# York MATH 2015 Multivariate and Vector Calculus: Topics and Practice

> A study guide for York University MATH 2015: partial derivatives, grad, div and curl, multiple integrals, line and surface integrals, Green, Gauss, Stokes.

Source: https://practicecalculus.com/courses/york-university-math-2015
Updated: 2026-11-08

## At a glance

- **Course code:** SC/MATH 2015 3.00
- **Credits:** 3.00 (one semester)
- **Prerequisites:** One of SC/MATH 1021 or 1025; one of SC/MATH 1010, 1014 or 1310; or SC/MATH 1505 plus permission of the course coordinator
- **Exclusions:** SC/MATH 2010, 2310, GL/MATH 2670, GL/MODR 2670, GL/MATH 3200
- **Covers:** Functions of several variables, arc length and curvature, partial derivatives, grad, div, curl and Laplacian, extrema and multivariate Taylor series, double and triple integrals, line and surface integrals, theorems of Green, Gauss and Stokes
- **Practice here:** [Calculus 3](/learn/calculus-3)

## Key takeaways

- MATH 2015 is a one-semester, 3-credit multivariable and vector calculus course at York for engineering and physical science students.
- It covers partial derivatives, the grad, div, curl and Laplacian operators, extrema and multivariate Taylor series.
- It includes double and triple integrals in various coordinate systems and line and surface integrals.
- It ends with the theorems of Green, Gauss and Stokes.

**MATH 2015** is Applied Multivariate and Vector Calculus at York University. It is a one-semester course designed for students in engineering and the physical sciences, and it covers a full **Calculus 3** syllabus in a single term: partial derivatives, vector operators, multiple integrals and the big theorems of vector calculus. This guide maps every MATH 2015 topic to free practice so you can keep up with the pace.

## What York MATH 2015 covers

York's Department of Mathematics and Statistics describes MATH 2015 as a course in the theory and applications of multivariable and integral calculus. The listed topics are functions of several variables; arclength and curvature; partial derivatives; the grad, div, curl and Laplacian operators; extrema and Taylor series for multivariate functions; double and triple integrals in various coordinate systems; line and surface integrals; and the theorems of Green, Gauss and Stokes. The department adds that, since the course extends single-variable techniques, understanding the prerequisite material is critical for success.

That is everything in a standard Calculus 3 course, from partial derivatives through Stokes' theorem.

## MATH 2015 topic map

| MATH 2015 topic | Practice | Read |
| --- | --- | --- |
| Functions of several variables | [Functions of several variables](/learn/calculus-3/partial-derivatives/functions-of-several-variables) | |
| Arc length and curvature | [Curves and motion](/learn/calculus-3/vectors-and-space/curves-and-motion) | [Parametric arc length](/learn/calculus-2/power-series-parametric-polar/parametric-arc-length) |
| Partial derivatives | [Partial derivatives](/learn/calculus-3/partial-derivatives/partial-derivatives), [second partials](/learn/calculus-3/partial-derivatives/second-partial-derivatives) | [Partial derivatives guide](/blog/partial-derivatives) |
| Gradient | [Gradient and directional derivatives](/learn/calculus-3/partial-derivatives/gradient-and-directional-derivatives) | [Gradient guide](/blog/gradient-and-directional-derivatives) |
| Divergence, curl and Laplacian | [Curl and divergence](/learn/calculus-3/vector-calculus/curl-and-divergence) | |
| Extrema | [Second derivative test](/learn/calculus-3/partial-derivatives/critical-points-and-second-derivative-test), [Lagrange multipliers](/learn/calculus-3/partial-derivatives/lagrange-multipliers) | [Lagrange multipliers guide](/blog/lagrange-multipliers) |
| Multivariate Taylor series | [Tangent planes](/learn/calculus-3/partial-derivatives/tangent-planes) | [Taylor's theorem](/theorems/taylors-theorem) |
| Double integrals | [Iterated integrals](/learn/calculus-3/multiple-integrals/iterated-integrals), [polar double integrals](/learn/calculus-3/multiple-integrals/polar-double-integrals) | [Double integrals guide](/blog/double-integrals) |
| Triple integrals and coordinate systems | [Triple integrals](/learn/calculus-3/multiple-integrals/triple-integrals), [cylindrical and spherical](/learn/calculus-3/multiple-integrals/cylindrical-spherical) | [Change of variables](/learn/calculus-3/multiple-integrals/change-of-variables) |
| Line integrals | [Line integrals of functions](/learn/calculus-3/vector-calculus/line-integrals-of-functions), [of vector fields](/learn/calculus-3/vector-calculus/line-integrals-of-vector-fields) | [Conservative fields](/learn/calculus-3/vector-calculus/conservative-fields) |
| Surface integrals | [Surface integrals and flux](/learn/calculus-3/vector-calculus/surface-integrals-and-flux) | |
| Green's theorem | [Green's theorem](/learn/calculus-3/vector-calculus/greens-theorem) | [Green's theorem statement](/theorems/greens-theorem) |
| Gauss and Stokes | [Stokes and divergence theorems](/learn/calculus-3/vector-calculus/stokes-and-divergence-theorems) | [Divergence theorem](/theorems/divergence-theorem), [Stokes' theorem](/theorems/stokes-theorem) |

