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Let Mm and N" be oriented manifolds. Endow M x N with the product orientation, that is, if TM MxN _ M and TN M xN 7 are the canonical projections on the elemen...

Question

Let Mm and N" be oriented manifolds. Endow M x N with the product orientation, that is, if TM MxN _ M and TN M xN 7 are the canonical projections on the elements of the product, adl and n respectively define orientations OH M and N_ then the orientation OH M x N is defined to be the orientation defined by TMW TNI: If a € 0"(M) and 8 € n"(N) have compact support, show thata x 8 := (Tira) ^ (TN8)has compact support and is a (m +n)-form o M x N. Then; prove Fubini $ theorem: a X

Let Mm and N" be oriented manifolds. Endow M x N with the product orientation, that is, if TM MxN _ M and TN M xN 7 are the canonical projections on the elements of the product, adl and n respectively define orientations OH M and N_ then the orientation OH M x N is defined to be the orientation defined by TMW TNI: If a € 0"(M) and 8 € n"(N) have compact support, show that a x 8 := (Tira) ^ (TN8) has compact support and is a (m +n)-form o M x N. Then; prove Fubini $ theorem: a X 8 = (J2) Ive)



Answers

Show that if a plane mirror is rotated an angle ? the images it produces rotate by an angle 2?.

Hello. We have a question in which you have been provided with tabular representations of F G and H. Okay, so we have to write GX and hX as transformation of effects. So let us just try and get the relationship between fx and G X. Okay, this is minus two, this is minus one. So simply it is blessed one plus one plus one. Okay and plus one like this plus one. So from effects too, G X affects two. G X X is moving her right word by one unit. Okay, and fx two G X before affects two G X. It is remaining the same so there is no transformation along Y axis. So G X will simply be written their heads F X minus one. Okay, this is G X. Now for ethics let us try and get and get it. This is x minus two, so value of X is keeping the same so there is no movement, no shifting along X axis but if this is minus two this is two minus 13 Okay, so it is getting added by four. So there is why only translation along with access? So a checks will simply be affects bless for So this is the relationship. Thank you so much.

To check this subspace. What we need to do is we need to verify three exams. The first axiom is zero, vector should be belonging to a subspace. W. I mean the set W If you won't, you too belongs to W. Then we should verify that you want to see. You too also belongs to W. What is the closure property? Third, if you belongs to W then land up. You should also belongs to W Where lambda. Is it number from the field? Okay, if all these three exams satisfied, then we call W is a subspace of the factor space week. No, these set off all environ mattresses over the field kit. All right now W is the set of all symmetric matrices. So what do you mean by a symmetric matrix? The matrix was transposed the same as it's A So that's called a symmetric matrix. All right. Now if is symmetric then that's null. Matrix belongs to W. Yes, of course. Because the non matrix transpose his narrative so yes, it's true. Second axiom some of two symmetric matters is A and B. Supposed to be a symmetric B is also symmetric. Then a place be whole transpose is equal to the property of transport. It is a transport transport. But he transposes A. Because asymmetric transpose is B. Because B symmetric diet implies a place to be. He's also symmetric so that implies equals B belonged to W. So second exam is very fine. Now coming to Torrance if is symmetric lambda E. We'll transport the property of transport is lambda and a transport is equal to the lambda. A. Because the symmetric it is lambda lambda. He will transpose islam day. So that means lambda is also symmetric so it belongs to doug. So all the three exams are verified. So yes. Set of all symmetric matrices is a subspace. The second one upper triangular mattresses. So what do you mean by an upper triangular matics? All the elements below the main diagnosed should be zeroes, for example A B C 000 123 So this is an upper triangular matrix below this. All the elements are zeros of is a pet Wrangler. He's a pet Wrangler and he's also a strangler. Then it let's be is it a veranda? Of course, yes. Because these zeros get started with the other metrics in the same place. For example 123 4045006 If you add with 178 089 00 10. So what do you guys? 000000000 So some of the upper triangular mattresses. Also an upper triangular markets. So this is very frank. Null matrix is not trying biometrics because null metrics by default, all the elements below the main diagnosis anyway, zeros. In fact all the elements are real zeros. So it belongs to going mathematics, The Time magazine. If you want to play any number with the matrix scale are multiple, zero into any number of zero. So these zeros will still be retained. So lambda you Orlando A is also an upper triangular matrix. So yes, the set of a particular mattress is a subspace. The third diagonal matrix, basically, we can just individually say that the diagonal matrices from the subspace because lower triangular mattresses also from subspace. Because what do you mean by lower strangler? All the elements about domain that loves you with the same reasoning as a crime. And what is the diagonal matrix? It is both the lower triangle ring up strangler. Since both our Wrangler and Wrangler mattresses from subspace set of diagonal mattresses. Also from subsidies. Simple reason. Right. And skill harm metrics is a special case of diagonal matrix, scalar matrix means all the diagnosed elements should be zero, sorry, All the diagnostic elements should be same and other diagnosed elements other elements should be basically zeros. So this is a special case of diarrhea magics. And since diagnosed mattresses from subspace, the set of the subset of it also forms the suspects. Alright, so this is a prison. So

Reflexive and transitive than ours? Asymmetric. I think this is the most straightforward of the three Bruce. Okay, so uh let R. B. Your reflexive and transitive. Um oh let X Y bien are and now what I'm going to do is I'm going to suppose that Y X I suppose why X is in our then since our is transitive, X. X is in our and why why is in our but that's a contradiction are is mhm. It reflexive. So Y X. What I supposed right here cannot be in our and thus our is anti symmetric. Oh, no, no, no. Asymmetric, sorry. Yeah.

Hi in this given problem, there is a plane mirror at which a ray of light is incident for which, first of all we draw the normal at this nearer and this is the real light incident at an angle or angle of incidents, I then it will be reflected back at the same angle with the normal. So this is i this will be our but actually angle are is equal to the angle of incidents. I now suppose the mirror is deviated by an angle theater, so suppose this is a new orientation of the mirror shown in red, so the normal down at its surface, it will also deviate from its original position by the same angle. T top counterclockwise, so angle as the ray of light remains the same hands, angle of incidence now becomes I minus theta. New angle off incidents let it be I dash, that is I minus theta here, this angle uh minus feet up so uh incident rate will remain the same but the reflected ray will have to change because every time reflected ray should be such that it should be making same angle of reflection with the normal. So the reflected ray will be just like this here, this is reflected right now and this angle here, that is angle r dash, the new angle of reflection, but that should be equal to i minus theta only because we know our dash is equal to I dash using laws of reflection. And here this angle, this deviation because this complete angles are close to from this are proceed to if we subtract our dish, so this deviation deviation in the reflected re means that deviation in image formed so that deviation will be that will be equal to are the original angle, this is the division, that will be equal to our minus our dish. Okay, saw for our we will put it, I hear then minus our dish for our dish, this is I minus tita here, this angle was theta. Sorry, this will not be I, this will not be are actually, this will be our plus theta here, this is the total angle are plus heater and from this are plus theta we have to subtract this our dish and that will remain as the aviation, they'll stop. So this is our plus theta minus our dish for our dish, that is I minus theta and our is also equal to I only saw deviation is equal to for our this is I, so this is I plus theater minus I minus theta. So expanding the record, it will be I plus theta canceling this eye deviation here will be twice of theater. Hence it is proved that if the mirror is deviated by An angle Tita image formed by eight will be deviated by to teeter from its original position. Thank you


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