Fundamentals of Electromagnetic Fields
Including numerous examples with complete, worked-out solutions, this book is designed to present the fundamental concepts of electromagnetic field theory as they relate to modern engineering applications.
TABLE OF CONTENTS Fundamentals of Electromagnetic Fields
1.6: DIFFERENTIATION OF A VECTOR FIELD
1.6 DIFFERENTIATION OF A VECTOR FIELD
Let R (u) be a vector depending upon a scalar variable u=u(x,y,z). Then,
where D u represents increments in u. The ordinary derivative of the vector R (u) with respect to the scalar u is given as 
, if the limit exists. Since [ d R (u)/du] is itself a vector depending upon variable u in the direction of ? R it is considered to be derivative of R (u) with respect to u if this limit exists, then we call such a vector field as differentiable.
FIG. 1 18: Differentiation of a vector field.
Continuous Vector Field
A vector function R=R x a x + R y a y+R z a z is said to be continuous if R x , R y , and R z scalar functions are continuous i.e.,
Differentiation Formulae
Partial Differentiation of a Vector Field
If A is a vector function of more than one scalar variable, say, x, y, z, then we write A= A (x,y,z). The partial derivative with respect to x is defined as:
Similarly, we can extend this concept to partial differentiation with respect to y and z.
Continuous Scalar Function
?(x,y,z) is called as continuous at (x,y,z) if,
Higher Derivatives
Higher derivatives and mixed partial derivatives can be defined in vector calculus as:
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