Introduction

In vector calculus, the gradient is a multi-variable generalization of the derivative. If f is a function of several variables, then the gradient vector field of f is given by:

(∇f) = (∂f/∂x, ∂f/∂y, ∂f/∂z)

where ∇f is the gradient operator and (x, y, z) are the coordinates of the point at which we wish to evaluate the gradient vector field. The value of the gradient vector field at a point P is thus a vector whose components are the partial derivatives of f at P.

The gradient vector field of a function

The gradient vector field of a function at a point is a vector that points in the direction of the steepest increase of the function at that point. It can be thought of as a generalization of the derivative to functions of more than one variable.

Definition of gradient vector field

A gradient vector field is a vector field that is generated by taking the gradient of a scalar function. The gradient of a scalar function is a vector whose components are the partial derivatives of the function. In three dimensions, the gradient vector field of a function f(x, y, z) is given by:

f'(x,y,z) =

The gradient vector field can be used to find the local maximum and minimum values of a function by finding the zeroes of the vector field. To do this, one needs to solve the following set of equations:

fx(x0,y0,z0) = 0
fy(x0,y0,z0) = 0
fz(x0) = 0

These equations are known as the gradient equations.

The gradient vector field of a function in three variables

The gradient vector field of a function in three variables is a vector field defined as follows:

Given a smooth function f(x,y,z), the gradient vector field of f is the vector field whose components are the partial derivatives of f with respect to x, y, and z:

The gradient vector field of f is denoted grad(f).

The gradient vector field of f(x,y,z)=2x^2+y^2+z^2

The gradient vector field of a function is a vector field that is perpendicular to the level surfaces of the function. The gradient vector field of a function at a point is the gradient of the function at that point.

The gradient vector field of f(x,y,z)=2x^2+y^2+z^2 in three variables

The gradient vector field of a function f(x,y,z) in three variables is given by:

grad f(x,y,z) = (2x^2+y^2+z^2 fx(x,y,z), 2x^2+y^2+z^2 fy(x,y,z), 2x^2+y^2+z^2 fz(x,y,z))

In this case, we have:

grad f(x,y,z) = (4xfx(x,y,z), 2yfy(x,y,z), 2zfz(x,y,z))

Conclusion

In conclusion, the gradient vector field of f(x,y,z) = 2x^2+y^2+z^2-(2x^2+y^2+z^2) is given by (0,0,0).