Abstract: Jacobian matrix and its determinant are important computational tools in science and technology. This paper summarizes the important roles of Jacobian matrix and its determinant in coordinate transformation, derivative transformation and linear approximation for functional integration, robot dynamics, thermodynamics, nonlinear dynamics, and stochastic dynamics to deepen the understanding of Jacobian matrix and its determinant.

Key words: Jacobian matrix, Jacobian determinant, coordinate transformation, derivative transformation, linear approximation