In this blog entry you can find lecture notes for Math2111, several variable calculus. See also the table of contents for this course. This blog entry printed to pdf is available here.

We discuss now derivatives of vector fields. To that end we introduce the *del *operator ∇. Throughout we assume that the vector field and the scalar field are , i.e. have at least one continuous partial derivative in each coordinate (similarly we write if the functions are assumed to have continuous partial derivative of order at least two).

In the following we write vectors either as or as (i.e. both have the same meaning).

**The del operator ∇**

We formally introduce the symbol as

which is an operator. That is, it makes sense when it acts or operates on real-valued functions. For instance, if , then

the gradient of .

Definition(Divergence)

Thedivergenceof a vector field , , is defined byVector fields for which are called

incompressible.

**Example **

Let . Hence we have , and . Then the divergence is given by

Definition(Curl)

Thecurlof a vector field , , is defined byVector fields for which are called

irrotational.

**Note**

The formula for the is easiest to remember in operator notation as the determinant of the following matrix:

**Example **

Let . Hence we have , and . Then the curl is given by

Definition(Laplace operator)

Let with continuous partial derivatives of second order in each variable. Then the Laplace operator acts on as follows

**Some common identities**

Theorem

For any vector field we havethat is, the divergence of the any curl is zero.

*Hint* Expand and use Clairaut’s theorem.

Theorem

For any function we havethat is, the curl of any gradient is the zero vector.

*Hint* Expand and use Clairaut’s theorem.

Some common identities

Let be vector fields and be scalar fields.

- for a constant
- at points where

**Note**

Let and be vector fields, where are scalar fields.

- The expression denotes an operator, in particular For instance
- By we mean
- Further