117b – Homework 4

Due Thursday, February 1 at 1:00 pm.

Homework 4

The second pdf contains some hints for problem 3.

Homework 4

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7 Responses to “117b – Homework 4”

  1. Jed Says:

    I believe there to be a typographical error in the definition of the \otimes product for 1d. Specifically, x_4y_3 term should carry a negative sign.

  2. Domenic Says:

    Also, maybe this is something way basic for everyone versed in topology, but it would be nice for me if you could clarify that the disks are open. I eventually just said, in part (e), “we must have C be an open disk; this does not change the area, so all important properties are preserved, but the openness will be required later.”

    Now to prove… 1e, 2, and 3. Woohoo…

  3. andrescaicedo Says:

    Jed,

    I think you may be right. Writing a vector (a,b,c,d) as
    a + bi + cj + dk the multiplication table should be

    i^2=j^2=k^2= -1

    ij=-ji=k

    jk=-kj=i

    ki=-ik=j

    *Sigh*. Sorry about this; I should have just written this table and distributivity instead of the huge identity I wrote.

  4. andrescaicedo Says:

    Domenic,

    Yes, the disk in part (g) -not (e)- is supposed to be open. Although this really doesn’t matter.

  5. Domenic Says:

    No idea what the bracket notation in the hint is supposed to mean. Floor? Ceiling? Round to nearest? Stuff in to a matrix and then take the determinant? Yeah… not much of an issue since I can’t seem to prove the identity anyway (50 more minutes to go!), but yeah.

  6. andrescaicedo Says:

    The bracket notation is the integer part (floor) function. It is the same notation used in lecture. (Edited.)

  7. Domenic Says:

    I must be doing something wrong then, since I have isolated additive terms u and z in the expansion. The fact that there are extra terms besides just u and z means that the ceiling must be > u + z, since any extra contribution besides u and z will push it up by at least 1.

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