VECTOR CALCULUS

BY ARJUN RASTOGI

DEFINITION

• Vector calculus (or vector analysis) is a branch of mathematics concerned with differentiation and integration of vector fields, primarily in 3-dimensional Euclidean space

Basic objects• Scalar fieldsA scalar field associates a scalar value to every

point in a space. The scalar may either be a mathematical number or a physical quantity

• Vector fieldsA vector field is an assignment of a vector to each

point in a subset of space. A vector field in the plane, for instance, can be visualized as a collection of arrows with a given magnitude and direction each attached to a point in the plane.

Vectors and pseudovectors

one further distinguishes pseudo vector fields and pseudo scalar fields, which are identical to vector fields and scalar fields except that they change sign under an orientation-reversing map .

Vector operations• scalar vector multiplication

multiplication of a scalar field and a vector field, yielding a vector field

Scalar vector additionaddition of two vector fields, yielding a

vector field:• dot product multiplication of two vector fields, yielding a

scalar field: 

• cross product multiplication of two vector fields, yielding a vector field: 

• scalar triple product the dot product of a vector and a cross product of two

vectors:

• vector triple product the cross product of a vector and a cross product of two

vectors: