Show that geodesics in the plane are straight lines.SolutionThe.pdf

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Show that geodesics in the plane are straight lines. Solution The length of the arc can be defined as ds=dx2+dy2ds=dx2+dy2 Consider the family of curves lying wholly on a given surface and passing through two given points both on the same plane. Among these curves there will be one for which the length of arc is a minimum. Such curves are known geodesics. Because of that we are required to minimize L=ds=dx2+dy2L=ds=dx2+dy2 where because of parameterization x=ucos(v)dx=ducos(v)usin(v)dvx=ucos(v)dx=ducos(v)usin(v)dv y=usin(v)dy=dusin(v)+ucos(v)dvy=usin(v)dy=dusin(v)+ucos(v)dv By rewriting the integral L=dx2+dy2=du2+u2dv2=dv(dudv)2+u2L=dx2+dy2=du2+u2dv2=dv(dudv)2+u2 The Euler Lagrange equation (for sake of simplicity u=dudvu=dudv) Luddv(Lu)=0Luddv(Lu)=0 uu2+u2ddv(uu2+u2)=0uu2+u2ddv(uu2+u2)=0 uu2+u2uuuu2(u2+u2)3/2=0uu2+u2uuuu2(u2+u2)3/2=0 uu2+u2uu(u2+u2)3/2=0uu2+u2uu(u2+u2)3/2=0 One solution is u=0u=0 and the other one u2+u2uu=0u=Asec(v+B)u2+u2uu=0u=Asec(v+B) where AA and BB are cosntants. In cartesian coordinates parametric equations for the geodesics are x=ucos(v)=Asec(v+B)cos(v)=Acos(v)cos(v+B)x=ucos(v)=Asec(v+B)cos(v)=Acos(v)cos(v+B) y=usin(v)=Asec(v+B)sin(v)=Asin(v)cos(v+B)y=usin(v)=Asec(v+B)sin(v)=Asin(v)cos(v+B) To eliminate vv we can use trigonometric identities and change of variable such that w=v+Bw=v+Bwithout loss of generality x=Acos(wB)cos(w)=Acos(B)Asin(B)sin(w)cos(w)sin(w)cos(w)=cot(B)xAsin(B)x=Acos(wB)cos (w)=Acos(B)Asin(B)sin(w)cos(w)sin(w)cos(w)=cot(B)xAsin(B) and y=Asin(wB)cos(w)=Acos(B)sin(w)cos(w)Asin(B)y=Asin(wB)cos(w)=Acos(B)sin(w)cos(w)Asi n(B) by replacing sin(w)cos(w)sin(w)cos(w) y=Acos(B)(cot(B)xAsin(B))Asin(B)y=Acos(B)(cot(B)xAsin(B))Asin(B) y=cot(B)x+A(cos(B)cot(B)sin(B))y=cot(B)x+A(cos(B)cot(B)sin(B)) which is a straight line..

Show that geodesics in the plane are straight lines.
Solution
The length of the arc can be defined as
ds=dx2+dy2ds=dx2+dy2
Consider the family of curves lying wholly on a given surface and passing through two given
points both on the same plane. Among these curves there will be one for which the length of arc
is a minimum. Such curves are known geodesics. Because of that we are required to minimize
L=ds=dx2+dy2L=ds=dx2+dy2
where because of parameterization
x=ucos(v)dx=ducos(v)usin(v)dvx=ucos(v)dx=ducos(v)usin(v)dv
y=usin(v)dy=dusin(v)+ucos(v)dvy=usin(v)dy=dusin(v)+ucos(v)dv
By rewriting the integral
L=dx2+dy2=du2+u2dv2=dv(dudv)2+u2L=dx2+dy2=du2+u2dv2=dv(dudv)2+u2
The Euler Lagrange equation (for sake of simplicity u=dudvu=dudv)
Luddv(Lu)=0Luddv(Lu)=0
uu2+u2ddv(uu2+u2)=0uu2+u2ddv(uu2+u2)=0
uu2+u2uuuu2(u2+u2)3/2=0uu2+u2uuuu2(u2+u2)3/2=0
uu2+u2uu(u2+u2)3/2=0uu2+u2uu(u2+u2)3/2=0
One solution is u=0u=0 and the other one
u2+u2uu=0u=Asec(v+B)u2+u2uu=0u=Asec(v+B)
where AA and BB are cosntants. In cartesian coordinates parametric equations for the geodesics
are
x=ucos(v)=Asec(v+B)cos(v)=Acos(v)cos(v+B)x=ucos(v)=Asec(v+B)cos(v)=Acos(v)cos(v+B)
y=usin(v)=Asec(v+B)sin(v)=Asin(v)cos(v+B)y=usin(v)=Asec(v+B)sin(v)=Asin(v)cos(v+B)
To eliminate vv we can use trigonometric identities and change of variable such that
w=v+Bw=v+Bwithout loss of generality
x=Acos(wB)cos(w)=Acos(B)Asin(B)sin(w)cos(w)sin(w)cos(w)=cot(B)xAsin(B)x=Acos(wB)cos
(w)=Acos(B)Asin(B)sin(w)cos(w)sin(w)cos(w)=cot(B)xAsin(B)
and
y=Asin(wB)cos(w)=Acos(B)sin(w)cos(w)Asin(B)y=Asin(wB)cos(w)=Acos(B)sin(w)cos(w)Asi
n(B)
by replacing sin(w)cos(w)sin(w)cos(w)
y=Acos(B)(cot(B)xAsin(B))Asin(B)y=Acos(B)(cot(B)xAsin(B))Asin(B)
y=cot(B)x+A(cos(B)cot(B)sin(B))y=cot(B)x+A(cos(B)cot(B)sin(B))

which is a straight line.

