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# Mathematics: Analysis in One Variable

Course · 15 credits

## Description

The overarching goal of the course is for students to develop understanding of central concepts, results and methods of analysis in one variable, and to apply these methods to solve standard calculus problems for functions in one variable. The course aims for students to develop the ability to communicate mathematics in speech and writing, as well as reading mathematical texts. The course aims additionally to prepare students for further studies in mathematics and natural sciences.

The course treats:

• The real numbers: axioms, examples of proofs of basic arithmetical rules.
• The elementary functions, polynomials, rational functions, the exponential function and the natural logarithm, the trigonometric functions and the inverse trigonometric functions; definitions, basic properties, and quantitative approximations using representations in terms of areas and arclengths.
• Sequences of numbers and their limits: formal definition of the limit, examples of proofs of their computational rules, visual representation of convergence of recursive sequences, quantitative approximations.
• Infinite series: applications and proofs of convergence tests, absolute convergence, quantitative approximations using partial sums and tail estimates.
• Functions and their limits: formal definition of the limit, proofs and applications of their computational rules, indeterminate forms and asymptote
• Continuity: continuity of elementary functions, the intermediate value theorem and the min-max theorem.
• Derivatives: definition, proofs and applications of computational rules, differentiation formulas for elementary functions, Rolle’s lemma, the mean value theorem and L’Hopital’s rule.
• Applications of the derivative: optimisation and graph sketching, techniques for establishing identities and inequalities.
• Indefinite integrals: proofs and applications of basic computational rules and integration methods, such as change of variables, partial integration and use of partial fraction decomposition.
• Definite integrals: Darboux integrability of monotone functions and functions with bounded derivative with related error estimates, the fundamental theorem of calculus, applications to arclength, rotational volumes and surfaces, numerical approximations of definite integrals.
• Improper integrals: convergence criteria for improper integrals for positive functions, absolute convergence, comparison to infinite series.
• Differential equations: direction fields, analytic solution methods for separable and linear first order differential equations, solution method for linear higher- order differential equations with constant coefficients, numerical approximations of solutions of initial value problems using Euler’s method.
• Taylor expansions: Taylor's formula with Lagrange’s formula for the error term, uniqueness theorem for Taylor polynomials, numerical approximations of function values and integrals using Taylor polynomials.
• In addition, materials on sets, functions and relations, induction, the binomial theorem, as well as variables, for-loops and if-statements in Python are covered at the beginning of the course.

## Course documents

Syllabus (PDF - new window)

## Contact

Director of Studies

Phone: +46 46-222 85 39

Email: Anna-Maria [dot] Persson [at] math [dot] lu [dot] se

## Requirements and selection

### Entry requirements

General requirements and studies equivalent of course Mathematics 4/D from Swedish Upper Secondary School.

### Selection criteria

Seats are allocated according to the following: The general average (GPA) of your higher secondary school leaving certificate: 34 %, The Swedish national university aptitude test: 34 %, number of previous ECTS at application deadline (up to 165): 32 %. If students have equal credentials, seats are allocated based on their results on The Swedish National University Aptitude Test. If this too is equal, seats are allocated based on a draw.

### English language requirements

Most of Lund University’s programmes require English Level 6 (unless otherwise stated under 'Entry requirements'). This is the equivalent of an overall IELTS score of 6.5 or a TOEFL score of 90. There are several ways to prove your English language proficiency – check which proof is accepted at the University Admissions in Sweden website. All students must prove they meet English language requirements by the deadline, in order to be considered for admission.

### Country-specific requirements

Check if there are any country-specific eligibility rules for you to study Bachelor's or Master's studies in Sweden:

Country-specific requirements for Bachelor's studies – universityadmissions.se

Country-specific requirements for Master's studies – universityadmissions.se

## Apply

### Start Autumn Semester 2024

Day-time Lund, part time 50%

In English

Study period

2 September 2024 - 19 January 2025

Application

You can only apply for this course in the 'Swedish student' application round. Find out more: Applying for studies – when to apply

Day-time Lund, part time 50%

In English

Study period

2 September 2024 - 19 January 2025

Application

You can only apply for this course in the 'Swedish student' application round. Find out more: Applying for studies – when to apply

### How to apply

Lund University uses a national application system run by University Admissions in Sweden. It is only possible to apply during the application periods.

When to apply for studies

### Step 1: Apply online

• Check that you meet the entry requirements of the programme or course you are interested in (refer to the section above on this webpage).
• Start your application – go to the University Admissions in Sweden website where you create an account and select programmes/courses during the application period.
Visit the University Admissions in Sweden website
• Rank your programme/course choices in order of preference and submit them before the application deadline.

### Step 2: Submit documents

Country-specific requirements for Bachelor's studies – universityadmissions.se
Country-specific requirements for Master's studies – universityadmissions.se

- official transcripts and high school diploma (Bachelor's applicants)
- official transcripts and degree certificate or proof that you are in the final year of your Bachelor's (Master's applicants)
- passport/ID (all applicants) and
- proof of English proficiency (all applicants).

• Prepare programme-specific documents if stated in the next paragraph on this webpage.
• Pay the application fee (if applicable – refer to the section below on this webpage) before the document deadline.

* Note that the process is different if you are applying as an exchange student or as a part of a cooperation programme (such as Erasmus+).
* If you have studied your entire Bachelor's programme in Sweden and all of your academic credits are in Ladok, you do not have to submit transcripts or your diploma when applying for a Master's programme. However, there may still be other documents you need to submit! See the link below.

## Tuition fees

### Non-EU/EEA citizens

Full programme/course tuition fee: SEK 42 500

First payment: SEK 42 500

Convert currency – xe.com

Citizens of a country outside of the European Union (EU), the European Economic Area (EEA) and Switzerland are required to pay tuition fees. You pay one instalment of the tuition fee in advance of each semester.

Tuition fees, payments and exemptions

### EU/EEA citizens and Switzerland

There are no tuition fees for citizens of the European Union (EU), the European Economic Area (EEA) and Switzerland.

### Application fee

If you are required to pay tuition fees, you are generally also required to pay an application fee of SEK 900 when you apply at the University Admissions in Sweden website. You pay one application fee regardless of how many programmes or courses you apply to.

*Note that there are no tuition or application fees for exchange students or doctoral/PhD students, regardless of their nationality.

## Scholarships & funding

### Lund University Global Scholarship programme

The Lund University Global Scholarship programme is a merit-based and selective scholarship targeted at top academic students from countries outside the EU/EEA.

Lund University Global Scholarship

### Swedish Institute Scholarships

The Swedish Institute offers scholarships to international students applying for studies in Sweden at Master's level.

Scholarship information on the Swedish Institute website

### Country-specific scholarships and funding options

Lund University has agreements with scholarship organisations and funding bodies in different countries, which may allow applicants to apply for funding or scholarships in their home countries for their studies at Lund University.

### External scholarships

Information about scholarships from external organisations