Hence for example a tf a t is the amount of fthat is swept through the boundary

# Hence for example a tf a t is the amount of fthat is

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are the speeds with which the respective boundary points are moving. Hence, for example, ∂a ∂t f ( a, t ) is the amount of f that is “swept through” the boundary location a as a moves in time. The General Transport Theorem Just as was true for the fundamental theorem of calculus, Leibnitz’s formula possesses higher- dimensional analogues. In particular, in three dimensions we have the following. Theorem 3.2 (General Transport Theorem) Let F be a smooth vector (or scalar) field on a region R ( t ) whose boundary is S ( t ) , and let W be the velocity field of the time-dependent movement of S ( t ) . Then d dt integraldisplay R ( t ) F ( x , t ) dV = integraldisplay R ( t ) F ∂t dV + integraldisplay S ( t ) F W · n dA . (3.9)
3.2. REVIEW OF PERTINENT VECTOR CALCULUS 57 The formula Eq. (3.9) is called the general transport theorem , and certain specific cases of it will be widely used in the sequel. Before going on to this, however, it is worthwhile to again check whether the indicated vector operations are consistent from term to term in this equation. It should be clear, independent of the order of F , that the first term on the right-hand side is consistent with that on the left-hand side since they differ only with respect to where (inside, or outside, the integral) differentiation by a scalar parameter is performed. It is the second term on the right that must be examined with more care. First, consider the case when F is a vector field, say of dimension three for definiteness. Then the first integral on the right is acting also on a 3-D vector field. We will assume that the velocity field W is also 3D (although this is not strictly necessary), and that the region R ( t ) is spatially 3D. Then the question is “What is the order of the integrand of this second integral?” But it is easy to see that the dot product of W and n leads to a scalar , a number—maybe different at each point of S ( t ), but nevertheless, a scalar; and a scalar times a vector is a vector. Hence, we obtain the correct order. Now suppose F is replaced with a scalar function F . Then clearly the left-hand side and the first term on the right-hand side of Eq. (3.9) are scalars. But now this is also true of the second integral because, as before, the dot product of W and n produces a scalar, and this now multiplies the scalar F . In fact, it should be clear that F could be a matrix (often called a “tensor” in fluid dynamics), and the formula would still work. Reynolds Transport Theorem The most widely-encountered corollary of the general transport theorem, at least in fluid dy- namics, is the following. Theorem 3.3 Let Φ be any smooth vector (or scalar) field, and suppose R ( t ) is a fluid element with surface S ( t ) traveling at the flow velocity U . Then D Dt integraldisplay R ( t ) Φ dV = integraldisplay R ( t ) Φ ∂t dV + integraldisplay S ( t ) Φ U · n dA . (3.10) This formula is called the Reynolds transport theorem , and just as was the case for F in the general transport theorem, Φ can have any order. But we will deal mainly with the scalar case in the sequel. There are two important things to notice in comparing this special case with the general formula. The first is that the integral on the left-hand side is now being differentiated with respect

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