York MATH 2015 Multivariate and Vector Calculus: Topics and Practice

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

Where MATH 2015 fits at York

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.