Vector analysis

Pagina 1

VECTOR ANALYSIS (Basic Theoretical Background) 1.1 VECTOR OPERATIONS Hereafter, vector quantities will be written in bold, while scalar quantities will be written in italic. 1.1.1 Vector operations The basic algebraic (non -differential) operations in vect or calculus are referred to as vector algebra , being defined for a vector space and then globally applied to a vector field, and consist of: scalar multiplication multiplication of a scalar field, a , and a vector field, v , yielding a vector field: a=bv ; vector addition addition of two vector fields, yielding a vector field: 12=b v +v ; dot product multiplication of two vector fields, yielding a scalar field: 12b = ⋅vv (or 12b = ×vv ) ; (the symbol × is alternative to the symbol ⋅ ) cross product multiplication of two vector fields, yielding a vector field: 12= ×bv v (or 12= ∧bv v ) ; (the symbol ∧ is alternative to the symbol × ) There are also two triple products: scalar triple product the dot product of a vector and a cross product of two vectors: 1 123 ()b = ⋅×vvv (or 11 23 ()b = ×∧v vv ) ; vector triple product the cross product of a vector and a cross product of two vectors: 1 1 23 ()= ××b v vv or 2 3 21 ()= ××b v vv Or alternatively: 11 23 ()= ∧∧bv vv or 2 3 21 ()= ∧∧b v vv ; although these are less often used as basic operations, as they can be expressed in terms of the dot and cross products. 1.2 SCALAR MULTIPLICATION Not to be confused with scalar product. In mathematics, scalar multiplication is one of the basic operations defining a vector space in linear algebra (or more generally, a module in abstract algebra). In an intuitive geometrical context, scalar multiplication of a real Euclidean vector by a positive real number multiplies the magnitude of the vector without changing its direction. The term…