Gradient

The gradient of a function is a vector pointing in the direction of most rapid ascent or descent. It is represented by the character del. To find the gradient of a function f, take:

del f = df/dx a_x + df/dy a_y + df/dz a_z

Where all derivatives are actually partial derivatives, and all vectors a are unit vectors pointing in the denoted directions.

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.

A gradient is in the very simplest of terms how steep a line is at a given point.

For straight lines this is nice and easy because the gradient stays the same along the whole line.

To calculate the gradient of a straight line you simply take any 2 points on that line (let’s call them A and B)and take the change in the y-axis values (how much higher or lower A is than B) and divide that value by the change in the x-axis values (how far A is left or right of B).

In other words you take delta y over delta x.

e.g. a line that went up 2 units whenever it went across (to the right) 1 unit would have a gradient of 2 {draw it and see if your not sure why}

Easy so far? Good.

For curved lines it gets a little more difficult but not much so don’t worry.

Curves do not have uniform gradients so you can’t say that a curve has the same gradient at all points on that curve but you can find the gradient of a curve at a given point on that curve.

In order to find the gradient exactly you must differentiate the curve but you can approximate the gradient as accurately as you fairly easily. All you need to do is zoom in on the bit of the curve you are interested in as you look at a smaller and smaller section of the curve it will begin to look straighter and straighter until it appears to be perfectly straight then you can find the gradient as described above. Note: Delta x and delta y will be tiny, the smaller they are the better your approximation.

If you have the line in the form of equation then all you need to do is take the point you are trying to find the gradient of as your point A and to plug a very slightly different value for x or y into the equation to find a suitable point B.

Gra"di*ent (?), a. [L. gradiens, p. pr. of gradi to step, to go. See Grade.]

1.

Moving by steps; walking; as, gradient automata.

Wilkins.

2.

Rising or descending by regular degrees of inclination; as, the gradient line of a railroad.

3.

Adapted for walking, as the feet of certain birsds.

 

© Webster 1913.

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Gra"di*ent, n.

1.

The rate of regular or graded ascent or descent in a road; grade.

2.

A part of a road which slopes upward or downward; a portion of a way not level; a grade.

3.

The rate of increase or decrease of a variable magnitude, or the curve which represents it; as, a thermometric gradient.

Gradient post, a post or stake indicating by its height or by marks on it the grade of a railroad, highway, or embankment, etc., at that spot.

 

© Webster 1913.