Applying the completeness theorem to this result, gives the existence of a model of T where the formula C < sub > T </ sub > is false.

* Applying the divergence theorem to the product of a scalar function g and a vector field F, the result is

* Applying the divergence theorem to the cross-product of two vector fields, the result is

Applying both sides to e < sub > j </ sub >, the result on each side is the jth partial derivative of f at p. Since p and j were arbitrary, this proves the formula (*).

Applying heat to the inner cable for soldering may weaken the steel, and although soft soldering is less strong than silver solder, a lower temperature is required to form the joint, and there is less likelihood of the inner cable being damaged as a result.

Applying Van Kampen's theorem gives the result.

Applying perspective ( pharmacoeconomic ) to analyze cost reduction, it has been shown that savings made by individual hospitals result in actual cost increases to the Provinces.

Applying this result repeatedly shows that there is a basis for V such that all elements of the Lie algebra are represented by upper triangular matrices.

Applying excessive force to the cube when twisting it may result in broken pieces.

Applying a result of MacIntyre on the model theory of p-adic integers, one deduces again that ζ < sub > G </ sub >( s ) is a rational function in p < sup >− s </ sup >.

Applying the result of the previous subsection to the displacement, the velocities in the two reference frames are related by the equation

Applying this method instead of Euler's method gives a much better result ( blue ).

Applying finding from affective forecasting research to happiness also raises methodological issues: should happiness measure the outcome of an experience, or the choice that is satisfied as a result of a forecast?

Applying a Newton or Picard iteration produces a system of linear equations which is nonsymmetric in the presence of advection and indefinite in the presence of incompressibility.

Applying linear algebra it will give explicit equations, one for each as polynomials of degree 2 in the.

Applying again the matrix operator on both side yields

Applying a conception of such illocutionary acts according to which they are ( roughly ) acts of saying something with the intention of communicating with an audience, he describes indirect speech acts as follows: " In indirect speech acts the speaker communicates to the hearer more than he actually says by way of relying on their mutually shared background information, both linguistic and nonlinguistic, together with the general powers of rationality and inference on the part of the hearer.

Applying only classical physics and the Bohr model of the atom makes both atomic and molecular ionization entirely deterministic ; that is, every problem will always have a definite and computable answer.

Applying thick wraps to both of his knees became a pre-game ritual, and by the end of his career, simply swinging a bat caused him to fall to one knee in pain.

Applying this to a space of dimension 0, which contains exactly one vector ( the zero vector ), the only candidate for a basis is the empty set ( both because 0 elements are required, and because the zero vector cannot be in any basis ).

Applying CGT to a position attempts to determine the optimum sequence of moves for both players until the game ends, and by doing so discover the optimum move in any position.

Applying both sides of this matrix equation to the component vector v < sub > 1 </ sub >( φ < sub > 1 </ sub >< sup >− 1 </ sup >( P )) and invoking ( 1 ) and ( 2 ) yields

Applying this rule often adds humor for both players and on-lookers.

Applying on both sides yields.

Applying the exterior connection to both sides gives the transformation law for ω:

Applying the Klein-Gordon operator on both sides, remembering that the four-moment is on-shell and that is the Green's function of the operator, we obtain:

Applying this to both the Earth's and the Moon's position does not disturb the positions relative to each other.

Applying the method hierarchically ( according to the data structure at hand ), optionally in both increasing and decreasing direction, yields highly efficient multidimensional range search which is important in both commercial and technical applications, e. g. as a procedure underlying nearest neighbour searches.

Applying the modern rule, both titles descend to heirs general, which left them in abeyance between the daughters of Sir Owen West, and this situation persists to the present day with their respective heirs.

Applying linearization techniques to the equation of motions for both species, to the equation of continuity, and Poisson's equation, and introducing the spatial and temporal harmonic operators, we can get the following expression:

Applying this approach, the control program uses all available scientific information regarding the San Francisco Estuary, the invasive cordgrasses, and the likely economic, sociological, and ecological consequences of both the invasion and the treatment program, to develop a management strategy that is effective, economical, and protective of public and environmental health.

Applying Leibniz's Law to the Ship of Theseus problem, S2 is the same as S1 if, and only if, S2 and S1 have all the same properties and relations.

Applying these relations to the strings above, one obtains

Applying the Curry – Howard isomorphism once more, inductive families correspond to inductively defined relations.

Applying duality, this means that a morphism in some category C is a monomorphism if and only if the reverse morphism in the opposite category C < sup > op </ sup > is an epimorphism.

Applying these ladder operators to the eigenstates of the total angular momentum, azimuthal angular momentum and energy operators, the eigenvalues of the first Casimir operator C < sub > 1 </ sub > are n < sup > 2 </ sup > − 1 ; importantly, they are independent of the l and m quantum numbers, making the energy levels degenerate.

Applying this value successively to the tones of a chromatic scale, starting from A above middle C with a frequency of 440 Hz, produces the following sequence of pitches:

* Wharton, C. Bradford, J. Jeffries, J. Franzke, M. Applying Cognitive Walkthroughs to more Complex User Interfaces: Experiences, Issues and Recommendations CHI ’ 92 pp381 – 388.

Applying the lessons of processor obsolescence learned during the Lockheed Martin F-22 Raptor, the Lockheed Martin F-35 Lightning II planned for processor upgrades during development, and switched to the more widely supported C ++ programming language.

The Meaning of Things: Applying Philosophy to Life, published in the U. S. as Meditations for the Humanist: Ethics for a Secular Age, is a book by A. C. Grayling.

Applying an annihilation operator followed by its corresponding creation operator returns the number of particles in the k < sup > th </ sup > single-particle eigenstate:

Applying the two sine waves at frequencies ω < sub > 1 </ sub >

Applying this estimate to any even power V < sup > k </ sup > of V and using the Künneth formula shows that the eigenvalues of Frobenius on the middle cohomology of a variety V of any dimension d satisfy

Applying Ohm's Law, the relationship between the input voltage, V < sub > in </ sub >, and the output voltage, V < sub > out </ sub >, can be found:

Applying the Tannakian formalism, one concludes that M < sub > num </ sub > is equivalent to the category of representations of an algebraic group G, which is called motivic Galois group.

Applying any additive functor to a split sequence results in a split sequence, so in particular R < sup > 1 </ sup > F ( A )

Applying this to the function defined by f () = log ( a < sub > 1 </ sub >), we obtain that

Applying Equation ( 1 ) and KCL at the OA's non-inverting input v < sub >+</ sub > gives

Applying the above bundle map ( pointwise ) to this section yields the pullback of α by φ, which is the 1-form φ < sup >*</ sup > α on M defined by

Applying the Ampère-Maxwell equation to surface S < sub > 1 </ sub > we find:

Applying the Nernst Equation above, one may account for these differences by changes in relative K < sup >+</ sup > concentration or differences in temperature.

Applying reduction of order in case of a multiple root m < sub > 1 </ sub > will yield expressions involving a discrete version of ln,

Applying the e < nowiki >'</ nowiki > th iterate of F to a ring which contains a field K of p < sup > e </ sup > elements gives us a fixed point set equal to K, similar to the example above.