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In mathematics, the gradient is a multi-variable generalization of the derivative. While a derivative can be defined on functions of a single variable, for functions of several variables, the gradient takes its place. The gradient is a vector field and is thus a particular case of the more general concept of a vector field.

The gradient of a function f(x,y), where x and y are independent variables, is denoted ∇f or ∇→f. It is defined as the vector whose components are the partial derivatives of f with respect to x and y:

### If z=f(x,y), then the gradient ∇z=(∂z/∂x,∂z/∂y). (2)

The above definition works in any number of dimensions. If f is a scalar field and u→=(u1,...,un)→ is a point in Rn then the directional derivative of f at u→ in the direction of v→=(v1,...,vn)→ is given by:

### Duf(u→;v→)=limh→0[f(u1+hv1,...,un+hnvn)−f(u1,...,un)]h=v1∂f∂x1+...+vn∂f∂xn (3)

At each point u=(u1,...un), the directional derivative measures how fast f changes when moving along v. In other words it gives us some idea about how f changes when we move from u in the direction given by v. But what if we want to know how fast f changes when moving in any direction? This brings us on to the gradient.

is the vector whose components are all the partial derivatives of f with respect to each variable xi. In other words it tells us how fast f changes when we move from u in any direction. We can think of it as giving us all the directional derivatives of f at once! Notice that if we take v=(vi,...vn)=vi=ei where ei=(0,...0,...1,...0,...0), i.e., vi only has a 1 in position i with 0s everywhere else then Duf(u;ei)=vi=ei=wi where wi=(wi,...wn)=(0,...0,...fi(ui),...0,...0). That is why we write ∇fi to mean partial derivative of fi with respect to xi or ∇xi=fi (5)

## Conclusion

So there you have it! Gradient explained in simple terms. Now go out there and wow your friends and family with your new-found knowledge!

## FAQ

### What is gradient of a line?

The gradient of a line is simply the slope of the line. It tells us how steep the line is and in which direction it is pointing.

### What does gradient mean in geometry?

In geometry, the gradient is a vector which points in the direction of the steepest ascent or descent of a function at a given point.

### Where is the gradient of a line?

The gradient of a line is always perpendicular to the line. This means that if you draw a line on a piece of paper, the gradient will point in a direction which is 90 degrees to the line, as shown in the picture below.

### What is a gradient simple definition?

A gradient is simply a measure of how much something changes over a given distance. For example, if you were to walk up a hill, the gradient would be the steepness of the hill.

## What is the gradient of a curve?

The gradient of a curve is the slope of the tangent to the curve at any given point. It tells us how steep the curve is and in which direction it is pointing.

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