The
gradient is an operation from
vector calculus, analogous to the
derivatve from function
calculus. It is the directional derivative of a scalar field in multiple dimensions. The result is a vector field that at each point is of the magnitude of and in the direction of the greatest instantaneous increase in the scalar field at that point.
In symbols:
∇s = v
"gradient of s equals v"
Here, ∇ is the directional derivative operator and is pronounced 'del.' In Cartesian coordinates in three dimensions
∇ = (i ∂/∂x + j ∂/∂y + k ∂/∂z)
where i, j, and k are the unit vectors in the x, y, and z directions respecively, and ∂/∂w is the partial derivative with respect to w.
so in cartesian coordinates in three dimensions
∇s = i ∂s/∂x + j ∂s/∂y + k ∂s/∂z.
As an example, if s were the three variable equation for the varying pressure in a material
(s = f(x,y,z)), then v would point in the direction of the most drastic infinitesimal increase in pressure at any point. v would then be a vector field:
v=(f(x,y,z)i + g(x,y,x)j + h(x,y,z)k)
The gradient can also be meaningful for scalar fields of any number of dimensions, for there is nothing mathematically special about three orthogonal dimensions. In addition, it can be calculated for scalar fields expressed in non-cartesian coordinate systems, as long as the operator ∇ is transformed properly.
If you want to find out the change in the value a scalar field when you move an infinitesimal amount in a certain direction, it is ∇s • dl, where dl = the infinitessimal change in direction (idx + jdy + kdz in cartesian coordinates). Then the total change in the value of the scalar field over a distance L is
∫ ∇s • dl. This is entirely analogous to ordinary function calculus, where the infinitesimal change in the value of a function is df/dx * dx, and the total change over an interval is ∫ df/dx * dx = ΔF. Likewise, a local maximum or minimum point of a scalar field is a point where ∇s = 0, again analogous to ordinary calculus.