In 3 dimensions, a differential 0-form is simply a function f ( x, y, z ); a differential 1-form is the following expression: a differential 2-form is the formal sum: and a differential 3-form is defined by a single term: ( Here the a-coefficients are real functions ; the " wedge products ", e. g. can be interpreted as some kind of oriented area elements,, etc.

In December 2004 Joe Pasquale, winner of the fourth series of ITV's I'm a Celebrity ... Get Me Out of Here !, became well known for his frequent use of the term " Jacobs ", for Jacob's Crackers, a rhyming slang term for knackers i. e. testicles.

Here, sgn ( p ) is the signature of each permutation ( i. e. + 1 if p is composed of an even number of transpositions, and − 1 if odd.

Here, the battery acts as a temporary energy store, allowing e. g. the implementation of regenerative braking and switching off the hydrocarbon engine when idling or stationary ( as used in automobiles such as the Toyota Prius ).

* Interactional: Here language is used to make contact with others and form relationships ( e. g. ' Love you, mummy ')

Here the most regular of irregularities, i. e., archaisms, will be delineated:

For example, the notion of gauge invariance forms the basis of the well-known Mattis spin glasses, which are systems with the usual spin degrees of freedom for i = 1 ,..., N, with the special fixed " random " couplings Here the ε < sub > i </ sub > and ε < sub > k </ sub > quantities can independently and " randomly " take the values ± 1, which corresponds to a most-simple gauge transformation This means that thermodynamic expectation values of measurable quantities, e. g. of the energy are invariant.

Here e < sub > a </ sup >< sup > μ </ sup > is the vierbein and D < sub > μ </ sub > is the covariant derivative for fermion fields, defined as follows

Here D is a traceless matrix such that αDe gives the derivative of e with respect to z.

Here, an " event " is a set of zero or more outcomes, i. e., a subset of the sample space.

Here, in each step of the XPath, the axis ( e. g. or ) is explicitly specified, followed by and then the node test, such as or in the examples above

Here, a number of its constituent sovereign nations have legalized traditional authority as manifested in the recognized hereditary transmission of chieftaincy in historically relevant regions of the continent ( e. g. the Asantehene of Ghana ).

( Here, e is the base of the natural logarithm, about 2. 71828 ..., and i is the imaginary unit, a square root of − 1.

A decade-and-a-half later, the reggae-rock band 311's hit single " I'll Be Here Awhile " " borrowed " several lines from " The Lust ..." ( i. e., " And if a person, place, or thing can deliver / I will quiver with delight ").

Here the coefficient A is an approximation of the short-wavelength ( e. g., ultraviolet ) absorption contributions to the refractive index at longer wavelengths.

Here, is density, u, v, and are the components of wind velocity, and is the 2-dimensional ( i. e. horizontal-component-only ) del.

Here the U ( 1 ) subgroup of U ( n ) consists of matrices of the form diag ( e < sup > iθ </ sup >, 1, 1, ..., 1 ).

Here there is the idea that God's heritage ( the created world ) is largely under the dominion of evil – i. e., it is " shared with Azazel " ( Abr.

Here | z | is the absolute value or modulus of the complex number z ; θ, the argument of z, is usually taken on the interval 0 ≤ θ < 2π ; and the last equality ( to | z | e < sup > iθ </ sup >) is taken from Euler's formula.

Here if the null hypothesis weren't rejected ( i. e. there is no reason to reject the hypothesis that A is at least as good as B ), the conclusion would be that treatment A should continue to be used.

Here, the real-life cast members were superimposed over pictures hanging in an art gallery and interacted with them in different ways ( e. g., spinning the canvas to put it right-side up, swinging the frame out as if it were a door, etc .).

Here h is the zero-field height ( or unreduced height ) of the barrier, e is the elementary positive charge, F is the barrier field, and ε < sub > 0 </ sub > is the electric constant.

Here the fiber over a point x in M is the set of all frames ( i. e. ordered bases ) for the tangent space T < sub > x </ sub > M.

Here R plays the role of im T and U is ker T, i. e.

Here, < sub > n </ sub > denotes the sample mean of the first n samples ( x < sub > 1 </ sub >, ..., x < sub > n </ sub >), s < sup > 2 </ sup >< sub > n </ sub > their sample variance, and σ < sup > 2 </ sup >< sub > n </ sub > their population variance.

Here, x is the angle that a line connecting the origin with a point on the unit circle makes with the positive real axis, measured counter clockwise and in radians.

Here Person ( x ) is interpreted to mean x is a person, Time ( y ) to mean that y is a moment of time, and Canfool ( x, y ) to mean that ( person ) x can be fooled at ( time ) y.

Here ( x, y | x ', y ') means that instead of ψ we are writing a different formula, in which x and y are replaced with x ' and y '.

Here, we have < tt >( λx. xx )( λx. xx ) → ( xx ):= λx. xx = ( x := λx. xx )( x := λx. xx ) = ( λx. xx )( λx. xx )</ tt >.

Here the summation is over all prime powers up to x.

Here m is the particle's mass and V ( x ) is the applied potential.

Here x is the node on which the splay operation is performed and root is the root node of the tree.

The Wildhearts song ' 29 x The Pain ', which namedrops the bands major influences, opens with the lyrics ' Here, sitting in my room, with the Replacements and Hüsker Dü '.

Here u ( x ) measures the distance from the equilibrium of the mass situated at x.

Here N < sub > x </ sub > is the number of x atoms per volume, λ the electron escape depth, θ the analyzer angle, T the transmission of the analyzer, I ( t ) the electron excitation flux at depth t, dΩ the solid angle, and δt is the thickness of the layer being probed.

Here u ( x, t ) is the temperature of the rod at position x and time t and α is a constant that depends on how fast heat diffuses through the rod.

Here π: P → X is required to be a smooth map between smooth manifolds, G is required to be a Lie group, and the corresponding action on P should be smooth.

Here FG: D → D and GF: C → C, denote the respective compositions of F and G, and I < sub > C </ sub >: C → C and I < sub > D </ sub >: D → D denote the identity functors on C and D, assigning each object and morphism to itself.

Here if f, g: X → Y are group homomorphisms, their coequalizer is the quotient of Y by the normal closure of the set

Here a function type is often denoted A → B, following mathematical convention, or B < sup > A </ sub >, based on the fact that there exist exactly B < sup > A </ sub > ( exponentially many ) set-theoretic functions mapping A to B.

Here I: Mon → Grp is the functor sending every monoid to the submonoid of invertible elements and K: Mon → Grp the functor sending every monoid to the Grothendieck group of that monoid.

Here, (( α → β ) → α ) is the type of the function f, which can either return a value of type α directly or apply an argument to the continuation of type ( α → β ).

Here, the choice is mostly between Latin letters with diacritics, as used in many Latin-based orthographies of other Slavic languages, and digraph combinations, as used in English: ж → ž / zh, ч → č / ch, ш → š / sh, щ → št / ŝ / sht.

Let V be a smooth vector field on a smooth manifold M. There is a unique maximal flow D → M whose infinitesimal generator is V. Here D ⊆ R × M is the flow domain.