CHAPTER 6 CURVILINEAR COORDINATES The vectors , and are mutually orthogonal and form a right-handed eˆ3 eˆ1 eˆ2 system. Note that unlike the unit vectors in the Cartesian system, the unit vectors iˆ, kˆ ˆj, , , eˆ3 or base vectors as they are called are not fixed in directions, but change eˆ1 eˆ2 as the position of P changes. (r ) If is a vector associated with P, then F (r )    F F eˆ1 F eˆ2 Fzeˆ3  vectors , and . eˆ1 eˆ2 eˆ3 2. Spherical Polar Coordinates P   Here is a function of three variables r, , . Z Y a) If  and  remain constant and r increases, P moves   is thus a vector P normal to the surface of the sphere at and the unit r eˆ1 p(r,,) eˆ2 X eˆ3 O vector , in that direction is therefore eˆ  eˆ1 r   P b) If and remain constant and increases, will move along the ‘meridian”
CHAPTER 6 CURVILINEAR COORDINATES The vectors eˆ1 , eˆ2 and eˆ3 are mutually orthogonal and form a right-handed system. Note that unlike the unit vectors iˆ, ˆj, kˆ in the Cartesian system, the unit vectors eˆ1 , eˆ2 , eˆ3 or base vectors as they are called are not fixed in directions, but change as the position of P changes. If F (r ) is a vector associated with P, then F (r )  F eˆ1  F eˆ2  Fzeˆ3  vectors eˆ1 , eˆ2 and eˆ3 . 2. Spherical Polar Coordinates Here P is a function of three variables r,  ,  . Z a) If  and  remain constant and r increases, P moves is thus a vector normal to the surface of the sphere at P and the unit X O   r eˆ1 p(r,,) eˆ2 eˆ3 Y vector eˆ , in that direction is therefore eˆ1  b) If r and  remain constant and  increases, P will move along the ‘meridian”