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

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    Re: MATH1231/1241/1251 SOS Thread

    Quote Originally Posted by InteGrand View Post
    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!
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    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?
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    Re: MATH1231/1241/1251 SOS Thread

    Quote Originally Posted by Flop21 View Post
    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?


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    Re: MATH1231/1241/1251 SOS Thread

    Quote Originally Posted by Flop21 View Post
    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?


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    Re: MATH1231/1241/1251 SOS Thread



    how do I do this one
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    Re: MATH1231/1241/1251 SOS Thread

    Quote Originally Posted by Flop21 View Post


    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.
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    Re: MATH1231/1241/1251 SOS Thread

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








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    Re: MATH1231/1241/1251 SOS Thread

    Quote Originally Posted by leehuan View Post
    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
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    If I am a conic section, then my e = ∞

    Just so we don't have this discussion in the future, my definition of the natural numbers includes 0.

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    Re: MATH1231/1241/1251 SOS Thread

    Quote Originally Posted by Paradoxica View Post
    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

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    Re: MATH1231/1241/1251 SOS Thread

    Quote Originally Posted by leehuan View Post
    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.
    If I am a conic section, then my e = ∞

    Just so we don't have this discussion in the future, my definition of the natural numbers includes 0.

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    Re: MATH1231/1241/1251 SOS Thread

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

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    Re: MATH1231/1241/1251 SOS Thread

    Quote Originally Posted by leehuan View Post
    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.
    If I am a conic section, then my e = ∞

    Just so we don't have this discussion in the future, my definition of the natural numbers includes 0.

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    Re: MATH1231/1241/1251 SOS Thread

    Quote Originally Posted by leehuan View Post
    This isn't helping sorry. Too rushed and you TLDRd it further. I don't see it....
    Essentially, here is a sketch.



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    Re: MATH1231/1241/1251 SOS Thread

    Quote Originally Posted by InteGrand View Post
    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???

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    Re: MATH1231/1241/1251 SOS Thread

    why are you still doing matricies
    "I have crippling depression" -Mahatma Gandhi
    Quote Originally Posted by Katsumi View Post
    lol

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    Re: MATH1231/1241/1251 SOS Thread

    Quote Originally Posted by turntaker View Post
    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
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    Re: MATH1231/1241/1251 SOS Thread

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

    are you doing things like intersection of lines, planes etc
    "I have crippling depression" -Mahatma Gandhi
    Quote Originally Posted by Katsumi View Post
    lol

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    Re: MATH1231/1241/1251 SOS Thread

    Quote Originally Posted by turntaker View Post
    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.
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    Re: MATH1231/1241/1251 SOS Thread

    Quote Originally Posted by Flop21 View Post
    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.
    "I have crippling depression" -Mahatma Gandhi
    Quote Originally Posted by Katsumi View Post
    lol

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    Re: MATH1231/1241/1251 SOS Thread

    Quote Originally Posted by Flop21 View Post
    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).
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    Re: MATH1231/1241/1251 SOS Thread

    Quote Originally Posted by turntaker View Post
    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

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    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)




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    Re: MATH1231/1241/1251 SOS Thread

    Quote Originally Posted by leehuan View Post
    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.

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    Re: MATH1231/1241/1251 SOS Thread

    Quote Originally Posted by InteGrand View Post
    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?


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