What is Gradient?

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:

∇f=(∂f/∂x,∂f/∂y). (1)

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. 

The gradient ∇f(u1,...,un)→=(∂f/∂x1,...,∂f/∂xn)→ (4)

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.