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Prove that a finite integral domain is a field?
A finite integral domain ( D ) has no zero divisors and is commutative. Since ( D ) is finite, for any non-zero element ( a \in D ), the set ( { a, 2a, 3a, \ldots, na } ) (where ( n ) is the number of elements in ( D )) must eventually repeat due to the pigeonhole principle. Thus, there exists an integer ( k ) such that ( ka = 0 ), but since ( D ) has no zero divisors, this implies ( k = 0 ) or ( a = 0 ), meaning every non-zero element has a multiplicative inverse. Therefore, ( D ) is a field.
What does the area under the curve equal?
The are under the curve on the domain (a,b) is equal to the integral of the function at b minus the integral of the function at a
What is the range of the function y x2 2x 4.?
The range depends on the domain. If the domain is the complex field, the range is also the whole of the complex field. If the domain is x = 0 then the range is 4.
What represent the surface integral of electric field intensity?
Electric flux.
What is the conditional expectation function?
It is the integral (or sum) of the joint probability distribution function of the two events, integrated over the domain in which the condition is met.
Can non integer numbers be part of the domain?
Yes, the domain represents all the x values on a graph. Since these x values can be non-integral numbers, the domain can contain non-integral numbers.
Is Every commutative ring with unity is an integral domain?
Nope. Take the commutative ring Z4 (the set {0,1,2,3} with modular arithmetic and multiplication). 1 is the identity element. But 2 x 2 = 4 = 0 while 2 is not the 0 element. So it's not an integral domain.
What does the area under the curve equal?
The are under the curve on the domain (a,b) is equal to the integral of the function at b minus the integral of the function at a
What is the domain of the cotangent function?
The cotangent function has domain all real numbers except integral multiples of pi./2(90degrees).
What is the surface integral of electric field?
The surface integral of the electric field is the flux of the electric field through a closed surface. Mathematically, it is given by the surface integral of the dot product of the electric field vector and the outward normal vector to the surface. This integral relates to Gauss's law in electrostatics, where the total electric flux through a closed surface is proportional to the total charge enclosed by that surface.
Discuss line integral and its real life applications?
A line integral can evaluate scalar and vector field functions along a curve/path. When applied on vector field, line integral is considered as measure of the total effect of the vector field along a specific curve whereas in scalar field application, the line integral is interpreted as the area under the field carved out by a particular curve.Line integral has many applications in physics. In mechanics, line integral is used to determine work done by a force in moving an object along a curve. In circuit analysis, it is used for calculating voltage.
What is the range of the function y x2 2x 4.?
The range depends on the domain. If the domain is the complex field, the range is also the whole of the complex field. If the domain is x = 0 then the range is 4.
What represent the surface integral of electric field intensity?
Electric flux.
What is Gaussian ring?
a Ring is called Gaussian Ring if: R is an Integral Domain. R is a Unique Factorization Domain (UFD), i.e. every non-zero non-unit element in R can be written as a product of irreducibles of R and The factorization into irreducibles is unique up to the order of the multiplication or the associates of the factors. Hope this has helped anyone Sagy
What value of a is required to normalize the function ?
To normalize a function, the value of a must be such that the integral of the function squared over its domain is equal to 1.
In order for a protein to be an integral protein it would have to be what?
An integral protein must be firmly embedded within a cell membrane, spanning across it from one side to the other. This positioning allows integral proteins to have both an extracellular and intracellular domain, contributing to their crucial role in cell signaling and transport.
What is domain trust relationship?
Each Domain in a active directory forest has a what kind trust relationship with every other domain in a forest?
