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Unformatted text preview: Lecture XXIX Mathematical Applications 1 Leibnitz’s Rule Leibnitz’s Rule : Let f ( x, t ) be a C 1 function defined for a ≤ x ≤ b . Then b b d ∂f f ( x, t ) dx = f t ( x, t ) dx, where f t ( x, t ) = . dt ∂t a a b In other words, if we define g ( t ) = A f ( x, t ) dx , then dg = b f t ( x, t ) dx. dt a Leibnitz’s rule also works for functions that have more parameters. For example, we have b b d f ( x, y, z, t ) dx = f t ( x, y, z, t ) dx. dt a a The rule can be generalized and applied to gradients. Indeed, let f ( x, y, z, t ) be a scalar field defined for all a ≤ t ≤ b and ( x, y, z ) ∈ D . Then if P is a point in D , we have that b b P fdt = P fdt. a A Let f ( x, y, z, t ) be a time-dependent mass-density function on a region D . Then, by Leibnitz’s rule, we have d ∂f fdV = dV. dt D ∂t D This means that the rate of change of the mass( the left-hand side of the equality) equals the integral of the rate of change of density( the right-hand side of the equality). equality)....
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- Two '04