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Homework Help: Gradient of a vector/function

  1. Jul 14, 2010 #1
    1. The problem statement, all variables and given/known data

    Find the gradients of the following functions:

    When I say gradient, I'm not just differentiating the functions, apparently I have to do it this way (because it's in my physics book)

    f(x,y,z) = x^2 + y^3 + z^4
    f(x,y,z) = x^2 y^3 z^4
    f(x,y,z) = e^x sin(y) ln(z)


    2. Relevant equations

    [tex]\nabla {\bf r} = \frac{\partial r}{\partial x} \hat{x} + \frac{\partial r}{\partial y} \hat{y} + \frac{\partial r}{\partial z} \hat{z} [/tex]

    I'm not just differentiating the functions

    where [tex] \frac{r}{|r|} [/tex] = gradient

    3. The attempt at a solution

    for the first question, finding the gradient of
    f(x,y,z) = x^2 + y^3 + z^4

    [tex]\nabla {\bf f(x,y,z) } = \frac{\partial r}{\partial x} \hat{x} + \frac{\partial r}{\partial y} \hat{y} + \frac{\partial r}{\partial z} \hat{z} [/tex]

    differentiating partially,[tex]\nabla[/tex]
    [tex] \nabla
    (x^2 + y^3 + z^4) = 2x \hat{x} + 3y^2 \hat{y} + 4z^3 \hat{z} [/tex]

    finding the magnitude of the function
    [tex] \sqrt{(x^2)^2 + (y^3)^2 + (z^4)^2} = \sqrt{x^4 + y^6 + z^8} [/tex]

    using the formula to find the gradient,
    [tex] \frac{r}{|r|} = \frac{\nabla (x^2 + y^3 + z^4) }{\sqrt{x^4 + y^6 + z^8}} = \frac{2x \hat{x} + 3y^2 \hat{y} + 4z^3 \hat{z}}{\sqrt{x^4 + y^6 + z^8}} [/tex]

    not sure if this is right or not, also I don't know if I can simplifier it further and if this satisfies as an answer,
    my book has no answers in the back to check, and solutions manuals are impossible to get to students, and the only way to find out if i'm right is to either post it here, or ask my lecturer, and I feel that this is a question that most people should be able to do, so it's embarrassing for me to ask my lecturer at tutorials
     
    Last edited: Jul 15, 2010
  2. jcsd
  3. Jul 14, 2010 #2
    [edit] fixed the latex
     
    Last edited: Jul 15, 2010
  4. Jul 14, 2010 #3

    lanedance

    User Avatar
    Homework Helper

    i may be missing something, but why not just use the cartesian form of the gradient

    [tex] \nabla = \begin{pmatrix}
    \frac{\partial }{\partial x} \\
    \frac{\partial }{\partial y} \\
    \frac{\partial }{\partial z}
    \end{pmatrix}[/tex]
     
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