SME
Calculus III (Multivariable)
Cylindrical and spherical coordinates, change of variables
Use this for “Write the point (3, 4, 12) in cylindrical and spherical coordinates.”
Diagram
OP = ⟨3, 4, 12⟩, with length |OP| = 13.
This is an example. Type your own values to start a new problem.
P
=
(
x
,
y
,
z
)
(r,
θ,
z)
=
(
r
,
θ
°,
z
)
(ρ,
θ,
φ)
=
(
ρ
,
θ
°,
φ
°)
Formulas
r
=
√
x
2
+
y
2
5
=
√
3
2
+
4
2
θ
=
atan2(
y
,
x
)
53.1301 = atan2(4, 3)
x
=
r
×
cos(
θ
°)
3 = 5 × cos(53.1301°)
y
=
r
×
sin(
θ
°)
4 = 5 × sin(53.1301°)
ρ
=
√
r
2
+
z
2
13
=
√
5
2
+
12
2
φ
=
cos⁻¹(
z
ρ
)
22.6199
=
cos⁻¹(
12
13
)
z
=
ρ
×
cos(
φ
°)
12 = 13 × cos(22.6199°)
r
=
ρ
×
sin(
φ
°)
5 = 13 × sin(22.6199°)
Assumptions
•
θ is measured in the xy-plane from the positive x-axis, counterclockwise seen from above, from 0° up to 360°.
•
φ is measured down from the positive z-axis, from 0° to 180°, the math convention.
•
r is the distance from the z-axis and ρ the distance from the origin, so r = ρ sin φ.
Step-by-step
WE KNOW
x = 3, y = 4, z = 12
FIND
distance from the z-axis (r), angle round the z-axis (θ), distance from the origin (ρ), angle down from the z-axis (φ)
Step 1 · Find distance from the z-axis (r)
Use r = √(x² + y²)
r
is
the
length
of
the
shadow
of
P
on
the
xy-plane:
Pythagoras
on
x
and
y
.
r
=
√
3
2
+
4
2
r
=
√
9 + 16
r
=
√
25
r
=
5
Step 2 · Find angle round the z-axis (θ)
Use θ = atan2(y, x)
tan
θ
=
y
x
;
when
x
<
0
add
180°
(and
360°
when
x
>
0
and
y
<
0),
so
θ
points
into
the
quadrant
of
(
x
,
y
).
θ
=
tan⁻¹(
y
x
)
θ
=
tan⁻¹(
4
3
)
θ
=
53.1301°
Step 3 · Find distance from the origin (ρ)
Use ρ = √(r² + z²)
r
across
and
z
up
are
the
legs
of
a
right
triangle;
ρ
is
its
hypotenuse,
from
the
origin
to
P.
ρ
=
√
5
2
+
12
2
ρ
=
√
25 + 144
ρ
=
√
169
ρ
=
13
Step 4 · Find angle down from the z-axis (φ)
Use φ = cos⁻¹(z ÷ ρ)
In
the
same
right
triangle
z
is
the
side
next
to
φ
and
ρ
the
hypotenuse,
so
cos
φ
=
z
ρ
.
φ
=
cos⁻¹(
12
13
)
φ
=
22.6199°
→
Cylindrical
(
r
,
θ
,
z
)
=
(5,
53.1301°,
12);
spherical
(
ρ
,
θ
,
φ
)
=
(13,
53.1301°,
22.6199°)
Check
Put the numbers back in. Both sides should match.
5
=
√
3
2
+
4
2
✓
53.1301 = atan2(4, 3) ✓
3 = 5 × cos(53.1301°) ✓
4 = 5 × sin(53.1301°) ✓
13
=
√
5
2
+
12
2
✓
22.6199
=
cos⁻¹(
12
13
)
✓
12 = 13 × cos(22.6199°) ✓
5 = 13 × sin(22.6199°) ✓
Topic check
See what you know: 5 questions on this topic
New questions each time. Pass to mark it complete.