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Is this ok?

  1. Oct 21, 2005 #1
    Hi,

    I started computing excercises on total differential and I would like to know if I'm doing it correctly. Could you please check it? Here it is:

    Does the function

    [tex]
    f(x,y) = \sqrt[3]{x^3+y^3}
    [/tex]

    have total differential in [0,0]?

    First I computed partial derivatives:

    [tex]
    \frac{\partial f}{\partial x} = \frac{x^3}{\sqrt[3]{(x^3 + y^3)^2}}
    [/tex]

    [tex]
    \frac{\partial f}{\partial y} = \frac{y^3}{\sqrt[3]{(x^3 + y^3)^2}}
    [/tex]

    I see that partial derivatives are continuous everywhere with the exception of the point [0,0].

    For the point [0,0] I have to compute partial derivatives from definition using the limit:

    [tex]
    \frac{\partial f}{\partial x}(0,0) = \lim_{t \rightarrow 0} \frac{f(t,0) - f(0,0)}{t} = \lim_{t \rightarrow 0} \frac{t}{t} = 1
    [/tex]

    [tex]
    \frac{\partial f}{\partial y}(0,0) = \lim_{t \rightarrow 0} \frac{f(0,t) - f(0,0)}{t} = \lim_{t \rightarrow 0} \frac{t}{t} = 1
    [/tex]

    So in the case that total differential in the point [0,0] exists, it must be of form:

    [tex]
    L(h) = \frac{\partial f}{\partial x}(0,0) h_1 + \frac{\partial f}{\partial y}(0,0) h_2 = h_1 + h_2
    [/tex]

    for any

    [tex]
    h = (h_1, h_2) \in \mathbb{R}^2
    [/tex]

    and must satisfy the limit

    [tex]
    \lim_{||h|| \rightarrow 0} \frac{f(0,0) + h) - f(0,0) - L(h)}{||h||} = 0
    [/tex]

    I can write it this way:

    [tex]
    \lim_{[h_1,h_2] \rightarrow [0,0]} \frac{\sqrt[3]{h_1^3 + h_2^3} - 0 - h_1 - h_2}{\sqrt{h_1^2 + h_2^2}}
    [/tex]

    When I put

    [tex]
    h_2 = kh_1
    [/tex]

    I can write

    [tex]
    \lim_{h_1 \rightarrow 0} \frac{ \sqrt[3]{h_1^3 + k^3h_1^3} - h_1kh_1}{\sqrt{h_1^2 + k^2h_1^2}} = \frac{\sqrt[3]{1+k^3} - 1 - k}{\sqrt{1 + k^2}} \neq 0
    [/tex]

    And thus I say that f doesn't have total differential in [0,0].

    Is this correct approach?

    Thank you for checking this out.
     
  2. jcsd
  3. Oct 21, 2005 #2

    HallsofIvy

    User Avatar
    Staff Emeritus
    Science Advisor

    You have the partial derivatives wrong- though it may just be a typo.
    [tex]f_x(x,y)= \frac{x^2}{^3\sqrt{(x^3+ y^3)^2}}[/tex]
     
  4. Oct 21, 2005 #3
    You're right, it's a typo. Is it ok otherwise?
     
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