Page 5 of 8 FirstFirst ... 34567 ... LastLast
Results 101 to 125 of 197
Like Tree91Likes

Thread: MATH2111 Higher Several Variable Calculus

  1. #101
    -insert title here- Paradoxica's Avatar
    Join Date
    Jun 2014
    HSC
    2016
    Gender
    Male
    Location
    Outside reality
    Posts
    2,436
    Rep Power
    4

    Re: Several Variable Calculus

    Quote Originally Posted by leehuan View Post
    Just a brief sketch-out please





    I know that f is not continuous at 0 but I'm not sure if that helps since we're talking about partial derivatives here.
    If we switch to polar co-ordinates, then the result is:



    It is then immediately obvious that the limit does not exist as r tends towards zero.
    Last edited by Paradoxica; 19 Mar 2017 at 10:17 PM.
    leehuan likes this.
    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.

  2. #102
    Rambling Spirit
    Join Date
    Dec 2014
    HSC
    N/A
    Gender
    Male
    Posts
    6,039
    Rep Power
    7

    Re: Several Variable Calculus

    Quote Originally Posted by leehuan View Post
    Just a brief sketch-out please





    I know that f is not continuous at 0 but I'm not sure if that helps since we're talking about partial derivatives here.
    You can try first computing an expression for ∂f/∂y (using first principles to find the value of this at the origin). Then try computing ∂2f/∂x∂y (i.e. ∂/∂x (∂f/∂y)) at the origin from first principles and show that the limit does not exist.
    Last edited by InteGrand; 19 Mar 2017 at 10:48 PM.
    leehuan likes this.

  3. #103
    Ancient Orator leehuan's Avatar
    Join Date
    May 2014
    HSC
    2015
    Gender
    Male
    Posts
    5,807
    Rep Power
    6

    Re: Several Variable Calculus

    Oh damn, looks like the brute way is the only way out of that one. Was hoping for a shortcut
    _________________________

    I got asked a question but I haven't seen the notation before





    What is that isolated nabla supposed to mean?

  4. #104
    Supreme Member seanieg89's Avatar
    Join Date
    Aug 2006
    HSC
    2007
    Gender
    Male
    Posts
    2,680
    Rep Power
    10

    Re: Several Variable Calculus

    Nabla phi I am sure you understand, that is the gradient of phi and is a vector-valued function. Which can be regarded as a triple of functions (f,g,h) from R^3 to R.

    Nabla cross (f,g,h) denotes the curl of the vector-valued function (f,g,h).

    If you write Nabla as (d/dx,d/dy,d/dz), then Nabla x (f,g,h) is defined the same way as the usual cross product between two 3-d vectors. (Just the first "vector" consists of a triple of differential operators, whilst the second consists of a triple of real-valued functions on R^3. So instead of multiplying real numbers (vector components) together as in the defn of the usual cross product, we are applying differential operators to functions.)

    eg first component is dh/dy-dg/dz.
    Last edited by seanieg89; 25 Mar 2017 at 10:57 AM.
    leehuan likes this.

  5. #105
    Supreme Member seanieg89's Avatar
    Join Date
    Aug 2006
    HSC
    2007
    Gender
    Male
    Posts
    2,680
    Rep Power
    10

    Re: Several Variable Calculus

    (The notation Nabla dot (f,g,h) is the divergence of a vector field and is understood in the same way.)
    leehuan likes this.

  6. #106
    New Member
    Join Date
    Mar 2017
    HSC
    2015
    Gender
    Male
    Posts
    2
    Rep Power
    1

    Re: Several Variable Calculus

    If you keep cheating you won't learn. Why are marks so important?

  7. #107
    Ancient Orator leehuan's Avatar
    Join Date
    May 2014
    HSC
    2015
    Gender
    Male
    Posts
    5,807
    Rep Power
    6

    Re: Several Variable Calculus

    Seriously? If I am stuck on a question why can I not ask for help? Accusing me of cheating on an assignment if I genuinely can't do my homework is just low.
    Paradoxica and wu345 like this.

