This is the course page for the UCL first year mathematics/BASc course MATH1403, *Mathematical Methods for Arts and Sciences*, which ran in the first term of the 2013–14 year.

I have no office hour in the second or third terms, but you are free to contact me to schedule a meeting at a mutually convenient time.

Handouts, coursework and useful resources are posted both here and on Moodle.

If you have any queries or problems, you can email me. I will also pick up anything on Twitter with the hashtag **#MATH1403**.

### Handouts

- Syllabus (29/05/2013)
- Handout 0: Something for the summer (26/06/2013)
- Handout 1: Syllabus and introduction (30/09/2013)
- Handout 2: Greek letters (30/09/2013)
- Handout 3: Number sets (07/10/2013)
- Handout 4: Hyperbolic identities (07/10/2013)
- Handout 5: Inverse functions (14/10/2013)
- Handout 6: Taylor series (17/10/2013)
- Handout 7: What
*is*integration? (22/10/2013) - Handout 8: Complex logarithms (21/11/2013)
- Handout 9: Second order ODE derivation (28/11/2013)
- Handout 10: Second order ODE cheatsheet (28/11/2013)
- Handout 11: Fourier series in pictures (05/12/2013)
- Handout 12: Fourier series orthogonality relations (09/12/2013)
- Handout 13: Revision advice (12/12/2013)

### Problem sheets

- Problem sheet 1 (due 5pm 09/10/2013)
- Problem sheet 2 (due 2.30pm 16/10/2013)
- Problem sheet 3 (due 5pm 23/10/2013)
- Problem sheet 4 (due 5pm 30/10/2013)
- Problem sheet 5 (due 5pm 13/11/2013)
- Problem sheet 6 (due 5pm 20/11/2013)
- Problem sheet 7 (due 5pm 27/11/2013)
- Problem sheet 8 (due 5pm 04/12/2013)
- Problem sheet 9 (due 5pm 11/12/2013)

### Lecture topics

- (30/09/2013)
**Differentiation**: Introduction, definition of differentiation, standard results, chain rule, product rule, quotient rule. - (01/10/2013) Standard results using the quotient rule, implicit differentiation, standard results using implicit differentiation, evil test.
- (03/10/2013) Stationary points, local and global maxima and minima, why we might not be able to differentiate a function.
- (07/10/2013) Introduction to hyperbolic functions, hperbolic identities, inverse hyperbolic functions.
- (08/10/2013) Differentiating hyperbolic functions.
- (10/10/2013) Differentiating inverse hyperbolic functions. The theory behind partial differentiation.
- (14/10/2013) A practical approach to partial differentiation
- (15/10/2013) Finding tangent planes to 3D graphs.
- (17/10/2013) Maclaurin series: intuition and how to find it.
- (21/10/2013) Taylor series: definition and examples.
- (22/10/2013) Using Taylor series to solve differential equations and perform otherwise hideous integrations.
**Integration**: what*is*integration? Partial fractions. - (24/10/2013) Integration using substitution.
- (27/10/2013) Integration by parts.
- (29/10/2013) Integration using reduction formulae / recurrence relations.
- (30/10/2013) Improper integrals (being careful with infinity)
- (11/11/2013)
**Complex numbers**: introduction, arithmetic, complex conjugates. - (12/11/2013) Argand diagram, modulus-argument form of a complex number. Multiplying complex numbers in modulus-argument form.
- (14/11/2013) Dividing complex numbers in modulus-argument form. Exponential form of a complex number. Euler’s identity.
- (19/11/2013) Exponential form of complex numbers, de Moivre’s theorem.
- (20/11/2013) Roots of unity, roots of complex numbers.
- (22/11/2013) Complex logarithms.
- (25/11/2013)
**Differential equations**: what is a differential equation?, separable equations, exact solutions, deriving the integrating factor. - (26/11/2013) Examples of solving first order ODEs using the integrating factor and substitution.
- (28/11/2013) Second order ODEs: derivation of solution form, some easy examples.
- (02/12/2013) Inhomogeneous second order ODEs, complementary function and particular integral, particular integral when it’s part of the complementary function, some examples.
- (03/12/2013) Examples, (Cauchy–)Euler’s equation.
- (05/12/2013)
**Fourier series**: motivation, periodic functions, trigonometric series. - (09/12/2013) Expressions for
*a*and_{n}*b*, Fourier series of a square wave._{n} - (10/12/2013) Fourier series of the sawtooth wave. Even and odd functions.
- (12/12/2013)
**Revision discussion**and chocolate fountains!

