CHAPTER 1 VECTOR SCALARS AND SCALAR FIELD If to each point (x,y,z) of a region R in space there corresponds a number or (x,  scalar , then is called a scalar function of position and we say that a scalar y, z) field  has been defined in R. Example 1.6 The temperature at any point within or on the earth’s surface at a certain time defines a scalar field. Example 1.7  (x,  y, z) x y z defines a scalar field. 3 2 (x, 3x 5 Example 1.8 Given the scalar field defined by    , find y, z) z xy 2 3  at the point (-1,-2,-3).  5    2 VECTOR FIELD If to each point (x,y,z) of a region R in space there corresponds a vector (x, , V y, z) then is called a vector function of position and we say that a vector field has V V been defined in R. Example 1.9 If the velocity at any point (x,y,z) within a moving fluid is known at a certain time, then a vector field is defined. Example 1.10 (x,  2  defines a vector field. A vector V y, z) xy yz x zkˆ ˆj 2iˆ 3 2 field that is independent of time is called a steady-state vector field. FURTHER EXAMPLES Example 1.11 Given two non-collinear vectors and , find an expression for any vector lying a b r D in the plane determined by and . a b Non-collinear vectors are vectors that are not parallel to R the same line. Thus when their initial points coincide, r A B they determine a plane. Let be any vector lying in r C b a O the plane of and and having its initial point a b
CHAPTER 1 VECTOR AND SCALARS SCALAR FIELD If to each point (x,y,z) of a region R in space there corresponds a number or scalar (x, y, z) , then  is called a scalar function of position and we say that a scalar field  has been defined in R. Example 1.6 The temperature at any point within or on the earth’s surface at a certain time defines a scalar field. Example 1.7  (x, y, z)  x 3 y z 2 defines a scalar field. Example 1.8 Given the scalar field defined by  (x, y, z)  3x 2 z xy 3  5 , find  at the point (-1,-2,-3).   2  5  VECTOR FIELD If to each point (x,y,z) of a region R in space there corresponds a vector V (x, y, z) , then V is called a vector function of position and we say that a vector field V has been defined in R. Example 1.9 If the velocity at any point (x,y,z) within a moving fluid is known at a certain time, then a vector field is defined. Example 1.10 V (x, y, z)  xy 2iˆ 2 yz 3 ˆj  x 2 zkˆ defines a vector field. A vector field that is independent of time is called a steady-state vector field. FURTHER EXAMPLES Example 1.11 Given two non-collinear vectors a and b , find an expression for any vector r lying in the plane determined by a and b . Non-collinear vectors are vectors that are not parallel to D R the same line. Thus when their initial points coincide, A r they determine a plane. Let r be any vector lying in the plane of a and b and having its initial point O a b C B