  8. #108
    Ancient Orator leehuan's Avatar
    Join Date
    May 2014
    HSC
    2015
    Gender
    Male
    Posts
    5,807
    Rep Power
    6

    Re: Several Variable Calculus

    So for a continuous mapping f: are the following always true

    (I didn't really define the domain and codomain of f so just assume whatever's convenient please)

    1) U is closed => f^-1(U) is closed
    2) U is open => f^-1(U) is open
    3) U is closed => f(U) is closed
    4) U is open = f(U) is open

    5) U is compact => f(U) is compact
    6) U is path-connected => f(U) is path connected

    Don't really need proof, just yes/no is sufficient

  9. #109
    Rambling Spirit
    Join Date
    Dec 2014
    HSC
    N/A
    Gender
    Male
    Posts
    6,039
    Rep Power
    7

    Re: Several Variable Calculus

    Quote Originally Posted by leehuan View Post
    So for a continuous mapping f: are the following always true

    (I didn't really define the domain and codomain of f so just assume whatever's convenient please)

    1) U is closed => f^-1(U) is closed
    2) U is open => f^-1(U) is open
    3) U is closed => f(U) is closed
    4) U is open = f(U) is open

    5) U is compact => f(U) is compact
    6) U is path-connected => f(U) is path connected

    Don't really need proof, just yes/no is sufficient
    1) True
    2) True
    3) False
    4) False
    5) True
    6) True
    leehuan and kawaiipotato like this.

  10. #110
    Ancient Orator leehuan's Avatar
    Join Date
    May 2014
    HSC
    2015
    Gender
    Male
    Posts
    5,807
    Rep Power
    6

    Re: Several Variable Calculus

    There was a question in my test that I could not do and I'm seeking a solution please


  11. #111
    Rambling Spirit
    Join Date
    Dec 2014
    HSC
    N/A
    Gender
    Male
    Posts
    6,039
    Rep Power
    7

    Re: Several Variable Calculus

    Quote Originally Posted by leehuan View Post
    There was a question in my test that I could not do and I'm seeking a solution please

    Yeah it is a boundary point. Here's some hints of a possible method. Note that S is just the graph of y = f(x) := sin(1/x) (x =/= 0). Use (or show) the fact that f attains the value 1 for arbitrarily small values of x > 0 to deduce that (0, 1) is a boundary point of S (noting that any ball around (0, 1) contains points not in S also, as there will be points in it with y-value greater than 1).
    Last edited by InteGrand; 7 Apr 2017 at 3:39 PM.
    leehuan and kawaiipotato like this.

  12. #112
    Ancient Orator leehuan's Avatar
    Join Date
    May 2014
    HSC
    2015
    Gender
    Male
    Posts
    5,807
    Rep Power
    6

    Re: Several Variable Calculus

    I get lost in this Einstein summation convention so can I see how this problem would be approached using it?

    Last edited by leehuan; 25 Apr 2017 at 8:21 PM.

  13. #113
    Rambling Spirit
    Join Date
    Dec 2014
    HSC
    N/A
    Gender
    Male
    Posts
    6,039
    Rep Power
    7

    Re: Several Variable Calculus

    Quote Originally Posted by leehuan View Post
    I get lost in this Einstein summation convention so can I see how this problem would be approached using it?

    leehuan likes this.

  14. #114
    Ancient Orator leehuan's Avatar
    Join Date
    May 2014
    HSC
    2015
    Gender
    Male
    Posts
    5,807
    Rep Power
    6

    Re: Several Variable Calculus



    Even with the diagram in front of me I still struggle to figure out my boundaries of integration.

    Last edited by leehuan; 30 Apr 2017 at 9:12 PM.

  15. #115
    Rambling Spirit
    Join Date
    Dec 2014
    HSC
    N/A
    Gender
    Male
    Posts
    6,039
    Rep Power
    7

    Re: Several Variable Calculus

    Quote Originally Posted by leehuan View Post


    Even with the diagram in front of me I still struggle to figure out my boundaries of integration.



    Last edited by InteGrand; 30 Apr 2017 at 9:41 PM.
    leehuan and kawaiipotato like this.

  16. #116
    Ancient Orator leehuan's Avatar
    Join Date
    May 2014
    HSC
    2015
    Gender
    Male
    Posts
    5,807
    Rep Power
    6

    Re: Several Variable Calculus

    I think spherical is preferred here (could be mistaken). But using the best approach, how would you determine the limits for this


  17. #117
    Rambling Spirit
    Join Date
    Dec 2014
    HSC
    N/A
    Gender
    Male
    Posts
    6,039
    Rep Power
    7

    Re: Several Variable Calculus

    Quote Originally Posted by leehuan View Post
    I think spherical is preferred here (could be mistaken). But using the best approach, how would you determine the limits for this

    Inspection would be the best way to determine the limits probably (you can draw a diagram to help if you decide to use inspection).

