- For ≠, let f=. Show that this function has a limit as→foron 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 ℝ
^{n}, v =.- v
_{1}v_{2} - v
_{2}sin - v
_{1}^{2}+ v_{2} - v
_{2}sin - v
_{1}

- v
- Here is a function of two variables. f= x
^{2}y + x^{2}. Find Dfdirectly 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= gHere x ∈ ℝ
^{n},f∈ ℝ^{m}and g∈ ℝ^{p}. where f,g are appropriately differentiable. Thus Dhresults from multiplication by a matrix. Using the chain rule, give a formula for the ij^{th}entry of this matrix. How does this relate to multiplication of matrices? In other words, you have two matrices which correspond to Dgand DfCall z = g,y = f. ThenExplain the manner in which the ij

^{th}entry of DhisThis is a review of the way we multiply matrices. what is the i

^{th}row of Dgand the j^{th}column of Df? - Find f
_{x},f_{y},f_{z},f_{xy},f_{yx},f_{zy}for the following. Verify the mixed partial derivatives are equal.- x
^{2}y^{3}z^{4}+ sin - sin+ x
^{2}yz

- x
- 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 f
_{xi}= 0 for each x_{i}. Hint: Consider f= h. Argue that h^{′}= 0 and then see what this implies about Df. - As an important application of Problem 7 consider the following. Experiments are done at n times,
t
_{1},t_{2},,t_{n}and at each time there results a collection of numerical outcomes. Denote by_{i=1}^{p}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, ∑_{i=1}^{p}^{2}is as small as possible. In other words, you want to minimize the function of two variables f≡∑_{i=1}^{p}^{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= fsolves the wave equation u
_{tt}− c^{2}Δu = 0. What about u= f? Here Δ u = u_{xx}. - Show that if Δu = λu where u is a function of only x, then e
^{λt}u solves the heat equation u_{t}− Δu = 0. Here Δu = u_{xx}. - Show that if f= o, then f
^{′}= 0. - Let fbe defined on ℝ
^{2}as follows. f= 1 if x≠0. Define f= 0, and f= 0 if y≠x^{2}. Show that f is not continuous atbut thatfor

an arbitrary vector. Thus the Gateaux derivative exists atin every direction but f is not even continuous there. - Let
Show that this function is not continuous at

but that the Gateaux derivativeexists and equals 0 for every vector

. - Let U be an open subset of ℝ
^{n}and suppose that f :× U → ℝ satisfiesare 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 sets.

- I found this problem in Apostol’s book [1]. This is a very important result and is obtained very
simply. Read it and fill in any missing details. Let
and

Note

Explain why this is so. Also show the conditions of Problem 14 are satisfied so that

Now use the chain rule and the fundamental theorem of calculus to find f

^{′}. Then change the variable in the formula for f^{′}to make it an integral from 0 to 1 and showNow this shows f

+ gis 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, ∫_{0}^{∞}e^{−t2 }dt,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. Thus - 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 x ∈, in fact x can be taken in, such that
You will need to recall simple theorems about the integral from one variable calculus.

- In this problem is a short argument showing a version of what has become known as Fubini’s
theorem. Suppose f ∈ C. Then
First explain why the two iterated integrals make sense. Hint: To prove the two iterated integrals are equal, let a = x

_{0}< x_{1}<< x_{n}= b and c = y_{0}< y_{1}<< y_{m}= d be two partitions ofandrespectively. Then explain whydo something similar for

and then observe that the difference between the sums can be made as small as desired by simply taking suitable paritions.

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