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case and vector
The coordinates of the vector are equal to the projections of the vector ( yellow ) onto the x-component basis vector ( green )-using the dot product ( a special case of an inner product, see below ).
A special case is the inner product of a vector with itself, which is square of its norm ( magnitude ):
The analogy in the case of rotating bucket is that the element of water surface will " slide " up or down the surface unless the normal to the surface aligns with the vector resultant formed by the vector addition F < sub > g </ sub > + F < sub > Cfgl </ sub >.
In the case where the vector field v and ∇ are interchanged:
( Note: ∇< sup > 2 </ sup > F represents, in this case, the vector formed by the individual Laplacian of each component of the vector in question )
where vertical bars |...| denote the vector magnitude, which in the case of r ( t ) is simply the radius R of the path.
In light of the physical interpretation, a vector field with constant zero divergence is called incompressible or solenoidal – in this case, no net flow can occur across any closed surface.
The determinant provides important information when the matrix is that of the coefficients of a system of linear equations, or when it corresponds to a linear transformation of a vector space: in the first case the system has a unique solution if and only if the determinant is nonzero, while in the second case that same condition means that the transformation has an inverse operation.
For the special case, this implies that the length of a vector is preserved as well — this is just Parseval's theorem:
The concept extends to an arbitrary number of elements of the random vector, in which case in general the iso-density contours are ellipsoids.
If the field of scalars of the vector space has characteristic p, and if p divides the order of the group, then this is called modular representation theory ; this special case has very different properties.
In this case, the group is also called a permutation group ( especially if the set is finite or not a vector space ) or transformation group ( especially if the set is a vector space and the group acts like linear transformations of the set ).
In the complex case of this theorem, the C-linearity assumptions demand, in addition to the assumptions for the real case, that for every vector x ∈ U, the vector be also in U and.
Two quarks can have their spins aligned, in which case the two spin vectors add to make a vector of length S = 1 and three spin projections ( S < sub > z </ sub > = + 1, S < sub > z </ sub > = 0, and S < sub > z </ sub > = − 1 ), called the spin-1 triplet.
For computing in either case, the system must be initialized, for example into the all-zeros string,, corresponding to the vector ( 1, 0, 0, 0, 0, 0, 0, 0 ).
There is no figure which has as full symmetry group the circle group, but for a vector field it may apply ( see the 3D case below ).

case and bundles
In each case, lines of the bundles are transformed by involutions within the pencils they determine with the multiple secant.
( the theorem can actually be further generalized to the case where the Lagrangian depends on up to the n < sup > th </ sup > derivative using jet bundles )
The preservation, use, and reconfiguration of mummy bundles and skeletal remains, like the later Inca, may suggest that this is the case.
In the last case, slow-moving arboreal mammals such as sloths, lorises and arboreal anteaters possess retia of the highly developed type known as vascular bundles.
case is generalized to gauge natural bundles in the form of a new geometric concept
In the case when there are just two non-zero bundles in the complex this implies that the symbol is an isomorphism off the zero section, so an elliptic complex with 2 terms is essentially the same as an elliptic operator between two vector bundles.
Conversely the index theorem for an elliptic complex can easily be reduced to the case of an elliptic operator: the two vector bundles are given by the sums of the even or odd terms of the complex, and the elliptic operator is the sum of the operators of the elliptic complex and their adjoints, restricted to the sum of the even bundles.
After Serre we recognise l ( K − D ) as the dimension of H < sup > 1 </ sup >( D ), where now D means the line bundle determined by the divisor D. That is, Serre duality in this case relates groups H < sup > 1 </ sup >( D ) and H < sup > 0 </ sup >( KD *), and we are reading off dimensions ( notation: K is the canonical line bundle, D * is the dual line bundle, and juxtaposition is the tensor product of line bundles ).
In the case of the complex line bundle, we are looking in fact also for circle bundles.
The case of the construction of vector bundles from data on a disjoint union of topological spaces is a straightforward place to start.
The language of fibre bundles allows this notion of a section to be generalized to the case when E is not necessarily a Cartesian product.
In December 2006, along with Larry Jones and his group Feed the Children, ABS participated in the " largest food distribution effort " in the U. S .. More than 10, 000 bundles of food were handed out in one day ( consisting of a case of chicken, juices, fruits and vegetable ) in Harlem, outside the Abyssinian Baptist Church on 138th Street.
In the first case three bundles of tow are burnt before the newly elected pontiff, who sits on the sedia gestatoria, while a master of ceremonies says: " Pater Sancte, sic transit gloria mundi " ( Holy Father, so passes the glory of the world ).
Furthermore, in this case the structure group of the normal bundles is the circle group ; it follows that the choice of embeddings can be canonically identified with the group of homotopy classes of maps from to the circle, which in turn equals the first integral cohomology group.
It is no longer true that the preimages π < sup > − 1 </ sup >( x ) must all look alike, unlike fiber bundles where the fibers must all be isomorphic ( in the case of vector bundles ) and homeomorphic.
The definition in that case is usually said with greater care ( using invertible sheaves or holomorphic line bundles ); see below.
Justification of the expected properties comes from the line bundle case, where V is a Whitney sum of line bundles.
In general a mechanism for reducing to that case comes from the splitting principle for vector bundles.
A hedonic regression equation treats these attributes ( or bundles of attributes ) separately, and estimates prices ( in the case of an additive model ) or elasticity ( in the case of a log model ) for each of them.
To define the Todd class td ( E ) where E is a complex vector bundle on a topological space X, it is usually possible to limit the definition to the case of a Whitney sum of line bundles, by means of a general device of characteristic class theory, the use of Chern roots ( aka, the splitting principle ).

