Vectors derivative operators

Conventionally, the vector derivative VaF of a function F(xa) of a vector variable xa is defined only for scalar values functions F and in order to calculate it, one expresses the vector derivative operator Va in some coordinates... 

It follows from Eq. (A 124) that the vector derivative operator changes the grade of the object it operates on by 1. For example, the vector derivative of the scalar X(xa) is a vector (because a X = 0 for any scalar X, so aX = a A X), and the vector derivative of the vector f(xa) is a scalar plus a bivector. The differentiation with respect to the vector variable xa greatly resembles the differentiation with respect to some scalar variable xa. For example, the vector differentiation is distributive,... 

However, the vector derivative operator does not commute with multivectors, and the product rule must be written as... 

The gradient operator is a vector derivative operator that produces a vector when applied to a scalar function. 

There are several vector derivative operators. We first define them in Cartesian coordinates. 

An operator is a symbol for carrying out a mathematical operation (see Chapter 9). There are several vector derivative operators. We first define them in Cartesian coordinates. 

The symbol V stands for the gradient operator, which is a vector derivative operator that is defined in Eq. (B-43) of Appendix B. The gradient operator is sometimes called del. The x component of S is given by... 

The vector derivative operators can be expressed in other coordinate systems. In spherical polar coordinates, the gradient of the scalar function / is... 

---