# Thread: First Year Mathematics B (Integration, Series, Discrete Maths & Modelling)

1. ## Re: MATH1231/1241/1251 SOS Thread Originally Posted by InteGrand Compute the partial derivatives (which they've done) and sub. in the given (x,y) point.

Ohh okay! They subbed the points in, explains things. Thanks!  Reply With Quote

2. ## Re: MATH1231/1241/1251 SOS Thread

Another tangent to surface question, how do I do this one: S: z^2+x^2+y^2 = 1., x0 = (1/3, 1/2, root(23)/6)

Find normal vector and equation of the tangent plane to the surface S at the point x0.

What is confusing me is the z. So do I move everything but the z to the RHS? Then solve?  Reply With Quote

3. ## Re: MATH1231/1241/1251 SOS Thread Originally Posted by Flop21 Another tangent to surface question, how do I do this one: S: z^2+x^2+y^2 = 1., x0 = (1/3, 1/2, root(23)/6)

Find normal vector and equation of the tangent plane to the surface S at the point x0.

What is confusing me is the z. So do I move everything but the z to the RHS? Then solve?  Reply With Quote

4. ## Re: MATH1231/1241/1251 SOS Thread Originally Posted by Flop21 Another tangent to surface question, how do I do this one: S: z^2+x^2+y^2 = 1., x0 = (1/3, 1/2, root(23)/6)

Find normal vector and equation of the tangent plane to the surface S at the point x0.

What is confusing me is the z. So do I move everything but the z to the RHS? Then solve?  Reply With Quote

5. ## Re: MATH1231/1241/1251 SOS Thread how do I do this one  Reply With Quote

6. ## Re: MATH1231/1241/1251 SOS Thread Originally Posted by Flop21  how do I do this one
Row-reduce that augmented matrix (get it into row-echelon form) and you should find a zero row at the bottom, with some linear expression involving x,y and z in this row in the right-hand augmented part. There will be solutions, i.e. we will have v be in S, if and only if this expression equals 0.  Reply With Quote

7. ## Re: MATH1231/1241/1251 SOS Thread

A hint on part c) please before I succumb to being 100% stuck. (Aside from conjug(x)=x)  Reply With Quote

8. ## Re: MATH1231/1241/1251 SOS Thread Originally Posted by leehuan A hint on part c) please before I succumb to being 100% stuck. (Aside from conjug(x)=x)

lol steven was doing this the other day

The point P is on the real axis so the conjugate of P is itself.

So, the conjugate distances from above will be (x-ω)(x-ω*) = x² - 2xcosθ +1

On the other hand, the product of all the conjugate pairs form all the irreducible quadratic factors of the degree n polynomial of unity.

Throw in the factor of (x+1) based on the parity of n.

Lastly, chuck in the 1-x factor which appears for all values of n.

This is equal to 1-x^n  Reply With Quote

9. ## Re: MATH1231/1241/1251 SOS Thread Originally Posted by Paradoxica lol steven was doing this the other day

The point P is on the real axis so the conjugate of P is itself.

So, the conjugate distances from above will be (x-ω)(x-ω*) = x² - 2xcosθ +1

On the other hand, the product of all the conjugate pairs form all the irreducible quadratic factors of the degree n polynomial of unity.

Throw in the factor of (x+1) based on the parity of n.

Lastly, chuck in the 1-x factor which appears for all values of n.

This is equal to 1-x^n
That was a bit too rushed. I had the idea of the quadratic factors but I don't see how they transform into 1-x^n  Reply With Quote

10. ## Re: MATH1231/1241/1251 SOS Thread Originally Posted by leehuan That was a bit too rushed. I don't get how the quadratic factors transform into 1-x^n
TL;DR

Factorise the nth polynomial of unity into it's complex factors and use the knowledge that x is inside the unit circle to obtain the distances you want.  Reply With Quote

11. ## Re: MATH1231/1241/1251 SOS Thread

This isn't helping sorry. Too rushed and you TLDRd it further. I don't see it....  Reply With Quote

12. ## Re: MATH1231/1241/1251 SOS Thread Originally Posted by leehuan This isn't helping sorry. Too rushed and you TLDRd it further. I don't see it....
...

x-ω

ω is one of the nth roots of unity

do I have to say more.  Reply With Quote

13. ## Re: MATH1231/1241/1251 SOS Thread Originally Posted by leehuan This isn't helping sorry. Too rushed and you TLDRd it further. I don't see it....
Essentially, here is a sketch.  Reply With Quote

