UCLA Math 32B, Winter 2026
Final
Instructor: David Beers
Thursday, March 19
Name:
UCLA ID Number:
The exam will take place from 3:00-5:50PM. Please read all of the following rules before proceeding.
• Points will be deducted if you do not show your work.
• Answers should be written on the test itself. Answers should be on either the front or back
side of the piece of paper the question was written on. E.g. the answers to question 2 should
be on the piece of paper with question 2 written on it.
• Calculator, notebooks, textbook, headphones, and any sort of cheat sheet are not allowed.
Phones should be turned off and put away. No talking or writing on the exam after
time is up while in the classroom. I will deduct at least 25 points for violations
of this rule.
• Good luck!
Polar to rectangular coordinates
x = r cos θ y = r sin θ
Rectangular to polar coordinates
p y
r= x2 + y 2 tan θ =
x
Moreover, −1 y
tan ( x )
x>0
π + tan−1 ( y )
x<0
x
θ= π
2
x = 0 and y > 0
π
−2 x = 0 and y < 0
Cylindrical coordinates
x = r cos θ y = r sin θ z=z
1
, Math 32B Final Thursday, March 19
Spherical coordinates
x = ρ sin ϕ cos θ y = ρ sin ϕ sin θ z = ρ cos ϕ
Integrals in different coordinates
dA = dx dy = r dr dθ
dV = dx dy dz = r dz dr dθ = ρ2 sin ϕ dρ dϕ dθ
Change of Variables
ZZ ZZ
f (x, y) dx dy = f x(u, v), y(u, v) Jac(G)(u, v) du dv,
D D0
with G : D0 → D given by G(u, v) = x(u, v), y(u, v) .
Operations on Vector Fields
D ∂f ∂f ∂f E
grad(f ) = ∇f = , ,
∂x ∂y ∂z
∂F1 ∂F2 ∂F3
div(F) = ∇ · F = + +
∂x ∂y ∂z
i j k D ∂F
∂ ∂ ∂ 3 ∂F2 ∂F1 ∂F3 ∂F2 ∂F1 E
curl(F) = ∇ × F = ∂x ∂y ∂z = − , − , −
∂y ∂z ∂z ∂x ∂x ∂y
F1 F2 F3
If F is two dimensional, i.e. F = ⟨F1 , F2 ⟩. Then div(F) is defined as div(⟨F1 , F2 , 0⟩), and curl(F) is
defined by
∂F2 ∂F1
curl(F) = curl(⟨F1 , F2 , 0⟩) = − k.
∂x ∂y
Line Integrals
Z Z b
f r(t) ∥r′ (t)∥ dt
f ds =
C a
Z Z Z b
F r(t) · r′ (t) dt
F · dr = F · T ds =
C C a
Line Integrals of Conservative Vector Fields
Z
F · dr = f (Q) − f (P ) if F = ∇f and C is oriented from P to Q
C
Parametrized Surfaces
∂G ∂G
Tu = Tv = N = Tu × Tv
∂u ∂v
Page 2 of 13
Final
Instructor: David Beers
Thursday, March 19
Name:
UCLA ID Number:
The exam will take place from 3:00-5:50PM. Please read all of the following rules before proceeding.
• Points will be deducted if you do not show your work.
• Answers should be written on the test itself. Answers should be on either the front or back
side of the piece of paper the question was written on. E.g. the answers to question 2 should
be on the piece of paper with question 2 written on it.
• Calculator, notebooks, textbook, headphones, and any sort of cheat sheet are not allowed.
Phones should be turned off and put away. No talking or writing on the exam after
time is up while in the classroom. I will deduct at least 25 points for violations
of this rule.
• Good luck!
Polar to rectangular coordinates
x = r cos θ y = r sin θ
Rectangular to polar coordinates
p y
r= x2 + y 2 tan θ =
x
Moreover, −1 y
tan ( x )
x>0
π + tan−1 ( y )
x<0
x
θ= π
2
x = 0 and y > 0
π
−2 x = 0 and y < 0
Cylindrical coordinates
x = r cos θ y = r sin θ z=z
1
, Math 32B Final Thursday, March 19
Spherical coordinates
x = ρ sin ϕ cos θ y = ρ sin ϕ sin θ z = ρ cos ϕ
Integrals in different coordinates
dA = dx dy = r dr dθ
dV = dx dy dz = r dz dr dθ = ρ2 sin ϕ dρ dϕ dθ
Change of Variables
ZZ ZZ
f (x, y) dx dy = f x(u, v), y(u, v) Jac(G)(u, v) du dv,
D D0
with G : D0 → D given by G(u, v) = x(u, v), y(u, v) .
Operations on Vector Fields
D ∂f ∂f ∂f E
grad(f ) = ∇f = , ,
∂x ∂y ∂z
∂F1 ∂F2 ∂F3
div(F) = ∇ · F = + +
∂x ∂y ∂z
i j k D ∂F
∂ ∂ ∂ 3 ∂F2 ∂F1 ∂F3 ∂F2 ∂F1 E
curl(F) = ∇ × F = ∂x ∂y ∂z = − , − , −
∂y ∂z ∂z ∂x ∂x ∂y
F1 F2 F3
If F is two dimensional, i.e. F = ⟨F1 , F2 ⟩. Then div(F) is defined as div(⟨F1 , F2 , 0⟩), and curl(F) is
defined by
∂F2 ∂F1
curl(F) = curl(⟨F1 , F2 , 0⟩) = − k.
∂x ∂y
Line Integrals
Z Z b
f r(t) ∥r′ (t)∥ dt
f ds =
C a
Z Z Z b
F r(t) · r′ (t) dt
F · dr = F · T ds =
C C a
Line Integrals of Conservative Vector Fields
Z
F · dr = f (Q) − f (P ) if F = ∇f and C is oriented from P to Q
C
Parametrized Surfaces
∂G ∂G
Tu = Tv = N = Tu × Tv
∂u ∂v
Page 2 of 13
