Notation For Differentiation - Notation in Vector Calculus

Notation in Vector Calculus

Vector calculus concerns differentiation and integration of vector or scalar fields particularly in a three-dimensional Euclidean space, and uses specific notations of differentiation. In a Cartesian coordinate o-xyz, assuming a vector field A is, and a scalar field is .

First, a differential operator, or a Hamilton operator ∇ which is called del is symbolically defined in the form of a vector,

,

where the terminology symbolically reflects that the operator ∇ will also be treated as an ordinary vector.

• Gradient: The gradient of the scalar field is a vector, which is symbolically expressed by the multiplication of ∇ and scalar field ,

,

,

.

• Divergence: The divergence of the vector A is a scalar, which is symbolically expressed by the dot product of ∇ and the vector A,

,

,

.

• Laplacian: The Laplacian of the scalar field is a scalar, which is symbolically expressed by the scalar multiplication of ∇2 and the scalar field φ,

,

where, is called a Laplacian operator.

• Rotation: The rotation, or, of the vector A is a vector, which is symbolically expressed by the cross product of ∇ and the vector A,

,

,

,

.

Many symbolic operations of derivatives can be generalized in a straightforward manner by the gradient operator in Cartesian coordinates. For example, the single-variable product rule has a direct analogue in the multiplication of scalar fields by applying the gradient operator, as in

Further notations have been developed for more exotic types of spaces. For calculations in Minkowski space, the D'Alembert operator, also called the D'Alembertian, wave operator, or box operator is represented as, or as when not in conflict with the symbol for the Laplacian.