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What was the lasting significance of the Crusades? your answer must have a minimum of 7 sentences. Solution The Crusades were major military expeditions formed together from a majority of Christian European nations during 1095-1099 BC to rescue Christians in the region of Palestine & Jerusalem from the hands of Mohammedans. The lasting significant of crusades are considered as crucial factors in the progression of modern civilization & history. The meticulous significance of crusades have specific influence on wealth & power distribution as utilitarian perspectives in the modern world & these are mainly influenced by political matters and catholic church. Later, the effects of lasting crusades have profound effect on the development of commerce, feudalism followed by development of intellectual development, social effects. Finally, the crusades have mainly influenced the people for material effects followed by famous Voyages of discovery..

What was the lasting significance of the Crusades your answer must .pdf

What was the lasting significance of the Crusades your answer must .pdf

What was the lasting significance of the Crusades your answer must .pdf

daniamantileonismc36

1. Screaming Pandas are a highly endangered if annoying creature that you want to conserve for future generations. If there are only 25 surviving pandas in your care: a) How much heterozygosity will be lost in the next generation? (Hint: use the formula given in lecture). b) What might the long-term consequences be of this loss of genetic diversity, and why? c) What if anything can you do to prevent this decrease in heterozygosity? formula given: Hg+1 = Hg (1 - (1/2N)) where N = population size 2. You are a zookeeper (no, I checked; you are). One day, you discover that a baby endangered black rhino born in your zoo carries a new mutant allele that gives it 20% higher fitness than normal (wild-type) rhinos. Since these animals are highly endangered in the wild, you consider releasing this lucky mutant into a natural population in Africa, hopefully to increase the average fitness of the surviving rhinos there. Would it make more sense to release this mutant, and the new allele it carries, into a population of 20 rhinos, or a population of 1,000 rhinos – in which population would the mutant allele likely do the most good? Explain your reasoning. 3. Islands often show extremely modified organisms, compared to their mainland counterparts -- examples include miniature hippos and rhinos, or gigantic birds, lizards and tortoises. (A) Give two reasons islands are likely to be strongly affected by genetic drift, and why this might lead to extreme modifications. (B) Explain why natural selection might change the organisms on an island from the ancestral mainland forms. [note: this question is basically asking you to think critically about the differences between selection and drift, and to recognize the ways each can affect a population that colonizes a new environment like an empty island.] Solution a) Calculation of Heterozygosity lost in the next generation need heterozygosity or allele freq in parent generation. b). Loss of genetic diversity would lead to inbreeding depression and may lead to extinctions under any selection pressure. c).Random mating between the individuals can be done to prevent the decrease in heterozygosity..

1. Screaming Pandas are a highly endangered if annoying creature tha.pdf

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daniamantileonismc36

1.Which plants contain lignified vascular tissue? (Select all that apply.) Gametophytes of bryophytes Gametophytes of seedless vascular plants Sporophytes of bryophytes Sporophytes of seedless vascular plants 2. Which plants produce multiple sporangia? (Select all that apply.) Gametophytes of bryophytes Gametophytes of seedless vascular plants Sporophytes of bryophytes Sporophytes of seedless vascular plants 3. Which of the following groups produce sporangia on leaves? (Mark all that apply.) Hepatophyta Lycophyta Bryophyta Anthocerophyta Monilophyta Gametophytes of bryophytes Gametophytes of seedless vascular plants Sporophytes of bryophytes Sporophytes of seedless vascular plants 2. Which plants produce multiple sporangia? (Select all that apply.) Gametophytes of bryophytes Gametophytes of seedless vascular plants Sporophytes of bryophytes Sporophytes of seedless vascular plants 3. Which of the following groups produce sporangia on leaves? (Mark all that apply.) Hepatophyta Lycophyta Bryophyta Anthocerophyta Monilophyta Solution 1. Sporophytes of seedless vascular plants Explanation- The xylem of seedless vascular plants is partially lignified which makes the wall more rigid. 2. Sporophytes of seedless vascular plants Explanation- The sporangia are present in clusters, the sori in ferns 3. Lycophyta, Monilophyta Explanation- Sporangia is borne on the adaxial surface of the leaves in Lycophytes. Sporania in monilophyta are present on leaves..

1.Which plants contain lignified vascular tissue (Select all that a.pdf

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1.Which plants contain lignified vascular tissue (Select all that a.pdf

daniamantileonismc36

What relatively recent scientific advancement has made mapping by linkage or classical genetic mapping approaches virtually obsolete? the use of synteny negative interference the inclusion of the X and Y chromosome in SNP experiments positive interference the genome sequence of a species Solution The answer is The use of synteny. Synteny - physical co-localization of genetic loci on the same chromosome within an individual or species..

What relatively recent scientific advancement has made mapping by lin.pdf

What relatively recent scientific advancement has made mapping by lin.pdf

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daniamantileonismc36

What characteristic of information is satisfied by an original signed document? A. Relevance. B. Reliability. C. Sufficiency. D. Usefulness Solution If any information is satisfied by an original signed document it means we can assume that this signed document is reliable. We can trust on them papers because these papers are signed so that is reliable papers. so characteristic of information is satisfied by an original signed document is b) Reliability. so answer is b) Reliability.

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What characteristic of information is satisfied by an original signe.pdf

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daniamantileonismc36

What are pseudo-assembly instructions and what are they used for? Solution pseudo instruction : they are the special commands in the assemly language which gives commands to the assembler about the location of the program .(it means that the where the program should be assembled at) the program can be assembled at 1)the module name 2)declartion of data 3)the title of the program 4)macros defnitions they doesnt produces excutable code unless a machine instruction is used they used for 1)they copy the content from one register to another register 2)they load in to register 3)they load address into register 4)they can load 32 bit offset word in to memory.

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