Edinburgh MATH08082 Further Analysis and Several Variable Calculus
A study guide for MATH08082 at Edinburgh: rigorous integration, partial derivatives, multiple integrals, Stokes and the divergence theorem, with practice.
University of Edinburgh Updated 3 min read
At a glance
- Course code
- MATH08082
- Credits
- 20 SCQF credits (10 ECTS), SCQF Level 8
- Taught
- Year 2, Semester 2
- Prerequisites
- MATH08081, Linear Algebra 1 (MATH08079) and Linear Algebra 2 (MATH08080); or MATH08080 and Accelerated Algebra and Analysis (MATH08083)
- Assessment
- Written exam 80%, coursework 20%
- Covers
- Rigorous single-variable integration, differentiation on R^n, the Jacobian, critical points, double and triple integrals, line and surface integrals, Stokes and divergence theorems
- Textbook
- Marsden and Tromba, Vector Calculus
- Practice here
- Calculus 3
Try it yourself
On this page
MATH08082 Further Analysis and Several Variable Calculus is the Year 2 course that continues the analysis sequence at the University of Edinburgh, taught in Semester 2 and worth 20 SCQF credits (10 ECTS). It picks up exactly where MATH08081 stops: first a rigorous look at integration in one variable, then a full course in multivariable calculus. This guide maps each topic to free practice so you can master the calculations while the lectures supply the theory.
What MATH08082 covers
The Edinburgh course catalogue (DRPS) describes three parts:
- Single-variable integration, treated rigorously and following the structure of Introduction to Mathematical Analysis, with a strong emphasis on both proof and calculus skills.
- Differential calculus in several variables: limits and continuity on \mathbb{R}^n, differentiation of scalar and vector-valued functions, gradient and divergence of vector fields, the Jacobian matrix as the matrix of the derivative, and critical points applied to extremal problems.
- Integral calculus in several variables: double and triple integrals, polar coordinates, line and surface integrals, and Stokes' theorem and the divergence theorem as the higher-dimensional Fundamental Theorem of Calculus.
The catalogue says the language of linear algebra is used to develop calculus in several variables, with a strong emphasis on calculation and geometry. The reading list is Marsden and Tromba, Vector Calculus.
MATH08082 topic map
| MATH08082 topic | Practice | Read |
|---|---|---|
| Riemann integration | Riemann sums | Riemann sums, definite integral properties |
| Fundamental Theorem of Calculus | FTC part 1 | FTC part 1 |
| Functions on \mathbb{R}^n | Functions of several variables | Vectors, dot and cross product |
| Partial derivatives | Partial derivatives | Partial derivatives, Clairaut's theorem |
| The derivative and the chain rule | Multivariable chain rule | Partial derivatives practice |
| Gradient | Gradient and directional derivatives | Gradient and directional derivatives |
| Critical points and extremal problems | Second derivative test | Lagrange multipliers |
| Double and triple integrals | Iterated integrals | Double integrals |
| Polar coordinates and change of variables | Polar double integrals | Double integrals practice |
| Line integrals | Line integrals of vector fields | Green's theorem |
| Divergence of vector fields | Curl and divergence | Divergence theorem |
| Surface integrals, Stokes and divergence theorems | Stokes and the divergence theorem | Stokes' theorem |
Where MATH08082 fits
| Course | When | Focus |
|---|---|---|
| MATH08081 Introduction to Mathematical Analysis | Year 1, Semester 2 | Sequences, series, limits, continuity, differentiation |
| MATH08079 and MATH08080 Linear Algebra 1 and 2 | Year 1 | Vectors, matrices, linear maps |
| MATH08082 Further Analysis and Several Variable Calculus | Year 2, Semester 2 | Rigorous integration and multivariable calculus |
Example problems at the MATH08082 level
1. A partial derivative and the gradient. For f(x, y) = x^2 y^3,
2. A double integral in polar coordinates. Over the unit disc D,
3. Classifying a critical point. For f(x, y) = x^2 - y^2, the only critical point is (0, 0). Since f_{xx}f_{yy} - f_{xy}^2 = (2)(-2) - 0 = -4 < 0, it is a saddle point.
How to do well in MATH08082
- Refresh linear algebra early. The Jacobian, change of variables and critical point tests all lean on matrices and determinants.
- Draw the region first. Most mistakes in double integrals come from getting the limits wrong, not from the integration.
- Practise the big theorems on easy examples. Checking Green's theorem on a square builds intuition for Stokes and the divergence theorem.
- Keep integration techniques sharp. Inner integrals often need substitution or parts.
Tip: think of Stokes' theorem and the divergence theorem as the Fundamental Theorem of Calculus in higher dimensions, exactly as the course does. Each relates an integral over a region to an integral over its boundary.
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