Last modified: Wednesday 14th April, 2021, 16:23
MAT257 Final Assessment Information and Rejected Questions
• The Final Assessment will take place on Tuesday April 20, 9AM-Noon (Toronto time), on Crowdmark (you
will get a link by email about one minute before the official starting time). Other than documented accessibility
matters, no exceptions!
• Our TAs Peter and Petr will hold extra pre-test office hours, in their usual zoom rooms. Peter on Saturday 9-12
and on Sunday 1-4 at Peter’s Zoom (password vchat), and Petr on Monday 9-11 at Petr’s Zoom (password
vchat).
• I will hold my regular office hours on Tuesday April 13, at 9-10 and 12-1 and then extra hours on Friday 9-11,
Sunday 9-11, and Monday 1-3, at http://drorbn.net/vchat.
• I will be available to answer questions throughout the exam, at my usual office (http://drorbn.net/vchat,
but I’ll add a waiting room). I will also be monitoring my regular email address ([email protected])
throughout the exam.
• There will be mishaps! I just hope that not too many. If you encounter one, document everything with specific
details, times, and screen shots, and send me a message by Wednesday April 21 at 7PM. I will deal with these
situations on a case by case basis.
• Don’t let unanswered questions and/or mishaps paralyze you! If you need an answer but for whatever reason
you cannot reach me, think hard, come up with what you think is the most reasonable answer/resolution,
document as best as you can (for example, by adding a note on your submission), and act following your
conclusions.
• Material: Everything excluding the material on Maxwell equations, with very light emphasis on the material
not covered in the previous term tests,
• Open book(s) and open notes but you can only use the internet (during the exam) to read the exam, to submit
the exam, and to connect with the instructor to ask clarification questions. No contact allowed with other
students or with any external advisors, online or in person.
• The format will be “Solve 7 of 7”, or maybe “6 of 6” or “8 of 8”.
• You will be required to copy in your handwriting and sign an academic integrity statement and submit it on
Crowdmark along with the rest of your exam. If you wish, you may save time by preparing the academic
integrity statement in advance as in this sample.
• You will be given an extra 20 minutes at the end of the exam to upload it and to copy/sign the academic
integrity statement.
• The vast majority of students will do honest work, and I appreciate that. Out of respect for the honest students
I will do my best to pursue and punish any cheating that may occur. Please! Life is much better if we don’t
need to go there.
• To prepare: Do the FA “rejects” available below, but more important: make sure that you understand every
single bit of class material so far!
• It is not the assessment I want! Class material and HW are important, but there won’t be questions straight
from class/HW. Many things in 2020/21 are not as we want them.
The following questions were a part of a question pool for the 2020-21 MAT257 Final Assessment, but at the
end, they were not included.
1. Suppose that the bounded functions f and g are integrable over some rectangle R ⊂ Rn. Show that f g, f 2, and
g2 are also integrable over R and that
∫
R
f g ≤
(∫
R
f 2
)1/2 (∫
R
g2
)1/2
.
2. Prove that the union of finitely many open sets in Rn is open, and give a counterexample to show that this
statement may not be true if the union is countably infinite.
3. If A and B are disjoint open sets in Rn, show that there exists disjoint closed subsets C and D of Rn such that
A ⊂ C and B ⊂ D.
4. Recall that the variation o( f , t) of a function f : R→ R at a point t ∈ R is defined to be
lim
r→0
sup {| f (x) − f (y)| : x, y ∈ Br(t)} .
Prove that if f is monotone on some interval [a, b] and P = (a = t0 < t1 < . . . < tn−1 < tn = b) is a partition of
[a, b], then
n∑
i=0
o( f , ti) ≤ | f (b) − f (a)|.
5. Show that the function f (x, y) = |xy|1/2 is continuous at a = (0, 0) and has both its partial derivatives exist at a,
yet it is not differentiable at a.
6. Prove that if a differentiable function f : Rn → R has a local max at some point a ∈ Rn, then f ′(a) = 0.
7. A function f : Rn → Rn satisfies |(x− y)− ( f (x)− f (y))| ≤ 13 |x− y| for every x, y ∈ Rn. Prove that f is surjective
(onto).
8. A function f : Rn → Rn satisfies |(x − y) − ( f (x) − f (y))| ≤ 1k |x − y| for every x, y ∈ Rn whose norm is less than
e−k. Prove that f is differentiable at 0 and compute its differential f ′(0).
9. A differentiable function f : Rn → Rk is said to be “submersive” at 0 if rank f ′(0) = k. Assume such a function
f is submersive at 0 and assume also that f (0) = 0, and show that there is a function g : Rn → Rn which is
defined, differentiable, and invertible near 0, and so that f (g(x1, . . . , xn)) = (x1, . . . , xk). (In other words, every
submersive function looks like the standard projection Rn → Rk near 0).
