*Before you read this post, you should understand* the dot product of vectors (appendix 2 post 17.13), the cross product of vectors (post 17.37) and partial differentiation (post 19.11).

I have been asked to write a post on the Navier-Stokes equation that was briefly introduced in post 17.15. Before I write it, I need to explain the mathematical operator **∇**. This post, and the two that follow it, are difficult topics so you may wish to ignore them. But I hope to use the ideas in this post to discuss electromagnetism, in later posts, as well as the Navier-Stokes equation, in post 20.36.

The operator **∇** is defined by the equation at the beginning of this post (highlighted in yellow). Here ** i**,

**and**

*j***are the orthogonal unit vectors defining an orthogonal Cartesian coordinate system (post 17.3) and**

*k**x*,

*y*and

*z*are the components of a vector in each of these directions, respectively;

*∂*/

*∂x*,

*∂*/

*∂y*and

*∂*/

*∂z*represent partial derivatives (post 19.11) with respect to

*x*,

*y*and

*z*, respectively.

**∇**is written in

**bold**because it’s a vector operator. The symbol ∇ is called

*del*or

*nabla*– “nabla” is supposed to be an ancient Greek word for a harp; I don’t know the origin of the word “del”.

Notice that **∇ **doesn’t represent a number – it represents a mathematical operation (like the integral sign in post 17.19). For example, we know (post 17.4) that

We could think of *d*/*dx* as an *operator* that acts on *x*^{2} to give the result 2*x*. But *d*/*dx* is not a vector operator – it is a scalar operator (see post 16.50 and 17.2 to find the difference between vectors and scalars).

Before we continue, let’s look at the square of the modulus of **∇**. According to appendix 2 of post 17.13

The final step arises because

** i.i** =

**=**

*j.j***= 1 (2)**

*k.k*and

** i.j** =

**,**

*k***= –**

*i.k***,**

*j***= –**

*j.i***,**

*k***=**

*j.k***,**

*i***= –**

*k.i***,**

*j***= –**

*k.j***(3)**

*i*(post 17.3). Equation 1 defines the Laplacian operator, previously defined in post 19.12.

Now let’s look at the operation of **∇** on *r* = *x*^{2}. Note that *x*^{2} is not a vector (it is a scalar) but we can still apply a vector operator to it. Then

Note that 2*x i* is the slope of

*r*(post 17.4) in the

**-direction. So it is the slope of the vector**

*i***= x**

*r*^{2}

**. A slope is sometimes called a gradient. So**

*i***∇**

*r*is sometimes called the

*gradient*of

**; this definition is usually written as**

*r*Note also that the operation of **∇** on a scalar gives a vector result.

When **∇** operates on a vector it can do it in two ways: forming a dot product (appendix 2 post 17.13) or forming a cross product (post 17.37).

Let’s look at the dot product first and, as an example, consider the vector ** r** = x

^{2}

**. Then**

*i*The final step arises from equations 2 and 3. Continuing

So **∇.**** r** produces a scalar result that increases as

*x*increases; the result diverges with increasing

*x*and is sometimes called the

*divergence*of

**, written**

*r*Finally, let’s think about the cross product of **∇** and a vector. This time we’ll consider the vector ** r** =

*y*

^{2}

**+**

*i**x*

^{2}

**, as an example. Then**

*j*The right hand side of this equation can be written as the determinant (post 17.37)

Evaluating the partial derivatives gives the result

Note that the vector ** r**, in this example, lies in the plane defined by the vectors

**and**

*i***so**

*j***∇**×

**is a vector perpendicular to**

*r***. The result is sometimes called the**

*r**curl*of

**(for reasons that need not concern us when we meet the Navier-Stokes equation) and is written**

*r*In conclusion, the vector operator **∇** can:

- act on a scalar to gives a vector result
- form a dot product with a vector to give a scalar result
- form a cross product with a vector to give a vector that is perpendicular to the original vector.

Related posts

19.11 Partial differentiation

17.3 Three-dimensional vectors

Follow-up posts