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


Department
Mathematics
Course Code
MAT237Y1
Professor
John Bland
Study Guide
Final

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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:UV, 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:UVas above, such that
– f is is a bijection (that is, both one-to-one and onto). This implies that f1:
VUexists.
Both fand f1are 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:URn, where Uis an open subset of Rn. If we define Vto
be the image of U(so that f:UVis onto), it may be hard to know whether f
is one-to-one (which would imply that f1exists) and if so, whether f1is 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.
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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
UVwhere
(2) U={(r, θ) : r > 0,|θ|< π}, V := R2\ {(x, 0) : x0}.
In this case, f1:VUexists 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:UVis a mapping of class C1.
Assume that aUis a point such that
(3) Df(a)is invertible,
and let b:= f(a). Then there exist open sets MUand NVsuch that
aMand bN,
fis one-to-one from Monto N(hence invertible), and
the inverse function f1:NMis of class C1.
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