KCL 4CCM112A Calculus II: Syllabus, Topics and Practice
A study guide for Calculus II (4CCM112A) at King's College London: partial derivatives, multiple integrals, vector fields, Stokes and Divergence theorems.
King's College London Updated 3 min read
At a glance
- Module code
- 4CCM112A
- Credits
- 15 credits, Level 4 (Year 1)
- Semester
- Spring
- Entry requirements
- The topics of 4CCM111A Calculus I
- Covers
- Surface sketching, partial derivatives, multiple integrals, geometry of curves and surfaces, vector fields, maxima and minima, generalised derivatives, Stokes' theorem and the Divergence Theorem
- Previous module
- 4CCM111A Calculus I
- Practice here
- Calculus 3
Try it yourself
On this page
4CCM112A Calculus II is the spring-semester calculus module for Year 1 mathematics students at King's College London, and a required module in the Mathematics BSc. It picks up where Calculus I leaves off and takes calculus into several dimensions. This guide sets out what King's says the module covers and maps each topic to free practice so you can learn every skill by doing it.
What KCL Calculus II covers
King's says the module aims to extend the methods of one-variable calculus to functions of many variables, that is, calculus on higher-dimensional spaces. Multiple integrals and partial derivatives let you make sense of the length of a curve, the area of a surface, and maxima and minima of functions of several variables. The final part presents the great integral theorems: Green's theorem, Stokes' theorem and the Divergence Theorem, which the module page calls a cornerstone of mathematics.
The official syllabus lists: surface sketching, partial derivatives, multiple integrals, geometry of curves, vector fields, geometry of surfaces, maxima and minima, generalised derivatives, Stokes' theorem and the Divergence Theorem.
The name can be confusing. At many universities "Calculus 2" means integration techniques and series in one variable. KCL Calculus II is multivariable calculus, which matches a Calculus 3 course elsewhere.
4CCM112A topic map
| 4CCM112A topic | Practice | Read |
|---|---|---|
| Surface sketching | Functions of several variables | Partial derivatives guide |
| Partial derivatives | Partial derivatives | Clairaut's theorem |
| Generalised derivatives and the chain rule | Multivariable chain rule | Gradient and directional derivatives |
| Gradient and tangent planes | Gradient and directional derivatives | Practice problems |
| Maxima and minima | Critical points and the second derivative test | Lagrange multipliers |
| Multiple integrals | Iterated integrals | Double integrals |
| Changing coordinates in integrals | Change of variables | Double integral problems |
| Geometry of curves | Curves and motion | Vectors, dot and cross product |
| Vector fields | Vector fields | Line integrals of vector fields |
| Geometry of surfaces | Surface integrals and flux | Cross product |
| Green's theorem | Green's theorem | Green's theorem |
| Stokes' theorem and the Divergence Theorem | Stokes and divergence theorems | Stokes' theorem, Divergence Theorem |
Year 1 calculus at King's
| Module | Semester | What it adds |
|---|---|---|
| 4CCM111A Calculus I | Autumn | Single-variable calculus from first principles, series and Taylor's theorem |
| 4CCM115A Sequences and Series | Autumn | Rigorous limits and convergence |
| 4CCM112A Calculus II | Spring | Several variables, multiple integrals and the integral theorems |
Example problems at the 4CCM112A level
1. Partial derivatives. For f(x, y) = x^2 y + \sin(xy),
2. Classifying a critical point. For f(x, y) = x^2 + 3y^2 - 2x, the only critical point is (1, 0). Since f_{xx} f_{yy} - f_{xy}^2 = 2 \cdot 6 - 0 > 0 and f_{xx} > 0, it is a local minimum.
3. A double integral.
4. The Divergence Theorem. For \mathbf{F} = (x, y, z) and the unit ball, \nabla \cdot \mathbf{F} = 3, so the outward flux through the sphere is 3 \cdot \frac{4}{3}\pi = 4\pi.
How to do well in KCL Calculus II
- Draw pictures. Surface sketching is on the syllabus for a reason. A quick sketch of the region usually tells you the limits of a multiple integral.
- Practise partial derivatives until they are automatic. Every later topic, from maxima and minima to the integral theorems, depends on them.
- Keep your Calculus I skills sharp. Multiple integrals are just repeated single integrals, so slips in u-substitution or integration by parts cost marks here too.
- Hand in every weekly sheet and use the tutorial to ask about anything you got stuck on.
- Revise with timed problems using practice exams before the summer exam.
Tip: Green's, Stokes' and the Divergence Theorem all say the same thing in different dimensions: an integral of a derivative over a region equals an integral over its boundary. Seeing that pattern makes all three easier to remember.
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Partial derivatives
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