Sentence examples for integrating with respect to 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,+∞).

Let's start with equation one and lets integrate with respect to x.

More precisely, we prove that the spectral shift function integrated with respect to the spectral parameter from −∞ to λ (from λ to +∞) is concave (convex) with respect to trace class perturbations.

These tractions are integrated with respect to each half of the interface width, yielding two interface lines along which the loading of the beams as well as the additional loading of the plate is defined.

These tractions are integrated with respect to each half of the interface width resulting two interface lines, along which the loading of the beams as well as the additional loading of the plate is defined.