remain closed, and the circulation about them must remain zero. Since a state of rest is a 

 state in which the circulation vanishes about all closed curves, the application of pressure to 

 the fluid can generate in it only irrotational motion. For this reason, the motion generated by 

 a moving ship in the surrounding water or by an airplane in the air is roughly irrotational, 

 except near the solid surfaces where friction plays a large role. 



6. THE VELOCITY POTENTIAL FOR IRROTATIONAL FLOW 



Only irrotational motion will be considered hereafter. Its mathematical treatment can 

 be greatly simplified by introducing the velocity potential. 



If the motion is irrotational within a singly connected region, the integral 



I 



ds 



taken along any path lying in the region and connecting two given points, P^ and P^, depends 

 only on the positions of the points P, and P^. The integral is defined as in the last section; 

 the chosen direction of motion along the path is from P^ toward P^' 



To prove the statement just made, note that any two paths B. and B^, as illustrated in 

 Figure 6a, when taken together form a closed curve around which the circulation vanishes, so 

 that 



r (Si) ^sdsA 



(B,) 9.^., 







But 



Figure 6b 



Figure 6 — Illustrating properties of the velocity potential. 



