Another way of computing this derivative is to view the composite function as the composite of and h. Applying the chain rule to this situation gives:

Applying the definition of the derivative gives:

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 this technique to the Pleiades gives an age of about 115 million years.

Applying this theorem when R is itself a PID gives us a way of building Dedekind domains out of PIDs.

Applying these three sorts of derivatives again to each other gives five possible second derivatives, for a scalar field f or a vector field v ; the use of the scalar Laplacian and vector Laplacian gives two more:

Applying the fundamental theorem of calculus twice gives

Applying separation of variables again to the second equation gives way to the pair of differential equations

Applying Itō's lemma with f ( Y ) = log ( Y ) gives

Applying this technique to the boundary layer equations gives the full turbulent boundary layer equations not often given in literature:

Applying a sinusoidal stress to a Maxwell model gives: where is the Maxwell relaxtion time.

Applying the conservation of energy, for steady-state conditions, and Newton ’ s law of cooling to the temperature nodes shown in Figure 2 gives the following set of equations.

Applying Van Kampen's theorem gives the result.

Applying these simplifying assumptions to the spatial components of the geodesic equation gives

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

Applying the quantum oracle gives

Applying the inverse system to gives the following.

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

Applying these distinctions to the categories in German wine classification gives the following overview.

Applying the same operators to the right hand side gives

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.

Applying a smear of pine tar on the wound gives the attacking hens something else to do.

Applying the system operator first gives

Applying this algorithm to our problem, and using 10-digit floating-point arithmetic, the solution of greater magnitude, as before, is The other solution is then

Applying the Doomsday algorithm involves three steps:

Applying Seifert's algorithm to this diagram, as expected, does produce a Seifert surface ; in this case, it is a punctured torus of genus g = 1, and the Seifert matrix is

Applying Lagrange's formula with the observation that Ω • r ( t ) = 0 at all times,

Applying higher pressure of the order of hundreds or thousands of kilobars induces a series of phase transformations, in particular with a tetragonal phase appearing at about 900 kbar.

Applying Chow's analysis to the case of the unexpected hanging ( again with the week shortened to two days for simplicity ), we start with the observation that the judge's announcement seems to affirm three things:

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 the rules and customs of prize law, the courts decided whether the Letter of Marque was valid and current, and whether the captured vessel or its cargo in fact belonged to the enemy ( not always easy, when flying false flags was common practice ), and if so the prize and its cargo were " condemned ", to be sold at auction with the proceeds divided among the privateer's owner and crew.

Applying power to one cathode surrounds it with an orange glow discharge.

Applying a pulse with specified width and voltages to the spades made the sheet advance to the next anode, where it stayed until the next advance pulse.

* Greening The Desert: Applying natural farming techniques in Africa, interview with Masanobu Fukuoka ( retrieved 30 November 2010 )

The Cholesky decomposition is commonly used in the Monte Carlo method for simulating systems with multiple correlated variables: The correlation matrix is decomposed, to give the lower-triangular L. Applying this to a vector of uncorrelated samples, u, produces a sample vector Lu with the covariance properties of the system being modeled.

Applying Itō's lemma with f ( S ) =

Applying to the French classics the rigorous method used with regard to the texts of the Middle Ages, he published the Pensées of Pascal, revised with the original manuscript ( 1887 – 1889 ), and the Provinciales ( 1891 ), edited with notes.

Applying this with, then and

Applying procedural fairness to an entire society therefore seems certain to create recognizable inefficiencies, and therefore be unfair, and ( by the equivalence of justice with fairness ) unjust.

Applying the rule of description, one occupies oneself with describing instead of explaining.

Applying again Bayes ' theorem, and assuming the classification between spam and ham of the emails containing a given word (" replica ") is a random variable with beta distribution, some programs decide to use a corrected probability:

Applying these methods in the workplace, such as Csikszentmihalyi did with Swedish police officers, can improve morale by fostering a sense of greater happiness and accomplishment, and in correlated to increased performance.

Applying the chaos theory to the second law of thermodynamics, the paradox of irreversibility can be explained in the errors associated with scaling from microstates to macrostates and the degrees of freedom used when making experimental observations.