# 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.

Source: https://practicecalculus.com/courses/university-of-edinburgh-math08082
Updated: 2026-11-08

## 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](/learn/calculus-3)

## Key takeaways

- MATH08082 is a Year 2 course that follows directly from Introduction to Mathematical Analysis (MATH08081).
- It starts with a rigorous treatment of integration in one variable, then develops differential and integral calculus in several variables.
- Linear algebra is used throughout: the derivative is the Jacobian matrix of partial derivatives.
- It ends with Stokes' theorem and the divergence theorem as the higher-dimensional Fundamental Theorem of Calculus.

**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](/courses/university-of-edinburgh-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:

1. **Single-variable integration**, treated rigorously and following the structure of Introduction to Mathematical Analysis, with a strong emphasis on both proof and calculus skills.
2. **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.
3. **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](/learn/calculus-1/integrals/riemann-sums) | [Riemann sums](/blog/riemann-sums), [definite integral properties](/rules/definite-integral-properties) |
| Fundamental Theorem of Calculus | [FTC part 1](/learn/calculus-1/integrals/fundamental-theorem-part-1) | [FTC part 1](/theorems/fundamental-theorem-of-calculus-part-1) |
| Functions on $\mathbb{R}^n$ | [Functions of several variables](/learn/calculus-3/partial-derivatives/functions-of-several-variables) | [Vectors, dot and cross product](/blog/vectors-dot-and-cross-product) |
| Partial derivatives | [Partial derivatives](/learn/calculus-3/partial-derivatives/partial-derivatives) | [Partial derivatives](/blog/partial-derivatives), [Clairaut's theorem](/theorems/clairauts-theorem) |
| The derivative and the chain rule | [Multivariable chain rule](/learn/calculus-3/partial-derivatives/multivariable-chain-rule) | [Partial derivatives practice](/practice-problems/partial-derivatives) |
| Gradient | [Gradient and directional derivatives](/learn/calculus-3/partial-derivatives/gradient-and-directional-derivatives) | [Gradient and directional derivatives](/blog/gradient-and-directional-derivatives) |
| Critical points and extremal problems | [Second derivative test](/learn/calculus-3/partial-derivatives/critical-points-and-second-derivative-test) | [Lagrange multipliers](/blog/lagrange-multipliers) |
| Double and triple integrals | [Iterated integrals](/learn/calculus-3/multiple-integrals/iterated-integrals) | [Double integrals](/blog/double-integrals) |
| Polar coordinates and change of variables | [Polar double integrals](/learn/calculus-3/multiple-integrals/polar-double-integrals) | [Double integrals practice](/practice-problems/double-integrals) |
| Line integrals | [Line integrals of vector fields](/learn/calculus-3/vector-calculus/line-integrals-of-vector-fields) | [Green's theorem](/theorems/greens-theorem) |
| Divergence of vector fields | [Curl and divergence](/learn/calculus-3/vector-calculus/curl-and-divergence) | [Divergence theorem](/theorems/divergence-theorem) |
| Surface integrals, Stokes and divergence theorems | [Stokes and the divergence theorem](/learn/calculus-3/vector-calculus/stokes-and-divergence-theorems) | [Stokes' theorem](/theorems/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 $f(x, y) = x^2 y^3$,

$$\nabla f = \left(2xy^3,\; 3x^2y^2\right)$$

**2. A double integral in polar coordinates.** Over the unit disc $D$,

$$\iint_D (x^2 + y^2)\,dA = \int_0^{2\pi}\int_0^1 r^2 \cdot r\,dr\,d\theta = 2\pi \cdot \frac{1}{4} = \frac{\pi}{2}$$

**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](/learn/calculus-3/multiple-integrals/general-regions) come from getting the limits wrong, not from the integration.
- **Practise the big theorems on easy examples.** Checking [Green's theorem](/learn/calculus-3/vector-calculus/greens-theorem) on a square builds intuition for Stokes and the divergence theorem.
- **Keep integration techniques sharp.** Inner integrals often need [substitution](/rules/u-substitution-rule) or [parts](/rules/integration-by-parts-rule).

> **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.

## Frequently asked questions

### What is MATH08082 at Edinburgh?

Further Analysis and Several Variable Calculus is a 20-credit Year 2 course. The catalogue describes it as a second course in mathematical analysis together with calculus in several variables, following on directly from Introduction to Mathematical Analysis.

### What are the prerequisites for MATH08082?

Students must have passed Introduction to Mathematical Analysis (MATH08081), Linear Algebra 1 (MATH08079) and Linear Algebra 2 (MATH08080), or Linear Algebra 2 together with Accelerated Algebra and Analysis (MATH08083).

### Can I take MATH08082 with Engineering Mathematics 2A or 2B?

No. The catalogue lists Engineering Mathematics 2A (SCEE08009) and Engineering Mathematics 2B (SCEE08010) as prohibited combinations.

### Is MATH08082 more theory or computation?

Both. The single-variable integration section is rigorous, while the several-variable sections put more weight on conceptual understanding, calculation and applications, with theory where appropriate. The catalogue stresses mastery of multivariable calculations.

### How should I prepare for MATH08082?

Review integration techniques and the Fundamental Theorem of Calculus, then get comfortable with partial derivatives and double integrals before the semester starts. The Calculus 3 path on PracticeCalculus covers all the computational topics.