14. ## Re: MATH1231/1241/1251 SOS Thread Originally Posted by InteGrand Row-reduce that augmented matrix (get it into row-echelon form) and you should find a zero row at the bottom, with some linear expression involving x,y and z in this row in the right-hand augmented part. There will be solutions, i.e. we will have v be in S, if and only if this expression equals 0.
I'm stuck on a similar one How do I find this vector???   Reply With Quote

15. ## Re: MATH1231/1241/1251 SOS Thread

why are you still doing matricies  Reply With Quote

16. ## Re: MATH1231/1241/1251 SOS Thread Originally Posted by turntaker why are you still doing matricies
lol what they did in 1131 was just adding/subtracting multiplying them, here they learn vector spaces, basis etc and eigenvalues/vectors  Reply With Quote

17. ## Re: MATH1231/1241/1251 SOS Thread Originally Posted by turntaker why are you still doing matricies
I believe this topic is called "linear combinations and spans" or the overall topic is "vector spaces".  Reply With Quote

18. ## Re: MATH1231/1241/1251 SOS Thread

are you doing things like intersection of lines, planes etc  Reply With Quote

19. ## Re: MATH1231/1241/1251 SOS Thread Originally Posted by turntaker are you doing things like intersection of lines, planes etc
we're doing things like vector spaces (e.g. show that the set is vector space), subspaces (e.g. show that the line segment defined by blah is not a subspace of R3 or find distinct members of the set blah). And I guess we are coming to a point involving matrices in sets or subspaces or whatever.

Soz don't really know wtf I'm talking about at this point lol.  Reply With Quote

20. ## Re: MATH1231/1241/1251 SOS Thread Originally Posted by Flop21 we're doing things like vector spaces (e.g. show that the set is vector space), subspaces (e.g. show that the line segment defined by blah is not a subspace of R3 or find distinct members of the set blah). And I guess we are coming to a point involving matrices in sets or subspaces or whatever.

Soz don't really know wtf I'm talking about at this point lol.
Nvm I was thinking about vectors not matricies. But the two are connected somehow.  Reply With Quote

21. ## Re: MATH1231/1241/1251 SOS Thread Originally Posted by Flop21 I'm stuck on a similar one How do I find this vector??? You essentially have two conditions:

x -(1/4)y + 0z = 0

and

0x -(3/4)y + z = 0

(assuming your answer was right, I didn't check it).

So the vector (x,y,z) is in the column space of A iff it satisfies those two conditions.

You can turn those conditions into a matrix and get it into row-echelon form (actually in this case, it already is in row-echelon form).

Then do the usual procedure of setting a non-leading column's variable to a free parameter (in this case, that variable is z, so set z = lambda say), then use back substitution as usual to get x and y in terms of lambda.

This will mean you'll end up with x, y, z in terms of lambda, which means you can get a vector b as desired. For the sake of example, if you ended up with x = 2lambda, y = -lambda, z = lambda, we'd have (x,y,z) = (2lambda, -lambda, lambda) = lambda (2,-1,1), and thus a vector we could choose for b would be (2,-1,1).  Reply With Quote

22. ## Re: MATH1231/1241/1251 SOS Thread Originally Posted by turntaker Nvm I was thinking about vectors not matricies. But the two are connected somehow.
In a way a matrix is just a ton of vectors smacked side by side  Reply With Quote

23. ## Re: MATH1231/1241/1251 SOS Thread

Am I doing something wrong with algebra or do I actually need to go for partial fractions? (Please don't complete the question)  Reply With Quote

24. ## Re: MATH1231/1241/1251 SOS Thread Originally Posted by leehuan Am I doing something wrong with algebra or do I actually need to go for partial fractions? (Please don't complete the question)

I think you made a simplification error in getting the second last line.  Reply With Quote

25. ## Re: MATH1231/1241/1251 SOS Thread Originally Posted by InteGrand I think you made a simplification error in getting the second last line.
Oh my bad I dropped a negative in that line which I later reintroduced. But wait

Factoring -v out I have

1 + 2/(1+v^2) = (1+v^2+2)/(1+v^2) = (3+v^2)/(1+v^2) right?

Sent from my iPhone using Tapatalk  Reply With Quote

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