Introduction

In mathematics, the gradient is a multi-variable generalization of the derivative. In particular, the gradient gives the direction of the steepest ascent of a function at a given point. The gradient is also a vector field, as opposed to a single vector, because it operates on multiple variables simultaneously. If f is a function of several variables, then the gradient vector field of f is denoted by grad f or ∇f.

What is a gradient vector field?

In mathematics, a gradient vector field, more commonly simply called a gradient field or a gradient, is a vector field that points in the direction of the greatest rate of increase of a given function. In other words, the gradient vector points in the direction of the steepest slope of the function.

Definition

In mathematics, a gradient vector field is a vector field that describes the direction and magnitude of the maximum rate of change of a scalar function. More specifically, the gradient of a scalar-valued differentiable function f(x,y,z) of several variables is the vector whose components are the n partial derivatives of f with respect to the n variables. These components are often written as follows:

∇f = (∂f/∂x, ∂f/∂y, ∂f/∂z).

The word “gradient” may also refer to the derivative operator itself. The operator ∇ is known as del or nabla.

Examples

A gradient vector field is a vector field that is derived from a scalar function. In other words, the gradient vector points in the direction of the steepest increase of the scalar function. The gradient vector can be thought of as a measure of how much the function changes in different directions.

For example, consider the function f(x,y,z) = 10x^2 + y^2 + z^2. The gradient vector of this function would be <20x,2y,2z>. Notice that the gradient vector always points in the direction of the steepest increase of the function. In this case, the steepest increase is along the x-axis, so the x-component of the gradient vector is 20x.

Similarly, if we consider the function g(x,y) = x^2 + y^2, then the gradient vector would be <2x,2y>. Once again, notice that the gradient vector points in the direction of steepest increase; in this case, it is along both the x- and y-axes.

The magnitude of a gradient vector can be thought of as a measure of how steeply a function increases. For example, if we consider the function h(x) = x^2, then the magnitude of its gradient vector would be |<2x>| = 2|x|. This makes sense because h(x) increases by 2 units for every unit increase in x. On the other hand, if we consider a constant function f(x)=c (where c is any constant), then its gradient vector would be <0>. This makes sense because a constant function does not change no matter what direction you go; it is constant!

How to find the gradient vector field of f fx y z 10 x2 y2 z2

There are many ways to find the gradient vector field of a function. You can use the definition of the gradient, the del operator, or the gradient operator. You can also use software like Mathematica or Maple to find the gradient vector field. Let’s go over a few examples.

Partial derivatives

The gradient vector field of a three-variable function f(x,y,z) is given by the formula:

[∂f/∂x, ∂f/∂y, ∂f/∂z]

where ∂f/∂x is the partial derivative of f with respect to x, and so on. In other words, the gradient vector field points in the direction of greatest change in f.

The gradient vector field

The gradient vector field of a function f is a vector field whose vectors are the gradients of f at each point. The gradient vector field of f is written as grad f or ∇f.

Conclusion

The gradient vector field of f(x,y,z)=10x^2+y^2+z^2-(x+y+z) is given by:

\vec{F}(x,y,z)=(20x-1,2y-1,2z-1)