Sentence examples for integrating with respect from inspiring English sources
where 3 α + 2 β ≤ 1, α ≥ 3, then the solution ( u, v, b ) can be extended smoothly beyond t = T. Proof Multiplying the first equation of (1.1) by u and integrating with respect to x on R 3, using integration by parts, we obtain 1 2 d d t ∥ u ( t ) ∥ L 2 2 + ∥ ∇ u ( t ) ∥ L 2 2 = ∫ R 3 b ⋅ ∇ b ⋅ u d x + χ ∫ R 3 ( ∇ × v ) ⋅ u d x. (4.1).
Multiplying the first equation of (1.1) by − Δ u and integrating with respect to x on R 3, and then using integration by parts, we obtain (4.22).
Integrating with respect to, we obtain (3.2).
Integrating with respect to over, we get (2.18).
Integrating with respect to over, we have (2.7).
Integrating with respect to s, we get (57).
In the expression of K, the term in θ is proportional to a three-parameter Beta-prime distribution, which makes it possible to integrate with respect to θ over [0,+∞). The integration with respect to η is then possible, over [0,+∞).
The logical outgrowth of these attempts to differentiate and integrate with respect to a Brownian motion process is the Ito (named for the Japanese mathematician Itō Kiyosi) stochastic calculus, which plays an important role in the modern theory of stochastic processes.
Now, we integrate with respect to, on the interval, the inequalities (3.5) and (3.6).
