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Math Help - Vector Fields...

  1. #1
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    Vector Fields...

    if f(x,y,z)= x^2+y^2-z

    let F be the vector field defined by F=▽f. Calculate ▽. F (scalar product) and ▽X F (cross product)

    Not all that sure about where F and ▽have come from, i have done scalar and cross products before but don't really know where to start on this, any help would be much appreciated.
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  2. #2
    is up to his old tricks again! Jhevon's Avatar
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    Quote Originally Posted by Ash_underpar View Post
    if f(x,y,z)= x^2+y^2-z

    let F be the vector field defined by F=▽f. Calculate ▽. F (scalar product) and ▽X F (cross product)

    Not all that sure about where F and ▽have come from, i have done scalar and cross products before but don't really know where to start on this, any help would be much appreciated.
    take \nabla = \left< \frac {\partial}{\partial x}, \frac {\partial}{\partial y}, \frac {\partial}{\partial z}\right> and \bold{F} = \left< \frac {\partial f}{\partial x}, \frac {\partial f}{\partial y}, \frac {\partial f}{\partial z} \right>

    note that the dot product gives you \text{div} \bold{F} and the cross-product gives you \text{curl} \bold{F}
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  3. #3
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    so if ▽= (2x+2y-1) this is multipied by f to get F? which is (2x+2y-1)(x^2+y^2-z)... i'm confused! whats the difference between d/dx and df/dx???
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  4. #4
    is up to his old tricks again! Jhevon's Avatar
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    Quote Originally Posted by Ash_underpar View Post
    so if ▽= (2x+2y-1) this is multipied by f to get F? which is (2x+2y-1)(x^2+y^2-z)... i'm confused! whats the difference between d/dx and df/dx???
    no, \nabla is exactly what i told you it was. it doesn't change. it is notation (conventionally) defined as i have it

    just write it out as you have it and you will "see" it

    \frac {\partial}{\partial x} by itself makes no sense. this is why i said it is a convention. \frac {\partial f}{\partial x} means the partial derivative of f with respect to x
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