case and principal
" As already indicated, the one exception is in the case of a priest celebrating Mass without other priests or as principal celebrant.
In the case of normal faults, the rock mass is pushed down in a vertical direction, thus the pushing force ( greatest principal stress ) equals the weight of the rock mass itself.
In the case of thrusting, the rock mass ' escapes ' in the direction of the least principal stress, namely upward, lifting the rock mass up, thus the overburden equals the least principal stress.
From late 1947 to May 1949, Fuchs gave Alexander Feklisov, his case officer, the principal theoretical outline for creating a hydrogen bomb and the initial drafts for its development as the work progressed in England and America.
An extension of Galois theory to the case of continuous symmetry groups was one of Lie's principal motivations.
Clinical Medicine and Pathophysiology: pathophysiology with emphasis on the principal physiological systems, clinical medicine, differential diagnosis, clinical correlation and case studies, chemical dependency, chronic pain management ;
In the case of the Earth, the principal sources of tidal force are the Sun and Moon, which continuously change location relative to each other and thus cause nutation in Earth's axis.
Most sections also have an assistant principal ( or co-principal or associate principal ), or in the case of the first violins, an assistant concertmaster, who often plays a tutti part in addition to replacing the principal in his or her absence.
The comic was reprimanded by the United Nations after featuring a strip called " The Thieving Gypsy Bastards " During the resulting court case, UK newspaper The Sun ran a story revealing that the principal Roma man who initiated the action against them was in fact also being tried for ( and was later found guilty of ) handling stolen property.
Reiner even started a contest with the crew during principal photography-whoever came up with the title won a case of champagne.
The simplest case to analyze is the one in which g is a principal axis ( eigenvector ) of, and the other two eigenvalues of are identical.
The principal surviving literary sources are Dio Cassius ( a contemporary and sometimes first-hand observer, but for this reign, only transmitted in fragments and abbreviations ), Herodian and the Historia Augusta ( untrustworthy for its character as a work of literature rather than history, with elements of fiction embedded within its biographies ; in the case of Commodus, it may well be embroidering upon what the author found in reasonably good contemporary sources ).
At the end of 1981, he finalized the Spatola case for trial, which enabled the prosecution to win 74 convictions, based on Falcone ’ s “ web of solid evidence, bank and travel records, seized heroin shipments, fingerprint and handwriting analyses, wiretapped conversations and firsthand testimony ” that proved that “ Sicily had replaced France as the principal gateway for refining and exporting heroin to the United States ”.
The principal packet structure of ARP packets is shown in the following table which illustrates the case of IPv4 networks running on Ethernet.
In the case when all future cash flows are incoming ( such as coupons and principal of a bond ) and the only outflow of cash is the purchase price, the NPV is simply the PV of future cash flows minus the purchase price ( which is its own PV ).
The principal Royal Canadian Mounted Police ( RCMP ) investigator on the case retired a year later.
In some countries, such as Pakistan, Israel, Thailand and several African nations, there is a strong relationship between the military and the principal national airlines, and many airline pilots come from the military ; however, that is no longer the case in the United States and Western Europe.
The term Japhetic was also applied by the early linguists ( brothers Grimm, William Jones, Rasmus C. Rask and others ) to what later became known as the Indo-European language group, on the assumption that, if descended from Japheth, the principal languages of Europe would have a common origin, which apart from Uralic, Kartvelian, Pontic, Dagestanian, and Basque, appears to be the case.
The principal argument favouring a migration during or after the transition period to NIA is the loss of the old system of nominal case, and its reduction to just a two-way case system, nominative vs. oblique.
The principal argument favouring a migration during or after the transition period to NIA is the loss of the old system of nominal case, and its reduction to just a two-way case system, nominative vs. oblique.

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