Recommended Syllabus
This is the recommended syllabus for the module detailed below. The module should
contain all the topics listed below in some form, but be aware that there may be
additional material covered that can also be examined.
MA225 Differentiation
Differentiable Functions (10 lectures)
(i) Point out that naı̈ve definition of the derivative of a function from Rm in terms
of limit of difference quotient is not possible because one cannot divide by a
vector.
(ii) Definitions of directional derivatives and partial derivatives.
(iii) The differential of a function from Rm to Rn as a linear transformation, uniqueness of the differential.
(iv) Examples: differential of a linear map, derivatives of matrix operations such
as M 7→ M −1 , M 7→ M t M , and M 7→ det M (at M0 = I).
(v) Affine linear approximation of a function by means of the derivative. Tangent
spaces to graphs.
(vi) Properties of differentiable functions: a vector function is differentiable if and
only if each of its components is differentiable; continuity; existence of directional derivatives.
(vii) Partial derivatives and the Jacobian matrix as matrix representation of the
derivative with respect to the standard basis of Rn .
(viii) Examples: a non-differentiable function all of whose partial derivatives exist;
a non-differentiable function all of whose directional derivatives exist.
(ix) Continuity of partial derivatives implies differentiability (with proof),
(x) Notion of C 1 maps.
(xi) Differential of a sum, product and quotient of differentiable functions.
(xii) Composition of differentiable functions and the chain rule (with proof).
(xiii) Examples, including a recall from Geometry & Motion of relation between
derivatives of F and f where F (r, θ) = f (r cos θ, r sin θ).
(xiv) Differential of an inverse function.
(xv) Mean value theorem for a function f : Rn → R. Corollary: dx f = 0 for all x
implies that f is locally constant.
(xvi) Lipschitz continuity of a differentiable function on a convex set.
(xvii) Partial derivatives of higher orders. Definition of C k functions.
(xviii) Commutativity of partial derivatives (with proof).
Implicit Function Theorem and Inverse Function Theorem (10 lectures)
This part contains two big theorems. Currently, (i) is proved using monotonicity
arguments. The proof of (ii) is based on the contraction mapping theorem and its
extension onto uniform families of contractions. It also requires openness of the set
of invertible linear maps.
Inverse function theorem is derived from the implicit function theorem. Each
theorem is illustrated by examples.
(i) Statement and proof of the implicit function theorem for a scalar function.
(ii) Statement and proof of the implicit function theorem for a system of equations.
(iii) Examples including perturbation theory.
(iv) Straightening of level sets away from a critical point as a theoretical application of the Implicit Function Theorem, i. e. if ∇f (x0 ) 6= 0 for some C 1
function f : Rn → R then there exists a diffeomorphism φ : U → V between
open subsets U and V of Rn such that x0 ∈ V and f ◦ φ(y1 , y2 , . . . , yn ) =
y1 ∀ (y1 , y2 , . . . , yn ) ∈ U.
(v) Statement and proof of the inverse function theorem.
(vi) Definition of a diffeomorphism.
(vii) Statement of the inverse function theorem using the coordinate notation.
(viii) Examples designed to illustrate importance of individual assumptions of the
inverse function theorem.
(ix) Optional topic: The set of local diffeomorphisms is open in C 1 topology but
not in C 0 topology.
Applications (10 lectures)
(i) Taylor’s theorem
(ii) Notion of a critical point, different types of critical points: maximum, minimum and saddle. (The two dimensional case was covered in Differential Equations, Term 1, Year 1. Temptation to mention the Morse Lemma must be
restricted to a statement. Recall of definiteness of symmetric bilinear forms
from the proposed Higher Linear Algebra)
(iii) Necessary conditions for a local minimum; sufficient condition for a local minimum (with proofs).
(iv) Definition of submanifold of Rn ; tangent (sub)space. Lots of examples: sphere,
orthogonal group, special linear group, etc.
(v) Conditional minimum, Lagrange multipliers.
Last updated 12th November 2008