1. ## gradient field

Hi I have a quiestion :
What thoes a gradient field means?

2. Originally Posted by MarianaA
Hi I have a quiestion :
What thoes a gradient field means?
for a scalar function $\displaystyle f(x,y)$ of two variables, we define $\displaystyle \nabla f$ as the following:

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

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

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

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

3. Originally Posted by Jhevon
for a scalar function $\displaystyle f(x,y)$ of two variables, we define $\displaystyle \nabla f$ as the following:

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

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

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

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

Ok Thanks