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 $f(x,y)$ 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 $\mathbb{R}^3$)

3. Originally Posted by Jhevon
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