# MAT237Y1 Study Guide - Final Guide: Inverse Function Theorem, Folland Aircraft, Multivariable CalculusExam

by OC1223278

Department

MathematicsCourse Code

MAT237Y1Professor

John BlandStudy Guide

FinalThis

**preview**shows page 1. to view the full**5 pages of the document.**University of Toronto

MAT237Y: Multivariable Calculus,

2017-8, Notes 9

1. summary

1.1. topics. These notes discuss the Inverse Function Theorem and coordinate transfor-

mations, covered at the end of the Fall Term. This corresponds to Section 3.4 in Folland’s

Advanced Calculus.

1.2. Tasks for students.

•Read these notes. Section 3.4 in Foland’s Advanced Calculus is an additional refer-

ence that you may ﬁnd useful.

•Start working on practice problems, see Section 3 below.

2. Discussion of material

2.1. Summary.

•In this section we are interested in functions f:U→V, where Uand Vare open

subsets of Rn. As Folland discusses (Section 3.4), with nice pictures to ilustrate the

point, such functions (whch he calls “transformations”) are best visualized using

“before and after” sketches, particularly when n= 2.

•We are especially interested in functions f:U→Vas above, such that

– f is is a bijection (that is, both one-to-one and onto). This implies that f−1:

V→Uexists.

–Both fand f−1are of class C1.

Such a transformation fmay sometimes be considered a change of coordinates.

•There are a few particular changes of coordinates that one encounters again and

again In math and related ﬁelds, such as the ones that deﬁne polar or spherical

coordinates. For these important examples, typically there are explicit formulas for

the inverses.

•Consider a general f:U→Rn, where Uis an open subset of Rn. If we deﬁne Vto

be the image of U(so that f:U→Vis onto), it may be hard to know whether f

is one-to-one (which would imply that f−1exists) and if so, whether f−1is C1.

However, the Inverse Function Theorem can provide (local) answers to these

questions. See below for details.

2.2. How to visualize a transformation. For this, it is a good idea to read Section 3.4

in Folland’s Advanced Calculus, at least through the middle of p. 136, and to look carefully

at the examples and the associated pictures. See also the practice problems below.

2.3. Some important coordinate systems.

1

###### You're Reading a Preview

Unlock to view full version

Only page 1 are available for preview. Some parts have been intentionally blurred.

2

2.3.1. polar coordinates in R2.As we know, polar coordinates (r, θ) are related to cartesian

coordinates (x, y) by

(1) x

y=rcos θ

rsin θ=f(r, θ).

For fto be a bijection between open sets, we have to restrict its domain and range. For

example, one comon choice (among many possibilities) is to specify that fis a function

U→Vwhere

(2) U={(r, θ) : r > 0,|θ|< π}, V := R2\ {(x, 0) : x≤0}.

In this case, f−1:V→Uexists and is of class C1. This follows from the implicit function

theorem (see below).

2.3.2. spherical coordinates in R3.Spherical coordinates (r, θ, ϕ) (r, θ) are related to carte-

sian coordinates (x, y) by

x

y

z

=

rcos θsin ϕ

rsin θsin ϕ

rcos ϕ

=f(r, θ, ϕ) = f(r, θ, ϕ).

See the practice problems.

As above, if we want fto be a bijection between open sets Uand V, it is necessary to

restrict the domain and range in some appropriate way.

2.3.3. cylindrical coordinates in R3.Cylindrical coordinates (r, θ, z) are related to cartesian

coordinates (x, y, z) by

x

y

z

=

rcos θ

rsin θ

z

=f(r, θ, ϕ) = f(r, θ, ϕ).

These are very closely related to polar coordinates in R2.

2.4. The Inverse Function Theorem. (also sometimes called the Inverse Mapping The-

orem)

The following theorem tells us when a transformation of class C1has a local inverse of

class C1.

Theorem 1 (Inverse function theorem).Let Uand Vbe open sets in Rn, and assume that

f:U→Vis a mapping of class C1.

Assume that a∈Uis a point such that

(3) Df(a)is invertible,

and let b:= f(a). Then there exist open sets M⊂Uand N⊂Vsuch that

•a∈Mand b∈N,

•fis one-to-one from Monto N(hence invertible), and

•the inverse function f−1:N→Mis of class C1.

###### You're Reading a Preview

Unlock to view full version