# Math 237 Multivariable Calculus

Fall 2015

MULTIVARIABLE CALCULUS The world is complex, multidimensional. Our environment can not be described by a single number, and it depends on numerous factors. We need functions of more than one variable and we need vectors. The goal of this course is for you to develop intuition about multivariable functions. Applications, visual representations, and computers all help.

How do you represent these objects? What are derivatives of two variable functions and what can they tell you? What do their integrals calculate? Why were these operations developed along with physics? These questions and more are addressed in the course, which gives attention to both mathematical foundations and applications. Computer assistance uses the Mathematica software.

TYPES OF FUNCTIONS One way to classify a function is by the number of input variables and output variables associated with it. We will go up to three variables. Here is a preview going up to two variables each for input and output, mentioning appropriate graphical representations we will study.

• 1 input, 1 output: $f(t) = t e^{-t}$. Ordinary function graph, a curve in the plane. This is the subject of single variable calculus. You are already an expert.
• 1 input, 2 outputs: $f(t) = (\cos t, \sin t)$. Parameterized curve in the plane. The example is a circle.
• 2 inputs, 1 output: $f(x, y) = x^2 + y^2$. Contour diagram; or function graph which is a surface in three dimensional space.
• 2 inputs, 2 outputs: $f(x, y) = (-y, x) = -y \vec i + x \vec j$. Vector field. Or  $f(r, t) = (r \cos t, r \sin t)$.  Coordinate change in the plane.

Course components

• Multivariable functions
• Partial derivatives
• Critical points and optimization
• Multiple integrals
• Parameterized curves and vector fields
• Line integrals and Green’s theorem
• Flux and the divergence theorem

Calculus is a great subject. I hope you will enjoy it as much as I do.

NOTE Beginning Spring 2015, the Math 237 unit on critical points and optimization has been replaced with a unit on sequences and series of 1-variable functions.  The course is renamed Applied Multivariable Calculus III.

Fall 2014