Vector Calculus Differential Forms

The first step in understanding vector calculus is the comprehension of scalar and vector differential operators - €”the subject of this chapter. The next step is becoming comfortable with the various integral forms to be able to convert between the differential and integral forms - the subject of the next chapter. Vector differential operators can be made to operate on scalar and vector fields in differing ways yielding other scalar, vector, and dyadic fields. Whereas a scalar differential operator operating on a scalar or vector field will yield another scalar or vector field, respectively, a vector differential operator can yield scalar, vector, or tensor fields depending on its formulative properties and depending upon the tensor rank of the operand - the field upon which the operator acts. A brief introduction to the first-order vector differential operators of gradient, curl, and divergence is given in Section 4.1. In addition, since these operators can be applied to tensors in general, some introductory rules of the gradient, divergence, and curl being applied to tensors are also discussed in this section.