Gradient: UNIZOR.COM - Physics4Teens - Waves - Field Waves

Notes to a video lecture on http://www.unizor.com

Gradient

Scientists involved in the mathematical representation of the laws of nature, starting with Newton and Leibniz, used differentiation to reflect the changing character of the world. Sometimes the mathematical representation of the results of research becomes too lengthy to deal with. To make the life easier and formulae shorter, different shorthands were invented.

In this lecture we will talk about operator ∇ called nabla - a convenient way to represent certain properties and characteristics of vectors.

First of all, ∇ is a set of three operators of partial differentiation (if needed, review the topic of partial derivatives in the "Calculus" part of the course "Math 4 Teens" on UNIZOR.COM), with partial differentiation performed for each of the three dimensions of space we live in, as represented by Cartesian coordinates.

More precisely, we use the symbol ∇ as a substitution for a triplet of operators of partial differentiation
∇ = {∂/∂x, ∂/∂y, ∂/∂z}

Consider a function F(x,y,z) defined in three dimensional space, our three operators on it result in three new functions:
∂F(x,y,z)/∂x,
∂F(x,y,z)/∂y,
∂F(x,y,z)/∂z

The result of each operator above is another function of three arguments and all three together can be interpreted as a vector of three functional components.

We used to think about three-dimensional vector as a set of three real numbers. In this case a triplet of operators also can be formally viewed as a vector (or, to distinguish, a pseudo-vector of operators or vector-operator).
Same with three functions that can be considered as a pseudo-vector of functions or vector-function.

The result of the application of a pseudo-vector ∇ to a function F(x,y,z) can be expressed as
∇F(x,y,z) or, even simpler, ∇F, assuming F stays for F(x,y,z) (there is nothing in between ∇ and F).
So, the expression ∇F(x,y,z) is a shorthand for a set of three functions ∂F(x,y,z)/∂x, ∂F(x,y,z)/∂y and ∂F(x,y,z)/∂z, as if a pseudo-vector ∇ is "multiplied" by a scalar F(x,y,z).

Let's emphasize that ∇F just helps to shorten the triplet of partial differentiation above, it's just a notation, a syntax, nothing more.

Now let's exemplify the use of a symbol ∇ in a particular physical case.
Consider a function F(x,y,z) that represents some physical characteristic at point (x,y,z) in space. It can be an air pressure or electric potential, or density of dust particle, or level of radiation etc.

Very often this function is called a scalar field.
Scalar because for each point (x,y,z) this function equals to a single number and field because, traditionally, when a function is defined in some area of a three-dimensional space, this area is called a field.

We are interested in how this characteristic changes in space. In particular, which direction from point (x,y,z) the change is the most significant.

Let's set a point P(x,y,z) as our base and move from it by an infinitesimal vector Δr={Δx,Δy,Δz} to point Q(x+Δx,y+Δy,z+Δz).

The value of our function will change during this move from F(x,y,z) at point P to F(x+Δx,y+Δy,z+Δz) at point Q.

Increment of a function, when all three of its arguments are changing, can be represented as three increments caused by a move along each coordinate axis:
F(x+Δx,y+Δy,z+Δz) −
− F(x,y,z) =
= [F(x+Δx,y,z) −
− F(x,y,z)] +
+ [F(x+Δx,y+Δy,z) −
− F(x+Δx,y,z)] +
+ F[(x+Δx,y+Δy,z+Δz) −
− F(x+Δx,y+Δy,z)]

A change in a value of a function, when only one argument is infinitesimally changing, can be represented as follows:
F(x+Δx,y,z) − F(x,y,z) ≅
≅ (∂F(x,y,z)/∂x)·Δx
F(x+Δx,y+Δy,z) −
− F(x+Δx,y,z) ≅
≅ (∂F(x,y,z)/∂y)·Δy
F(x+Δx,y+Δy,z+Δz) −
− F(x+Δx,y+Δy,z) ≅
≅ (∂F(x,y,z)/∂z)·Δz

Therefore,
F(x+Δx,y+Δy,z+Δz)−F(x,y,z)≅
≅ (∂F(x,y,z)/∂x)·Δx +
+ (∂F(x,y,z)/∂y)·Δy +
+ (∂F(x,y,z)/∂z)·Δz

Now is the time to apply ∇ syntax.
Notice that the expression
(∂F(x,y,z)/∂x)·Δx +
+ (∂F(x,y,z)/∂y)·Δy +
+ (∂F(x,y,z)/∂z)·Δz
can be viewed as a scalar (dot) product of two vectors
∇F(x,y,z) = {∂F(x,y,z)/∂x, ∂F(x,y,z)/∂y, ∂F(x,y,z)/∂z}
and Δr = {Δx, Δy, Δz}

Therefore,
F(x+Δx,y+Δy,z+Δz)−F(x,y,z)≅
≅ (∇F(x,y,z) · Δr)
where (a·b) denotes the scalar product of two vectors.

From the properties of a scalar product of two vectors we know that
|(a,b)| = |a|·|b|·cos(φ)
where φ is an angle between these two vectors.
If lengths of these vectors are fixed, the maximum absolute value of their scalar product will be if cos(φ)=1.

From this we conclude that function F(x,y,z) will have the largest increment
F(x+Δx,y+Δy,z+Δz)−F(x,y,z)
if vectors ∇F(x,y,z) and Δr are collinear.

The vector
∇F(x,y,z) = {∂F(x,y,z)/∂x, ∂F(x,y,z)/∂y, ∂F(x,y,z)/∂z}
is called a gradient of a scalar field F(x,y,z) and it points to a direction of the largest by absolute value change of the values of this field.