Using the techniques of differential calculus, the velocity and acceleration as a function of time can be found:

Using differential calculus, we can determine the limit, or the value that Δy / Δx approaches as Δy and Δx get closer to zero ; it follows that this limit is the exact slope of the tangent.

where the reference variables may be functions of the number of particles N. Using the definition of the heat capacity at constant volume for the first differential and the appropriate Maxwell relation for the second we have:

Using sensors and pumps, a relatively small amount of energy ( i. e. that used to power a pump and controller ) can harvest a far larger amount of available thermal energy by switching on as soon as a useful temperature differential becomes present.

Using differential scanning calorimetry to study the stability to oxidation of samples generally requires an airtight sample chamber.

Using the first variation of,, in place of yields the first variation of, ; this is similar to how the differential is obtained from the gradient.

* Using principal bundles, a connection can be realized as a Lie algebra-valued differential form.

Using the differential form of Faraday's law,, this gives

Using a partial differential equation ( PDE )- based method and solving the PDE equation by a numerical scheme, one can segment the image.

Using this can simplify the differential equation

Using this approximation, the equation near becomes a differential equation:

Using the Laplace transform, letting, and applying to the differential equation we get

Using the same 1600 cc engine and a four or five-speed manual gearbox from the Corcel, came with the differential of Corcel 1: 4 to better use to advantage the force of the engine.

Using the above definition of the potentials and applying it to the other two Maxwell's equations ( the ones that are not automatically satisfied ) results in a complicated differential equation that can be simplified using the Lorenz gauge where A is chosen to satisfy:

Using the results from either Maxwell-Boltzmann statistics, Bose-Einstein statistics or Fermi-Dirac statistics, and considering the limit of a very large box, the Thomas-Fermi approximation is used to express the degeneracy of the energy states as a differential, and summations over states as integrals.

Using Taylor series to represent the in and out flux terms across the boundaries of the control volume, and using the divergence theorem to turn the flux across the boundary into a flux over the entire volume, the final form of the groundwater flow equation ( in differential form ) is:

Using the convenience that Laplace transforms follow the rules of linearity, one can solve the above example for y by performing a Laplace transform on both sides of the differential equation, substituting in the initial values, solving for the transformed function, and then performing an inverse transform.

Using the superconnection formalism of Quillen, they obtained a refinement of the Riemann – Roch formula, which links together the Thom classes in K-theory and cohomology, as an equality on the level of differential forms.

Using a branch of mathematics known as tuple calculus, he demonstrated that such a system could support all the operations of normal databases ( inserting, updating etc.

Using these two examples, we can make some useful observations that motivate the major ideas in lambda calculus.

Using the probability calculus of Bayes Theorem, Salmon concludes that it is very improbable that the universe was created by the type of intelligent being theists argue for.

Using geometry and calculus, some simple combinations of sources are easily solved ; others are not.

Using a method called " calculus of statement ," Lentz helps the team to mitigate the pressure harming the plant operators.

Using the fundamental theorem of calculus often requires finding an antiderivative.

Using this in the sum, one can evaluate the remaining integral using the calculus of residues and arrive at the desired generating function.

Using the limit notation of calculus, the volume may be expressed as:

Using tensor calculus, proper time is more rigorously defined in general relativity as follows: Given a spacetime which is a pseudo-Riemannian manifold mapped with a coordinate system and equipped with a corresponding metric tensor, the proper time experienced in moving between two events along a timelike path P is given by the line integral

Using the parametrization of the hyperbola with hyperbolic angle, the theory of special relativity provided a calculus of relative motion with the one-parameter group indexed by rapidity.

* Using process calculus to model biological systems ( stochastic-calculus, BioAmbients, Beta Binders, BioPEPA, Brane calculus ).

Using calculus, Gershun could compute the irradiance falling on points () on a surface.

Using the functional calculus for larger classes of functions, we can for example define rigorously the " square root " of the ( negative ) Laplacian operator − Δ or the exponential

Using the bounded functional calculus, one can prove part of the Stone's theorem on one-parameter unitary groups:

Using typed lambda calculus to implement formal systems on a machine.

Using modern predicate calculus, we quickly discover that the statement is ambiguous.

Using launch systems presently available, we are developing satellites to scout the world's weather ; ;

Hence S breaks up into uncountably many orbits under G. Using the axiom of choice, we could pick a single point from each orbit, obtaining an uncountable subset X of S with the property that all of its translates by G are disjoint from X.

Using the identity, we have

Using Zorn's lemma, it can be shown that every field has an algebraic closure, and that the algebraic closure of a field K is unique up to an isomorphism that fixes every member of K. Because of this essential uniqueness, we often speak of the algebraic closure of K, rather than an algebraic closure of K.

Using the above theorem it is easy to see that the original Borsuk Ulam statement is correct since if we take a map f: S < sup > n </ sup > → ℝ < sup > n </ sup > that does not equalize on any antipodes then we can construct a map g: S < sup > n </ sup > → S < sup > n-1 </ sup > by the formula

Using the right-hand rule, we expect the curl to be into the page.

Using the product rule in reverse, we get:

Using now the power series definition from above we see that for real values of x

Using the composition of functions we can rewrite this statement as follows:

Using the definition of free variables, we may now define a capture-avoiding substitution.

Using our senses of sight, hearing, touch, smell, and taste, we perceive colors, sounds, textures, etc.

* Using the above methods for simple polygons, we discover that particular regions within the polygon may have their area multiplied by a factor which we call the density of the region.

Using this information we can perform a series of exchanges on the input array that puts it in order, moving items only once.

Using =, we find that the period of precession is given by:

Using relativistic relations for energy and momentum, we have

Using the chosen basis, we may write

Using the previous example, we would get the following:

Using this, we can get the relation, again choosing the right sign:

Using this Carnapian approach, Maher identifies a proposition which we intuitively ( and correctly ) know to be false, but which we easily confuse with the paradoxical conclusion.

Using Le Chatelier's principle, we can predict that the amount of methanol will increase, decreasing the total change in CO.