Multivariate Calculus - Bruce E. Shapiro

48 LECTURE 7. CYLINDRICAL AND SPHERICAL COORDINATES
To find (r, θ, z)given (x, y, z ):
Example 7.1 Convert the equation
to cylindrical coordinates
y = r sin θ
z = z
r = √ x 2 + y 2
tan θ = y/x or θ = tan −1 (y/x)
z = z
x 2 − y 2 + 2yz = 25
Solution.We have x = r cos θ, y = r sin θ and z remains unchanged. Therefore the
equation can be written as
25 = x 2 − y 2 + 2yz = (r cos θ) 2 − (r sin θ) 2 + 2zr cos θ
With some factoring and application of a trigonometric identity:
Example 7.2 Convert the equation
25 = r 2 (cos 2 θ − sin 2 θ) + 2zr cos θ
25 = r 2 cos 2θ + 2zr cos θ.
r 2 cos 2θ = z
from Cylindrical to Cartesian coordinates.
Solution. With some algebra,
z = r 2 cos 2θ = r 2 (cos 2 θ − sin 2 θ)
= r 2 cos 2 θ − r 2 sin 2 θ
= (r cos θ) 2 − (r sin θ) 2
= x 2 − y 2
Example 7.3 Convert the expression
r = 2z sin θ
from Cylindrical coordinates to Cartesian coordinates.
Solution. Multiply through by r to give
r 2 = 2zr sin θ
Then use the identities r 2 = x 2 + y 2 and y = r sin θ to get
x 2 + y 2 = 2zy.
Revised December 6, 2006. Math 250, Fall 2006