Advanced Calculus Final Exam
MATH-404-604 Advanced Calculus
Final Examination
Spring 2024
Instructions: This is a take-home exam. It is due at midnight on May 3rd - please submit it via Canvas.
You may use the written notes provided to you, the notes you take in class, the homework sets, and the two
midterms - nothing else. You are not allowed to talk to anyone regarding this exam. Please show all your
work. Thank you and good luck!
Question 1
Let f, g : R → R be twice differentiable. Define u(x, y) := f(xy) and v(x, y) := f(x − y) + g(x + y). Show
that
x
∂u
∂x
− y ∂u
∂y
= 0,
∂2v
∂x2
− ∂
2v
∂y2
= 0
Question 2
Let f : R2 → R3 be given by x = uv, y = u2 − v2, z = u+ v. Let p = (1, 1) and q = f(p) = (1, 0, 2). Let S
be the image of this map. Let u = [3,−2, 3].
(a) Compute the Jacobian Jf(p) of f at the point p.
(b) Show that the columns of Jf(p) span TqS.
(c) Find an equation for the tangent space TqS.
(d) Verify that u ∈ TqS.
(e) Find a curve in S with velocity u.
(f) Estimate f(0.95, 1.01).
Question 3
The one-form ω = (x− a)yex dx+ (xex + z3) dy + byz2 dz is closed. Let C be the curve parameterized by
γ(t) = (t, cos 2t, cos t) for t ∈ [0, π].
(a) Find a and b given that ω is closed.
(b) Since ω it is closed and smooth in all of R3, by the Poincare´ Lemma it must be exact. With a and b
computed above, find a function f such that ω = df .
(c) Evaluate
∫
C
ω.
(d) Use (c) to evaluate
∫
C
xyex dx+ (xex + z3) dy + 4yz2 dz.
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Advanced Calculus Final Exam
Question 4
Let D be a region in the plane whose boundary is a simple and smooth curve C oriented counterclockwise.
Assume that the area of D is 5. Let
f : D ⊂ R2 → R3, f(u, v) = (u, v, 1− u− v)
be a parameterized surface S = f(D) whose boundary is ∂S = f(C). Let ω = −2z dx− 4x dy + 8y dz be a
one-form.
(a) Compute α = dω, α1 = f
∗(α), ω1 = f∗(ω).
(b) Evaluate the line integral of ω along ∂S.
(c) Evaluate the line integral of ω1 along C.
(d) Relax, deep breath and take a good look at the calculations of this problem. What general statements
can you formulate?
Question 5
Find the volume of the solid bound by the ellipsoid
x2
a2
+
y2
b2
+
z2
c2
= 1
by integrating the two-form zdx ∧ dy along the surface. (Hints: (1) review question 2 part c in exam 2 to
find out how to get the parametrization of an ellipsoid from that of a sphere, (2) see example 28 in page 78
of the online version of the notes for the calculation of the integral.)
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Advanced Calculus Final Exam
Question 6
A polygon P has vertices p1(x1, y1), p2(x2, y2)), ..., pn(xn, yn), in counterclockwise order. The area A of this
polygon may be evaluated by integrating the two form dx ∧ dy. Use Green’s Theorem to show the Shoelace
Formula
A =
(x1y2 − y1x2) + (x2y3 − x3y2) + ...+ (xn−1yn − xnyn−1) + (xny1 − x1yn)
2
MATH-604 question
We have seen that the line integral of an exact 1-form along a closed curve (i.e. a curve without boundary)
is zero. Let ω be an exact two-form that is smooth in R3. Show that its surface integral along a closed
surface (i.e. a surface without a boundary) is zero.
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