A.1 Vector Differential Calculus

Derivatives of a vector function of the space coordinates can be performed in any combination of the coordinate directions. A useful differentiation operator is the del operator, denoted by the symbol ?, or del, defined in Cartesian coordinates as

The operator often appears in partial differential equations in its squared form

To illustrate some of the uses of the del operator, consider a unit vector a with direction cosines a _x, a _y, a _z. Then, the operation a ? acting on a vector F gives the derivative of F in the direction of a, or

The operator v ? is encountered often in fluid mechanics, particularly where v is the velocity vector. The operator itself is not a vector, since v ? ? ? v, and is referred to as a pseudo-vector.

Frequent uses of the ? operator include its operations on a scalar and also in vector multiplications. For instance, if is a scalar function of the coordinates, then

is called the gradient of . Its magnitude tells us relative information about the spacing of lines of constant . Where the magnitude...