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Homework Help: Totaly lost

  1. Nov 20, 2008 #1
    missed a lecture and now have this homework problem and dont even know what the upside down triangle symbol indicates, can someone please give me a hand getting started, thanks

    consider the vector function q=(1/4X^4 y^2 z, x^3 yz^6 - cosh(xz), 1/7x^3 z^7)

    calculate f(x,y,z)=upside down triangle . q
  2. jcsd
  3. Nov 20, 2008 #2

    D H

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    Staff Emeritus
    Science Advisor

    That upside down triangle is the nabla symbol and is typically called "del". "Del" is an operator analagous to the derivative operator d/dx except that del takes partial derivatives. In Cartesian 3-space,

    [tex]\boldsymbol{\nabla} \equiv
    \hat{\boldsymbol x} \frac{\partial}{\partial x} +
    \hat{\boldsymbol y} \frac{\partial}{\partial y} +
    \hat{\boldsymbol z} \frac{\partial}{\partial z}

    When applied to a scalar function f(x,y,z), the del operator yields the gradient of the function:

    [tex]\boldsymbol{\nabla} f(x,y,z) \equiv
    \hat{\boldsymbol x} \frac{\partial f(x,y,z)}{\partial x} +
    \hat{\boldsymbol y} \frac{\partial f(x,y,z)}{\partial y} +
    \hat{\boldsymbol z} \frac{\partial f(x,y,z)}{\partial z}

    The operator definition of del looks like a vector. With a little abuse of notation, it can be applied to vector functions as a dot product (yielding a scalar) and a cross product (yielding a vector):

    [tex]\boldsymbol{\nabla} \cdot \boldsymbol{f}(x,y,z) \equiv
    \hat{\boldsymbol x} \frac{\partial f_x(x,y,z)}{\partial x} +
    \hat{\boldsymbol y} \frac{\partial f_y(x,y,z)}{\partial y} +
    \hat{\boldsymbol z} \frac{\partial f_z(x,y,z)}{\partial z}

    and similarly for the cross product. The expression [itex]\boldsymbol{\nabla} \cdot \boldsymbol{f}(x,y,z)[/itex] is called the divergence of f(x,y,z) while [itex]\boldsymbol{\nabla} \times \boldsymbol{f}(x,y,z)[/itex] is called the curl.
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