### Relevant Khan Academy/MIT videos

- (L1) KA: Definition of differentiation
- (L1) KA: Standard results: derivative of
*x*^{n} - (L1) KA: Chain rule (more, even more)
- (L1) KA: Product rule
- (L1) KA: Quotient rule (note he argues that the quotient rule is unnecessary, since it comes directly from the product rule. I find that people tend to make mistakes with negative powers which the quotient rule avoids)
- (L2) KA: Implicit differentiation (more, even more, even more still)
- (L3) KA: Maxima and minima
- (L4-6) KA: Hyperbolic functions
- (L7-8) KA: Partial differentiation (more) (note he approaches the topic using vectors, which is a sensible thing to do but since vectors are not part of our course, I have avoided doing)
- (L9-11) KA: Taylor series: intuition, Maclaurin series of cos(
*x*), of sin(*x*), of*e*.^{x} - (L11) KA: Partial fractions (more, even more)
- (L12) KA: Integration using substitution (more, even more, even more still)
- (L13) KA: Integration by parts

(L13) MIT: Integration by parts (up to example 2) - (L14) MIT: Reduction formulae (examples 3 and 4)
- (L15) MIT: Improper integrals
- (L16) KA: Introduction to
*i*, the square root of -1 - (L16) KA: Complex numbers (part 0, part 1, part 2)
- (L16) KA: Complex conjugate example
- (L17-19) KA: Argand diagram, modulus-argument form of complex numbers, exponential form of complex numbers
- (L20) KA: Roots of complex numbers
- (L22) KA: What is a differential equation?
- (L22) KA: Separable differential equations (more)
- (L22) KA: Solving first order ODEs using the integrating factor (more)
- (L23) KA: Solving first order ODEs using a change of variables (more)
- (L24) KA: Homogeneous second order ODEs: if
*y*_{1}and*y*_{2}are solutions, showing that*Ay*_{1}+*By*_{2}is a solution - (L24) KA: Homogeneous second order ODEs: general solution for real, distinct roots of the auxiliary equation (note the way he just chooses
*y*=*e*is a fudge: we derived it properly!)^{rx} - (L24) KA: Homogeneous second order ODEs: showing that the cos-sin form for complex roots of the auxiliary equation is equivalent to the
*e*form (part 2 with an example) (note he calls the auxiliary equation the ‘characteristic equation’)^{λx} - (L25) KA: Homogeneous second order ODEs: specific solution for real, distinct roots of the auxiliary equation with two boundary conditions (more)
- (L25) KA: Homogeneous second order ODEs: specific solution for complex roots of the auxiliary equation with two boundary conditions
- (L25) KA: Homogeneous second order ODEs: specific solution for repeated real roots of the auxiliary equation with two boundary conditions
- (L25-26) KA: Inhomogeneous second order ODEs: derivation of particular integral + complementary function form of solution and example (more, even more, even more still) (note he calls this the method of undetermined coefficients—everything has to have a name, I guess. I’ve never heard it called this, though. Maybe an American thing.)
- (L26) YouTube: Euler’s method for solving second order equations of the form
*x*^{2}*y”*+*axy’*+*by*=*f*(*x*) - (L27) MIT: Deriving Fourier series
- (L28) MIT: Fourier series of a square wave
- (L28-29) MIT: Manipulating Fourier series