传递现象-Transport-phenomenon-Lesson-1---Vectors-and-Tensors-in-Cartesian-Coordinates(I).ppt

v Preparative Knowledge Concept of Vectors The concept of vectors came from the physical quantities which posses two characteristics, magnitude direction Examples: Velocity, Force, Momentum, Displacement Geometrical Description of Vectors A vector is described geometrically as a directed line segment, which is an intuitive analogy to the displacement. v Algebraic Operations (1)from a geometrical viewpoint Addition Subtraction By parallelogram Addition subtraction Algebraic Operations (2)from a geometrical viewpoint Addition Subtraction By triangle Addition subtraction Algebraic Operations (3)from a geometrical viewpoint Multiplication by a scalar The magnitude is altered by a factor |k|. The direction keeps the same or reversed, depending on the sign of k. Algebraic Operations (4)from a geometrical viewpoint Dot product of two vectors The dot product of two vectors is a scalar. Not associative: Commutative: Distributive: Algebraic Operations (5)from a geometrical viewpoint Cross product of two vectors The cross product of two vectors is a new vector perpendicular to both the vectors by right hand rule. Not associative: Not Commutative: Analytical Description of Vectors A vector can be expressed, in Cartesian coordinates, as the following sum: v are unit vectors in the directions of coordinate axes xi , named as base vectors Analytical Description of Vectors v vi are known as the components of the vector v Algebraic Operations (1)from an analytical viewpoint Addition Subtraction Multiplication by a scalar Algebraic Operations (2)from an analytical viewpoint Dot product of two vectors Cross product of two vectors Algebraic Operations (3)from an analytical viewpoint , named as permutation symbol, has the following properties: * * * * * * * * * * * * * * * * * * * * * * * * * *