, let f =
. Show that this function has a limit as
on an arbitrary straight line through
. Next show
that this function fails to have a limit at
- Here are some scalar valued functions of several variables. Determine which of
these functions are o. Here
v is a vector in ℝn, v = .
- v2 sin
- v12 + v2
- v2 sin
- Here is a function of two variables. f =
x2y + x2. Find Df directly from the
definition. Recall this should be a linear transformation which results from
multiplication by a 1
× 2 matrix. Find this matrix.
- Let f =
. Compute the derivative directly from the definition. This
should be the linear transformation which results from multiplying by a 2
× 2 matrix.
Find this matrix.
- You have h =
x ∈ ℝn,f
∈ ℝm and g
∈ ℝp. where f,g are
appropriately differentiable. Thus Dh results from multiplication by a matrix.
Using the chain rule, give a formula for the
ijth entry of this matrix. How
does this relate to multiplication of matrices? In other words, you have two
matrices which correspond to Dg and
z = g
,y = f
Explain the manner in which the ijth entry of Dh is
This is a review of the way we multiply matrices. what is the ith row of Dg and the
jth column of Df?
- Find fx,fy,fz,fxy,fyx,fzy for the following. Verify the mixed partial derivatives are
- x2y3z4 + sin
- sin +
- Suppose f is a continuous function and f : U → ℝ where U is an open set and suppose
that x ∈ U has the property that for all y near x, f
≤ f. Prove that if
f has all
of its partial derivatives at x, then fxi = 0 for each
xi. Hint: Consider
h. Argue that
h′ = 0 and then see what this implies about
- As an important application of Problem 7 consider the following. Experiments are done
at n times, t1,t2,
,tn and at each time there results a collection of numerical
outcomes. Denote by
i=1p the set of all such pairs and try to find numbers a
and b such that the line x = at + b approximates these ordered pairs as well as possible
in the sense that out of all choices of a and b, ∑
2 is as small as
possible. In other words, you want to minimize the function of two variables
2. Find a formula for a and b in terms of the given
ordered pairs. You will be finding the formula for the least squares regression line.
- Let f be a function which has continuous derivatives. Show that u =
solves the wave equation
utt − c2Δu = 0. What about u =
u = uxx.
- Show that if Δu = λu where u is a function of only x, then eλtu solves the heat
equation ut − Δu = 0. Here Δu = uxx.
- Show that if f =
f′ = 0.
- Let f be defined on
ℝ2 as follows. f = 1 if
x≠0. Define f = 0, and
f = 0 if
y≠x2. Show that f is not continuous at but that
for an arbitrary vector. Thus the Gateaux derivative exists at
f is not even continuous there.
Show that this function is not continuous at but that the Gateaux derivative
exists and equals 0 for every vector .
- Let U be an open subset of ℝn and suppose that f :
× U → ℝ satisfies
are all continuous. Show that
all make sense and that in fact
Also explain why
is continuous. Hint: You will need to use the theorems from one variable calculus
about the existence of the integral for a continuous function. You may also want to use
theorems about uniform continuity of continuous functions defined on compact
- I found this problem in Apostol’s book . This is a very important result and is
obtained very simply. Read it and fill in any missing details. Let
Explain why this is so. Also show the conditions of Problem 14 are satisfied so
Now use the chain rule and the fundamental theorem of calculus to find f′
change the variable in the formula for f′ to make it an integral from 0 to 1 and
Now this shows f +
g is a constant. Show the constant is
π∕4 by letting x → 0.
Next take a limit as x →∞ to obtain the following formula for the improper integral,
In passing to the limit in the integral for g as x →∞ you need to justify why that
integral converges to 0. To do this, argue the integrand converges uniformly to 0
for t ∈ and then explain why this gives convergence of the integral.
- The gamma function is defined for x > 0 as
Show this limit exists. Note you might have to give a meaning to
if x < 1. Also show that
How does Γ for
n an integer compare with !?
- Show the mean value theorem for integrals. Suppose f ∈ C
. Then there exists
, in fact x can be taken in
, such that
You will need to recall simple theorems about the integral from one variable
- In this problem is a short argument showing a version of what has become known as
Fubini’s theorem. Suppose f ∈ C
First explain why the two iterated integrals make sense. Hint: To prove the
two iterated integrals are equal, let a = x0 < x1 <
< xn = b and
c = y0 < y1 <
< ym = d be two partitions of and
Now use the mean value theorem for integrals to write
do something similar for
and then observe that the difference between the sums can be made as small as desired
by simply taking suitable paritions. We will do Fubini’s theorem better later in the