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Divergence, The divergence of a vector point function F is denoted by div F and is defined as below., , Let PeRi+hj+hk, eeae (iR+jF, shry= SF, Fy oh, x by & ax, , R-VE, divF _— a az, , It is evident that div F is scalar function., PHYSICAL INTERPRETATION OF DIVERGENCE, , Let us consider the case of a fluid flow. Consider a small rectangular parallelopiped of, dimensions dv, dy, dz parallel to x,y and z axes respectively., , , , > A A A, Let V’ =I i+)', j +1, k be the velocity of the, fluid at P(x, , 2) ‘, , , , , , , , , , , , 6 R, . Mass of fluid flowing in through the face ABCD in unit time, = Velocity x Area of the face = V’, (dy dz) 5 ! ss, Mass of fluid flowing out across the face PORS per unit time “ c >, = V(r + dy) @) >,, , i, = (real a, , , , Net decrease in mass of fluid in the parallelopiped

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corresponding to the flow along x-axis per unit time, , , , , , = 1, ay de-(r, + Teas) by de, _ WV, . ., = ar dx dy dz (Minus sign shows decrease), , a,, Similarly, the decrease in mass of fluid to the flow along y-axis = “git dy dz, , oO,, and the decrease in mass of fluid to the flow along z-axis = = dy dz, , , , , , Total decrease of the amount of fluid per unit time = (S++ a a, Thus the rate of loss of fluid per unit vob = Ws, Hy We, per unit volume = —" oy a, = (2-32 +82 Jer, +9V, +80) = OF advil, ae oy x Yy z + =div, If the fluid is compressible, there can be no gain or loss in the volume element. Hence, , div? = 0 wD), and I is called a Solenoidal vector function., Equation (1) is also called the equation of continuity or conservation of mass., , Divergence of a Vector:, , , , Le F (x,y,z) is a vector . Then Divergence of a vector F is defined as; Div F=V.F, , , , Solenoid Vector:, , , , A vector F is said to be Solenoid, if Div F = 0., , Properties of the Divergence:, , , , Ifa & D are two vector & is a scalar function then, () Div(@+b)=V.(¥ +b)=V.T4V.b, , (ii) Div(9X)=V.(97)=@(V.a)4+(VoQ).a