Calculus 2 tutoring, online: from the domain to the result.
Functions of several variables, double and triple integrals, curves, vector fields and surfaces: Calculus 2 changes the way you reason, because geometry comes before calculation. We work 1-to-1 online on exercises from your course, from sketching the domain to checking the result.
Let’s start from where you are stuck now.
Who asks us for help with Calculus 2.
- You passed Calculus 1 but several variables leave you lost: partial derivatives, gradient, Hessian matrix.
- You get multiple integrals wrong because of the domain or the change of variables, not the calculation itself.
- You are preparing the oral: the theorems of Green, divergence and Stokes, conservative fields, Fourier series.
- You are resitting and want to understand why the written paper went badly even though you knew the formulas.
How we work on Calculus 2.
- The sketch before the calculation: for every exercise we build the domain, the curve or the surface together, and only then choose coordinates and order of integration.
- Hypotheses next to results: every time a theorem is used we write down what its validity depends on, the same discipline the oral requires.
- Timed practice with a practice test and a guided review of the mistakes, before the exam.
What Calculus 2 covers.
Contents change from one course and one lecturer to another: this is the structure we meet most often. Your course syllabus has the last word: in the first lesson we read it together and work on that.
- Several variables
- Limits, continuity and partial derivatives
- Gradient, differentiability and tangent plane
- Free maxima and minima with the Hessian matrix
- Constraints and Lagrange
- Constrained extrema and Lagrange multipliers
- Implicit functions and level curves
- Extrema on a closed domain: interior and boundary
- Multiple integrals
- Double integrals, reduction formulas
- Change of variables: polar, cylindrical, spherical
- Triple integrals, volumes and centroids
- Curves, fields and surfaces
- Line integrals and the work of a field
- Conservative fields and potentials
- Flux, the divergence, Green and Stokes theorems
- Series and equations
- Power series and radius of convergence
- Fourier series
- Linear second-order differential equations
How the exam is usually structured.
The Calculus 2 written paper usually has three or four exercises of different kinds: a double or triple integral, a maxima and minima problem, an exercise on a vector field or a surface and, depending on the course, a series or a differential equation. Here too the format depends on the lecturer, and past papers are the best guide.
Unlike Calculus 1, almost no exercise is solved with a single technique: the step that decides the mark is often the first one, that is, drawing the domain and understanding the geometry of the problem. Whoever starts calculating straight away tends to integrate over the wrong region and obtain a clean but wrong number.
The oral, when there is one, revolves around the theorems, the difference between a conservative and an irrotational field, and the hypotheses on domains. That is why in lessons we always keep a sketch and a hypothesis next to every result.
Three mistakes that cost marks.
- Forgetting the change-of-variables factor
Switching to polar, cylindrical or spherical coordinates changes the area or volume element: you need the factor r in polar and cylindrical, ρ² sin φ in spherical. Without the Jacobian the working is tidy and the result is wrong.
- Saying «irrotational, hence conservative»
Zero curl is a necessary condition; to be sufficient the domain must be simply connected. On a domain with a hole, such as the plane minus the origin, a field can have zero curl and no potential. Check the domain before writing «the potential exists».
- Looking for extrema only in the interior
On a closed domain finding the stationary points is not enough: the maximum and the minimum can sit on the boundary, where the gradient does not vanish. You need to parametrise the boundary, or use Lagrange, and compare all the candidates.
An exam-style exercise, step by step.
A double integral in polar coordinates
Let us integrate over the quarter of an annulus
Before calculating, we draw the domain: it is the part in the first quadrant between two circles of radius and . The natural choice is polar coordinates.
Set , . The domain becomes , and the area element is : the factor is the Jacobian, the one that gets forgotten. Also .
The domain is a rectangle in the variables , so the integral separates:
Without the factor we would have integrated instead of and obtained instead of : a clean number, but a wrong one.
Timed Calculus 2 practice test.
A test that mixes the topics of the syllabus the way a real exam sitting does, with marking and weak points. It is free and you do not need an account to try it.
Frequently asked questions about Calculus 2.
Is Calculus 2 harder than Calculus 1?
It is different rather than harder: less repetitive calculation, more geometry and more hypotheses to check. Students who were good at the calculations of Calculus 1 often stumble at first, because here choosing the method comes before the calculation.
Do I need to be able to draw domains?
Yes, and it is the first skill we train: domain, curve or surface drawn by hand before writing an integral. With a correct sketch the choice of coordinates and limits becomes almost automatic.
Are the Gauss and Stokes theorems only for the oral?
No, they also appear in the written paper, often as the quickest way to compute a flux or a circulation. The point is not remembering the formula but knowing when the hypotheses hold and how to orient the surface and its boundary.
Do you also cover Fourier series and differential equations?
Yes, they are part of the typical Calculus 2 syllabus and we treat them with the same method: exam exercises, written steps and a final check of the result.
The other exams we cover.
- Calculus 1
Limits, derivatives, integrals and series: exam-style exercises and an oral on the hypotheses of the theorems.
See the page → - Linear algebra
Matrices, vector spaces, eigenvalues and quadratic forms: the vocabulary matters as much as the calculations.
See the page → - Physics 1
Mechanics and thermodynamics: the method for setting up the problem, symbols before numbers.
See the page →
If you first need the basics of the previous course, see the page on Calculus 1.
For the full picture go to university tutoring and the maths syllabus.
From the domain to the result, together.
Message me on WhatsApp with your course and your last paper: we start from the exercises where you got stuck.