10. (a) A subset A ⊂ R is known to have content 0. Is it necessarily true that ∂A also has content 0?
(b) A subset B ⊂ R is known to have measure 0. Is it necessarily true that ∂B also has measure 0?
11. If f is a bounded function defined on a rectangle R ⊂ Rn and if supp f (the closure of {x ∈ R : f (x) , 0}) is a
set of measure 0, show that f is integrable on R and that
∫
R
f = 0.
12. If f is a bounded non-negative function defined on a rectangle R ⊂ Rn and if ∫
R
f = 0, show
(a) For every b > 0, the set {x ∈ R : f (x) ≥ b} has content 0.
(b) The set {x ∈ R : f (x) > 0} has measure 0.
13. Let f : R→ R be a smooth function. Prove that there is a smooth function h : R2 → [0, 1] such that h(x, f (x)) =
1 for every x ∈ R, yet always, if x, y ∈ R and |y − f (x)| ≥ 1, then f (x, y) = 0.
14. Let m : [0, 1] → Mn×n(R) be a path in the space of n × n matrices, and suppose that for every t ∈ [0, 1] the
columns of m(t) make a basis of Rn. Show that the bases m(0) and m(1) define the same orientation of Rn.
15. Show that if F =
∑
i fi(p)(p, ei) and G =
∑
iGi(p)(p, ei) are smooth vector fields on Rn, then there is a third
smooth vector field H =
∑
i hi(p)(p, ei) on Rn such that
DF ◦ DG − DG ◦ DF = DH,
where DF : Ω0(Rn) → Ω0(Rn) is the operation of directional derivative in the direction of F, which maps
smooth functions on Rn to smooth functions on Rn (and likewise for DG and DH).
16. An exploration problem: a 3-vector on R4txyz is a function F : R
4
txyx → R3xyz. It can be regarded as a time-
dependent vector field on R3, and so it makes sense to write grad, curl, and div in this context, and also ∂t = ∂∂t .
Of course, you also need to consider “scalar functions” f : R4 → R, to talk about grad and div. Can you
interpret the sequence
Ω0(R4)
d−→ Ω1(R4) d−→ Ω2(R4) d−→ Ω3(R4) d−→ Ω4(R4)
in this language of scalar functions, 3-vectors, grad, curl, div, and ∂t?
17. Prove that the form xdydz + ydzdx + zdxdy is closed but not exact on the 2-dimensional unit sphere S 2 ⊂ R3xyz.
18. ω is a smooth 3-form on R7, and we know that the integral of ω over every 3-cube in R7 vanishes. Prove that
ω itself vanishes.
19. We will say that a 1-form ω on Rn is “precise” if its integral over any 1-cube depends only on the boundary of
that 1-cube (namely, ∂c1 = ∂c2 =⇒
∫
c1
ω =
∫
c2
ω. Show that a 1-form ω is precise if and only if it is exact.
20. Suppose M is a k-dimensional manifold in Rn, and suppose F is a smooth vector field on M (so in particular
F(x) ∈ TxM for every x ∈ M). Show that there is some vector field G on some open set A ⊃ M (in particular,
G(x) ∈ TxRn for every x ∈ A) such that G restricted to M is F. You may need to use one of the precise
definitions of a manifold, and something to make the local go global.
21. A smooth vector field E defined on R3 is known to satisfy div E = 0 outside of D31/2, the 3-dimensional closed
ball of radius 1/2 in R3, and it is also known that
∫
∂D31
(E ·n)dA = 257, where everything is taken with “standard
conventions”: orientations, positive normals, and area forms. Compute
∫
∂D32
(E ·n)dA and ∫
∂D31(p)
(E ·n)dA, where
D31(p) denotes the closed ball of radius 1 about a point p, and p is a point of ∂D
3
2.
22. A subset B of R3 is the union two infinite lines positioned as on the figure on the left below, and in addition,
oriented loops R1, R2, and Gi for i = 1, 2, 3, 4, 5 are also given as in the same figure. A vector field F is
also given, and it is known to be smooth away from B and to satisfy curl F = 0 on R3 \ B. It is known that∫
R1
(F · T )ds = pi and ∫
R2
(F · T )ds = e. Compute ∫
Gi
(F · T )ds for i = 1, 2, 3, 4, 5. Hint. You may want to also
think about 2D subsets of R3 that are shaped like masks and/or tubes as in the figure below on the right
Please watch this page for changes — I may add to it later. Last modified: Wednesday 14th April, 2021, 16:23

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