Concordia MAST 218 Multivariable Calculus I: Topics and Practice

MAST 218 is Multivariable Calculus I at Concordia University. The calendar describes it as an introductory course in the theory of functions of several variables, addressing differential and vector calculus. In practice, it takes derivatives into higher dimensions: curves, surfaces, partial derivatives and optimization with constraints. This guide maps every MAST 218 topic to free practice so you can learn each idea by working problems.

What Concordia MAST 218 covers

The Concordia calendar lists vector geometry; lines and planes; curves in \mathbb{R}^n; vector functions; vector differential calculus; functions of several variables, the chain rule and implicit differentiation; extrema, classification of extremal values and Lagrange multipliers. It is a 3-credit lecture course, and the prerequisites are MATH 204 and MATH 205, or equivalent.

That is the differential half of a typical Calculus 3 course. Integration in several variables is in MAST 219, which covers line and surface integrals, change of variables with Jacobians, and the theorems of Green, Stokes and Gauss.

MAST 218 topic map

MAST 218 and MAST 219

Before MAST 218 you need MATH 205 (which follows MATH 203) and MATH 204, Vectors and Matrices.

Example problems at the MAST 218 level

1. A line in space. The line through (1, 2, 3) in the direction \langle 2, -1, 4 \rangle is \mathbf{r}(t) = \langle 1 + 2t,\; 2 - t,\; 3 + 4t \rangle.

2. The chain rule. If z = x^2 y with x = \cos t and y = \sin t:

3. Classifying a critical point. For f(x, y) = x^2 - y^2, the only critical point is (0, 0), and D = f_{xx}f_{yy} - f_{xy}^2 = -4 < 0, so it is a saddle point.

4. Lagrange multipliers. Maximise f = xy on x + y = 10. From \nabla f = \lambda\nabla g: y = \lambda and x = \lambda, so x = y = 5 and the maximum is 25.

How to do well in MAST 218

• Bring your linear algebra. Vectors, dot and cross products and planes from MATH 204 are used from the first week.
• Draw tree diagrams for the chain rule. They show every path from the output variable to the input variable.
• Classify with the second derivative test. Compute D at each critical point and state clearly whether it is a maximum, minimum or saddle.
• Check constraint answers. With Lagrange multipliers, substitute your points back into the constraint and compare the values of f.

Tip: the gradient is perpendicular to level curves. That one fact explains tangent planes and Lagrange multipliers.