from to moves from to then the bug experiences a change in the scalar value, (the total differential). If the bug is moving with a velocity the change in particle position is and we may write where is the gradient of the scalar field p. So: If the bug is just moving with the flow, the same formula applies, but now the velocity vector,v, is that of the flow, u. The last parenthesized expression is the substantive derivative of the scalar pressure. Since the pressure p in this computation is an arbitrary scalar field, we may abstract it and write the substantive derivative operator as See also Continuum mechanics Cauchy momentum equation Special relativity Substantive derivative References James R. Welty, Charles E. Wicks, Robert E. Wilson & Gregory Rorrer Fundamentals of Momentum, Heat, and Mass Transfer Notes External links PDFs Integral Approach to the Control Volume analysis of Fluid Flow Continuum mechanics Thermodynamics