University Calculus III Worksheets

Which sets match your course code

University Calculus III is not one course with one number. The material is split across different courses in the public calendars of Montreal universities, and each orders and weights it its own way. Find your code below for the sets that cover its material, then check your own outline for what your exam includes.

• ENGR 233 — Vector Functions and Space Curves, Functions of Several Variables, Partial Derivatives, Directional Derivatives and the Gradient, Double Integrals, Triple Integrals and Change of Variables, Vector Fields and Line Integrals, Surface Integrals and Integral Theorems
• MAST 218 — Vector Functions and Space Curves, Functions of Several Variables, Partial Derivatives, Directional Derivatives and the Gradient, Optimization in Several Variables
• MAST 219 — Double Integrals, Triple Integrals and Change of Variables, Vector Fields and Line Integrals, Surface Integrals and Integral Theorems
• MAT1115 — Functions of Several Variables, Partial Derivatives, Directional Derivatives and the Gradient, Optimization in Several Variables, Double Integrals, Triple Integrals and Change of Variables
• MAT1400 — Functions of Several Variables, Partial Derivatives, Directional Derivatives and the Gradient, Optimization in Several Variables, Double Integrals, Triple Integrals and Change of Variables
• MAT1410 — Vector Functions and Space Curves, Vector Fields and Line Integrals, Surface Integrals and Integral Theorems
• MAT165 — Vector Functions and Space Curves, Functions of Several Variables, Partial Derivatives, Directional Derivatives and the Gradient, Optimization in Several Variables, Double Integrals, Triple Integrals and Change of Variables, Vector Fields and Line Integrals
• MATH 222 — Vector Functions and Space Curves, Functions of Several Variables, Partial Derivatives, Directional Derivatives and the Gradient, Optimization in Several Variables, Double Integrals, Triple Integrals and Change of Variables
• MTH1101 — Functions of Several Variables, Partial Derivatives, Directional Derivatives and the Gradient, Optimization in Several Variables

Notes and unit tests — in the bundle

Every University Calculus III unit has three printable documents, not one: the practice worksheet above, which is free, and two that come with the University Calculus III bundle — a set of unit notes stating everything the unit defines, and a full end-of-unit test to sit closed-book. See what each one covers:

What the 9 sets of unit notes cover  What the 9 unit tests look like

What University Calculus III covers

Calculus III is the course where every idea from the two one-variable courses comes back with more than one variable in it. A curve in space is a vector function, and its derivative is a tangent vector rather than a slope. A function of two variables has a partial derivative in each direction, a gradient that points uphill, and a tangent plane where there used to be a tangent line. The second derivative test becomes a statement about a determinant, and a constrained maximum is found with a Lagrange multiplier rather than by substitution.

Integration follows the same arc. A double integral is an iterated integral whose limits carry the shape of the region, so most of the marks in that set are in describing the region — and in reversing the order when the first order cannot be done. Triple integrals add cylindrical and spherical coordinates, and the change-of-variables formula explains where their extra factors come from. The last two sets take the integral along a curve and across a surface, and close with Green's, Stokes' and the divergence theorem, which are three versions of one idea.

The course stands on CEGEP Calculus I, Calculus II and Linear Algebra. Vectors, lines and planes, and one-variable sequences and series are not re-taught here — they are in those courses' sets. The courses that carry this material at university split it differently: some stop before line integrals, some take vector calculus in a second term, and one engineering course meets line integrals before double integrals. The list of course codes on this page says which sets each one covers.