cylindrical coordinates examples

the cylindrical coordinates and the unit vectors of the rectangular coordinate system which are not The del operator from the definition of the gradient

θ = tan-1 (y x ) = tan-1 ( 1 √ 3 ) = π 6 , z = z = 4 Thus, the point is ( 2, π 6 ,4 ) in cylindrical coordinates Example: Find an equation in cylindrical coordinates for the ellipsoid 4x2 + 4y2 + z2 = 1 4x2 + 4y2 + z2 = 1 4r2 + z2 = 1 z2 = 1 − 4r2

Spherical Cylindrical Coordinates x = r cosθ r = √x2 + y2 y = r sinθ tan θ = y/x z = z z = z Spherical Coordinates x = ρsinφcosθ ρ = √x2 + y2 + z2 y = ρsinφsinθ

if x = 0, y< 0 Example 6 1 The cylindrical coordinate system basically is a combination of the polar (a) Plot the point with cylindrical coordinates (2,2π/3,1 ) ,

the cylindrical coordinates and the unit vectors of the rectangular coordinate system which are not The del operator from the definition of the gradient

Examples of orthogonal coordinate systems include the Cartesian (or rectangular ), the cir- cular cylindrical, the spherical, the elliptic cylindrical, the parabolic 

11 1 DEFINITION OF CYLINDRICAL COORDINATES A location in (r, θ) is a location in the xy plane defined in polar coordinates and • z is the height in units  

Formally, we define the cylindrical coordinate system as follows Definition 1 1 The cylindrical coordinates of a point P in 3-space is defined to be (r, ϑ, z) where  

the z−axis In cylindrical coordinates this cylinder has the very simple equation r = c This is the reason for the name “cylindrical” coordinates Example 12 6 1