  18. #118
    Ancient Orator leehuan's Avatar
    Join Date
    May 2014
    HSC
    2015
    Gender
    Male
    Posts
    5,807
    Rep Power
    6

    Re: Several Variable Calculus

    Quote Originally Posted by InteGrand View Post
    Inspection would be the best way to determine the limits probably (you can draw a diagram to help if you decide to use inspection).
    I was able to deduce that

    It's always rho that gets me.

    Unless I picked the order of integration wrong.

  19. #119
    Rambling Spirit
    Join Date
    Dec 2014
    HSC
    N/A
    Gender
    Male
    Posts
    6,039
    Rep Power
    7

    Re: Several Variable Calculus

    Quote Originally Posted by leehuan View Post
    I was able to deduce that

    It's always rho that gets me.

    Unless I picked the order of integration wrong.
    It might be easier to use cylindrical coordinates. The way rho varies is different depending on what range phi is in. When phi is such that your point in the region lies within the "ice cream cone" part of the region, then rho will vary from 0 to a. For phi larger than the angle make by the cone to the vertical, rho will vary from 0 to the value of rho at the point on the sphere x^2 + y^2 + (z-a)^2 = a^2 with this phi value. You can find these using the relations between Cartesian and Spherical Coordinates.
    leehuan likes this.

  20. #120
    Ancient Orator leehuan's Avatar
    Join Date
    May 2014
    HSC
    2015
    Gender
    Male
    Posts
    5,807
    Rep Power
    6

    Re: Several Variable Calculus



    I might be misinterpreting the question but I thought that they're basically asking for 1/4 * area of curve of intersection, which happens to be an ellipse through (0,0,8) passing through (0,8,4) and (8,0,4).

    If I'm right, how do I use surface integrals to get to the answer of ? And if I'm wrong, how do I get back on the right path?

  21. #121
    Rambling Spirit
    Join Date
    Dec 2014
    HSC
    N/A
    Gender
    Male
    Posts
    6,039
    Rep Power
    7

    Re: Several Variable Calculus

    Quote Originally Posted by leehuan View Post


    I might be misinterpreting the question but I thought that they're basically asking for 1/4 * area of curve of intersection, which happens to be an ellipse through (0,0,8) passing through (0,8,4) and (8,0,4).

    If I'm right, how do I use surface integrals to get to the answer of ? And if I'm wrong, how do I get back on the right path?




    Last edited by InteGrand; 6 May 2017 at 8:36 PM.
    leehuan likes this.

  22. #122
    Ancient Orator leehuan's Avatar
    Join Date
    May 2014
    HSC
    2015
    Gender
    Male
    Posts
    5,807
    Rep Power
    6

    Re: Several Variable Calculus

    Quote Originally Posted by InteGrand View Post




    Yep makes perfect sense, thanks as always

    Out of curiosity, in general does setting x and y to be the parameters make things easier? Or are there scenarios where you'd pick a different choice of parameters.

  23. #123
    Rambling Spirit
    Join Date
    Dec 2014
    HSC
    N/A
    Gender
    Male
    Posts
    6,039
    Rep Power
    7

    Re: Several Variable Calculus

    Quote Originally Posted by leehuan View Post
    Yep makes perfect sense, thanks as always

    Out of curiosity, in general does setting x and y to be the parameters make things easier? Or are there scenarios where you'd pick a different choice of parameters.


    leehuan likes this.

  24. #124
    Ancient Orator leehuan's Avatar
    Join Date
    May 2014
    HSC
    2015
    Gender
    Male
    Posts
    5,807
    Rep Power
    6

    Re: Several Variable Calculus


  25. #125
    Rambling Spirit
    Join Date
    Dec 2014
    HSC
    N/A
    Gender
    Male
    Posts
    6,039
    Rep Power
    7

    Re: Several Variable Calculus

    Quote Originally Posted by leehuan View Post
    leehuan likes this.

Page 5 of 8 FirstFirst ... 34567 ... LastLast

Thread Information

Users Browsing this Thread

There are currently 1 users browsing this thread. (0 members and 1 guests)

Posting Permissions

  • You may not post new threads
  • You may not post replies
  • You may not post attachments
  • You may not edit your posts
  •