are functions of n variables x = (x<sub>1</sub>,...,x<sub>n</sub>) and r âÂÂ¥ 0 is an integer then the r<sup>th</sup> transvectant of these functions is a function of n variables given bywhereis Cayley's é process, and the tensor product means take a product of functions with different variables x<sup>1</sup>,..., x<sup>n</sup>, and the trace operator Tr means setting all the vectors x<sup>k</sup> equal.
The zeroth transvectant is the product of the n functions.The first transvectant is the Jacobian determinant of the n functions.The second transvectant is a constant times the completely polarized form of the Hessian of the n functions.
When , the binary transvectants have an explicit formula:which can be more succinctly written aswhere the arrows denote the function to be taken the derivative of. This notation is used in Moyal product.