British scientists, including Maxwell, had relied on Hamilton's quaternions in order to express the dynamics of physical quantities, like the electric and magnetic fields, having both a magnitude and a direction in three-dimensional space. Following W. K. Clifford in his Elements of Dynamic (1888), Gibbs noted that the product of quaternions could be separated into two parts: a one-dimensional (scalar) quantity and a three-dimensional vector, so that the use of quaternions involved mathematical complications and redundancies that could be avoided in the interest of simplicity and to facilitate teaching. In his Yale classroom notes he defined distinct dot and cross products for pairs of vectors and introduced the now common notation for them. Through the 1901 textbook Vector Analysis prepared by E. B. Wilson from Gibbs notes, he was largely responsible for the development of the vector calculus techniques still used today in electrodynamics and fluid mechanics.