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Math Help - gradient field

  1. #1
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    gradient field

    Hi I have a quiestion :
    What thoes a gradient field means?
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  2. #2
    is up to his old tricks again! Jhevon's Avatar
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    Quote Originally Posted by MarianaA View Post
    Hi I have a quiestion :
    What thoes a gradient field means?
    for a scalar function f(x,y) of two variables, we define \nabla f as the following:

    \nabla f = f_x(x,y) ~\bold{i} + f_y(x,y)~ \bold{j}

    similarly, for a function f(x,y,z) of three variables,

    \nabla f = f_x(x,y,z) ~\bold{i} + f_y(x,y,z)~\bold{j} + f_z(x,y,z)~\bold{k}

    we call \nabla f the gradient of f and it is a vector field on \mathbb{R}^2 or \mathbb{R}^3 for two or three variables respectively. specifically, it is called a gradient vector field, as it assigns each point (x,y) in domain f(x,y) (respectively each point (x,y,z) in the domain of f(x,y,z)) with a 2-dimensional vector in \mathbb{R}^2 (respectively, a 3-Dimensional vector in \mathbb{R}^3)
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  3. #3
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    Quote Originally Posted by Jhevon View Post
    for a scalar function f(x,y) of two variables, we define \nabla f as the following:

    \nabla f = f_x(x,y) ~\bold{i} + f_y(x,y)~ \bold{j}

    similarly, for a function f(x,y,z) of three variables,

    \nabla f = f_x(x,y,z) ~\bold{i} + f_y(x,y,z)~\bold{j} + f_z(x,y,z)~\bold{k}

    we call \nabla f the gradient of f and it is a vector field on \mathbb{R}^2 or \mathbb{R}^3 for two or three variables respectively. specifically, it is called a gradient vector field, as it assigns each point (x,y) in domain f(x,y) (respectively each point (x,y,z) in the domain of f(x,y,z)) with a 2-dimensional vector in \mathbb{R}^2 (respectively, a 3-Dimensional vector in \mathbb{R}^3)

    Ok Thanks
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