In the previous chapter the fluid dynamic equations are presented in symbolic formation. These formulations are valid for all possible coordinate systems. For practical purposes or, to say, for finding solutions for realistic flow problems, in particular with the aid of modern numerical methods, a frame of reference has to be defined based on a specific coordinate system. This leads to the Fluid dynamic equations component formfluid dynamic equations in component form. In this chapter we present the continuity equation ( 3.13 ), the momentum equations ( 3.15 ), ( 3.16 ), ( 3.23 ) and the energy equations ( 3.24 ), ( 3.25 ), ( 3.33 ), ( 3.35 ), ( 3.38 ) in the three well-known coordinate systems: \(\text{ Cartesian } \; x,y,z, \; \text{ cylindrical } \; r,\varphi ,z, \; \text{ spherical } \; r, \theta ,\varphi .\) The component presentation of the governing equations in the various coordinate systems requiresBird, R.B. a deepStuart, W.E. insightLightfoot, E.N. intoAris, R. theSommerfeld, A. vector/tensorKlingbeil, E. calculusTeichmann, H. asBronstein, I.N. it can be found in the literature , see Sect. 4.1 . The inspection of the equations treated in Chap. 3 reveals that 11 vector/tensor operations must be provided. For that purpose we use the neutral scalar, vector and tensor quantities \(c, \, \textbf{a}, \, {\bar{\bar{\vartheta }}}\) . These operations are presented each in the beginning of the Sects. 4.1 , 4.2 , 4.3 .