Results 1 to 3 of 3

Math Help - Vector Calculus

  1. #1
    Newbie
    Joined
    Jul 2008
    Posts
    11

    Vector Calculus

    1) Show that the vector field F = (3x^2)i + (5z^2)j + (10yz)k is irrotional. Then find a potential function f(x,y,z) such that gradient f = F.

    (Ok, so i got the part of showing that F is irrotional. But when i tried figuring out to find the function i got something weird. I am not sure if i got this right. My answer is: f(x,y,z) = x^3 + 5z^2 + 5yz^2 - 5z^2.)

    2) Let F(x,y,z) = (8xz)i + (1-6yz^3)j + (4x^2 - 9y^2z^2)k. Show that integral with lower terminal C of (F.dr) is independent of path by finding a potential function f for F.

    If you could please help me with these problems much would be appreciated. Thank you
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Flow Master
    mr fantastic's Avatar
    Joined
    Dec 2007
    From
    Zeitgeist
    Posts
    16,948
    Thanks
    5
    Quote Originally Posted by davidson89 View Post
    1) Show that the vector field F = (3x^2)i + (5z^2)j + (10yz)k is irrotional. Then find a potential function f(x,y,z) such that gradient f = F.

    (Ok, so i got the part of showing that F is irrotional. But when i tried figuring out to find the function i got something weird. I am not sure if i got this right. My answer is: f(x,y,z) = x^3 + 5z^2 + 5yz^2 - 5z^2.)

    [snip]
    Your answer is easily checked by taking grad(f) .... and seen to be wrong.

    Solve the following system of simultaneous pde's (all derivatives are partial derivatives):

    df/dx = 3x^2 .... (1)

    df/dy = 5z^2 .... (2)

    df/dz = 10yz .... (3)


    From (1): f = x^3 + g(y, z).

    From (2): dg/dy = 5z^2 => g = 5z^2 y + h(z).

    Therefore f = x^3 + 5z^2 y + h(z).

    From (3): 10 zy + dh/dz = 10yz => dh/dz = 0 => h = constant. Take h = 0.

    Therefore f = x^3 + 5z^2 y.
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Flow Master
    mr fantastic's Avatar
    Joined
    Dec 2007
    From
    Zeitgeist
    Posts
    16,948
    Thanks
    5
    Quote Originally Posted by davidson89 View Post
    [snip]
    2) Let F(x,y,z) = (8xz)i + (1-6yz^3)j + (4x^2 - 9y^2z^2)k. Show that integral with lower terminal C of (F.dr) is independent of path by finding a potential function f for F.

    If you could please help me with these problems much would be appreciated. Thank you
    Confirm that curl F = 0, that is rot F = 0. This is sufficient to prove path independence. Nevertheless:

    curl F = 0 => F = grad f.

    Find f by solving (all derivatives are partial derivatives):

    df/dx = 8xz .... (1)

    df/dy = 1 - 6yz^3 .... (2)

    df/dz = 4x^2 - 9y^2 z^2 .... (3)


    From (1): f = 4x^2 z + g(y, z)

    From (2): dg/dy = 1 - 6yz^3 => g = y - 3y^2 z^3 + h(z).

    Therefore f = 4x^2 z + y - 3y^2 z^3 + h(z).

    From (3): 4x^2 - 9 y^2 z^2 + dh/dz = 4x^2 - 9y^2z^2 => dh/dz = 0. Take h = 0.

    Therefore f = 4x^2 z + y - 3y^2 z^3.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Vector Calculus (Position Vector)
    Posted in the Calculus Forum
    Replies: 6
    Last Post: August 23rd 2011, 01:43 PM
  2. Vector Calculus
    Posted in the Calculus Forum
    Replies: 1
    Last Post: July 25th 2011, 09:09 AM
  3. Vector Calculus
    Posted in the Calculus Forum
    Replies: 1
    Last Post: August 30th 2010, 06:21 PM
  4. vector calculus - vector feilds
    Posted in the Calculus Forum
    Replies: 3
    Last Post: February 25th 2010, 01:17 AM
  5. vector calculus
    Posted in the Calculus Forum
    Replies: 8
    Last Post: September 13th 2009, 06:28 PM

Search Tags


/mathhelpforum @mathhelpforum