For example, if the water is 25 ft ( 8 m ) deep, and the anchor roller is 3 ft ( 1 m ) above the water, the scope is the ratio between the amount of cable let out and 28 ft ( 9 m ).

Let the function g ( t ) be the altitude of the car at time t, and let the function f ( h ) be the temperature h kilometers above sea level.

They were let out to contractors, like the other works mentioned above, and when they were completed, the censors had to see that the work was performed in accordance with the contract: this was called opus probare or in acceptum referre.

To illustrate the above formulas, let x, y be given as:

It is easy enough to " translate " between polar and rectangular coordinates in the plane: let, as above, direction and distance be and respectively, then we have

Suppose that and let tend to from above.

Support acts for the first concert included The Church, Mental as Anything, The Party Boys, The Sunnyboys, and Midnight Oil — who acknowledged, " Hooks were the only Australian band they would let top the bill above them ".

As above, we let ( V, g ) be an n-dimensional complex vector space equipped with a nondegenerate bilinear form.

If we let Δx and Δy be the distances ( along the x and y axes, respectively ) between two points on a curve, then the slope given by the above definition,

Although the three-peg version has a simple recursive solution as outlined above, the optimal solution for the Tower of Hanoi problem with four pegs ( called Reve's puzzle ), let alone more pegs, is still an open problem.

I would walk and hitch and sail away from this dark city to the bright spaces of the wet west coast, and there throw myself into the tall, glittering seas beyond Iona ( with its cargo of mouldering kings ) to let the gulls and seals and tides have their way with my remains, and in my dying moments look forward to an encounter with Staffa ’ s six-sided columns and Fingal ’ s cave ; or I might head south to Corryvrecken, to be spun inside the whirlpool and listen with my waterlogged deaf ears to its mile-wide voice ringing over the wave-race ; or be borne north, to where the white sands sing and coral hides, pink-fingered and hard-soft, beneath the ocean swell, and the rampart cliffs climb thousand-foot above the seething acres of milky foam, rainbow-buttressed.

Niamh let him borrow Embarr, who could run above ground, and made him promise not to get off of the horse or touch Irish soil.

But his brother Tāne ( or Tāne-mahuta ) disagrees, suggesting that it is better to push them apart, to let Rangi be as a stranger to them in the sky above while Papa will remain below to nurture them.

Initialisms similar to the one above include GIYF (" Google is your friend ") and LMGTFY (" let me google that for you ").

Since 1998 these letters have been recognised as representing the sixth and seventh lines of the Fourth Eclogue of Virgil, which reads Redeunt Saturnia Regna, Iam Nova Progenies Caelo Demittitur Alto, meaning ' The Golden Ages are back, now a new generation is let down from Heaven above '.

The three-ship flew above 5000 feet until near the target area, then let down to tree top level for the run-in to the target, flying an in-trail formation with a 1000 feet between aircraft.

To complete the proof, let F be any total computable function, and construct h as above.

*" Inge Morath was, above all, a traveller ... er approach to a story was ' to let it grow ', without any apparent concern for narrative structure, trusting in her experience and interests to shape her work rather than in an editorial formula ... She unsentimentally made pictures that were guided by her relationship to a place.

However, none of the largest predatory ( Brontornis ), possibly omnivorous ( Dromornis ) or herbivorous ( Aepyornis ) flightless birds of the Cenozoic ever grew to masses much above 500 kg, and thus never attained the size of the largest mammalian carnivores, let alone that of the largest mammalian herbivores.

: Meredith Phillips, who was let go by bachelor Bob Guiney ( see above ), chose Ian Mckee.

:( 5 ) Jews must " rise above the heavy burden of historical memories which have made it difficult for them to achieve any real understanding, let alone an appreciation, of Christianity.

Let f = ( X < sub > 1 </ sub >, …, X < sub > n </ sub >) be a basis of vector fields, and as above let G be the matrix of coeffients

In Surat Al-Anbiya (" The Prophets ", 21: 96 – 100 ) in fact, we read that Allah threatens to open " the dim of Gog and Magog " to let those people descend from above.

Reference counting can be used to let the reader perform removal so, even if the same thread performs both the update step ( step ( 2 ) above ) and the reclamation step ( step ( 4 ) above ), it is often helpful to think of them separately.

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

If we attempt to use the above formula to compute the derivative of f at zero, then we must evaluate 1 / g ′( f ( 0 )).

When g ( x ) equals g ( a ), then the difference quotient for is zero because f ( g ( x )) equals f ( g ( a )), and the above product is zero because it equals f ′( g ( a )) times zero.

Because the above expression is equal to the difference, by the definition of the derivative is differentiable at a and its derivative is f ′( g ( a )) g ′( a ).

If f is a function of as above, then the second derivative of is:

In this situation, the chain rule represents the fact that the derivative of is the composite of the derivative of f and the derivative of g. This theorem is an immediate consequence of the higher dimensional chain rule given above, and it has exactly the same formula.

Given a function f ∈ I < sub > x </ sub > ( a smooth function vanishing at x ) we can form the linear functional df < sub > x </ sub > as above.

Differential calculus is a subfield of calculus concerned with the study of the rates at which quantities change while integral calculus is defined informally to be the area of the region in the xy-plane bounded by the graph of f, the x-axis, and the vertical lines and, such that areas above the axis add to the total, and the area below the x axis subtract from the total.

Let ( X < sub > i </ sub >, f < sub > ij </ sub >) be an inverse system of objects and morphisms in a category C ( same definition as above ).

is defined informally to be the area of the region in the xy-plane bounded by the graph of f, the x-axis, and the vertical lines and, such that area above the x-axis adds to the total, and that below the x-axis subtracts from the total.

This then opens the question as to what sort of function from a countable set to a countable set, such as f and g above, can actually be constructed.

We assume that A is an m-by-n matrix over either the real numbers or the complex numbers, and we define the linear map f by f ( x ) = Ax as above.

Part of the theoretical material has to do with evolutionary areas ( topics e and f above ), the rest relates especially to the problem of classification.

For instance if, then for ' f ( 5 )= 3 ' to appear, terms like ' g ( 5 )= 6 ' and ' h ( 5, 6 )= 3 ' must occur above.

Then ( X × Y, ( π < sub > 1 </ sub >, π < sub > 2 </ sub >)) is a terminal morphism from Δ to the object ( X, Y ) of D × D. This is just a restatement of the above since the pair ( f, g ) represents an ( arbitrary ) morphism from Δ ( Z ) to ( X, Y ).

As noted above, f ( r ) is the inverse transform of its Fourier transform F ( q ); however, such an inverse transform is a complex number in general.

The function f ( r ) is real if and only if the second integral I < sub > sin </ sub > is zero for all values of r. In turn, this is true if and only if the above constraint is satisfied

Any frequency component above f < sub > s </ sub >/ 2 is indistinguishable from a lower-frequency component, called an alias, associated with one of the copies.

For example, by using the above central difference formula for and and applying a central difference formula for the derivative of at x, we obtain the central difference approximation of the second derivative of f:

hence the above Newton interpolation formula ( by matching coefficients in the expansion of an arbitrary function f ( x ) in such symbols ), and so on.