## Where MATH 2015 fits at York

| Course | Title | Role |
| --- | --- | --- |
| [MATH 1300](/courses/york-university-math-1300) or MATH 1013 | First calculus course | Differential calculus |
| [MATH 1310](/courses/york-university-math-1310) or MATH 1014 | Second calculus course | Integral calculus and series |
| MATH 2015 | Applied Multivariate and Vector Calculus | Several variables and vector calculus |

You also need a linear algebra course (MATH 1021 or MATH 1025) before MATH 2015.

## Example problems at the MATH 2015 level

**1. Divergence and curl.** For $\mathbf{F} = \langle x^2, xy, z \rangle$: $\nabla\cdot\mathbf{F} = 2x + x + 1 = 3x + 1$ and $\nabla\times\mathbf{F} = \langle 0, 0, y \rangle$.

**2. The Laplacian.** For $f = x^2 - y^2$, $\nabla^2 f = 2 - 2 = 0$, so $f$ is harmonic.

**3. The Divergence Theorem.** For $\mathbf{F} = \langle x, y, z \rangle$ over the unit ball, $\nabla\cdot\mathbf{F} = 3$, so the outward flux is $3\cdot\frac{4}{3}\pi = 4\pi$.

**4. Green's theorem.** The area inside a closed curve $C$ is $\frac{1}{2}\oint_C (x\,dy - y\,dx)$. For the unit circle this gives $\pi$.

## How to do well in MATH 2015

- **Review before the term.** York stresses the prerequisites. Integration techniques and vectors are used in almost every problem.
- **Know which theorem fits.** Green's theorem for plane curves, Stokes' theorem for curves bounding surfaces, and the Divergence Theorem for closed surfaces.
- **Track orientation.** Signs in line and surface integrals depend on direction. State the orientation you use.
- **Draw the region every time.** Bounds for triple integrals are much easier to read off a sketch.

> **Tip:** if $\nabla\times\mathbf{F} = \mathbf{0}$ on a simply connected region, the field is conservative and line integrals depend only on the endpoints.

## Frequently asked questions

### What is MATH 2015 at York?

MATH 2015 is Applied Multivariate and Vector Calculus, a one-semester course designed for students in engineering and the physical sciences. York's department lists functions of several variables, arclength and curvature, partial derivatives, grad, div, curl and Laplacian operators, extrema and Taylor series for multivariate functions, double and triple integrals in various coordinate systems, line and surface integrals, and the theorems of Green, Gauss and Stokes.

### What are the prerequisites for MATH 2015?

One of SC/MATH 1021 or SC/MATH 1025 (linear algebra), and one of SC/MATH 1010, SC/MATH 1014 or SC/MATH 1310; or SC/MATH 1505 plus permission of the course coordinator.

### MATH 2015 vs MATH 2310?

They are credit exclusions of each other, along with MATH 2010. MATH 2015 is the applied version aimed at engineering and the physical sciences.

### How should I prepare for MATH 2015?

York's description stresses that understanding the prerequisite material is critical. Review single-variable integration techniques and vectors from linear algebra before the term starts.