Our existence lies in a perpetual state of change. An apple falls from a tree, clouds move across expansive farmland blocking out the sun for days, meanwhile satellites zip around the Earth transmitting and receiving signals to our cell phones. The calculus was invented to develop a language to accurately describe and study the change we see. The ancient Greeks began a detailed study of change but they were scared to wrestle with the infinite and so it was not until the 17th century that maverick mathematicians, Isaac Newton and Gottfried Leibniz, among others, tamed infinity and gave birth to an extremely successful branch of mathematics called
the Calculus. Though just a few hundred years old, the Calculus has become an indispensable research tool in both the natural, social sciences and Economics and Buisness.
The power of calculus lies in its power to reduce complicated problems to simple rules and procedures. While these procedures can be (and often are) taught with little regard to the underlying mathematical concepts or their practical uses, our emphasis will be on understanding all of these: concepts, procedures and uses. We will engage in the full mathematical process, which includes searching for patterns, order and reason; creating models of real world situations to clarify and predict better what happens around us; understanding and explaining ideas clearly; and applying the mathematics we know to solve unfamiliar problems.
This course extends the thread of mathematical inquiry of the dual topics of differentiation and
integration for functions of several variables. Whereas functions of one variable may be visualized as curves on the plane, we may visualize a real-valued function of two-variables as a surface in
three-dimensional space. But what does it mean to take a derivative or a defnite integral of a surface? We will address this question and more in this course. Like in Calculus I &p; II, we will see that there is a beautiful geometry associated with functions of several variables.
Our frst task will be to develop a useful language for easily describing geometric objects in two and three dimensions using vectors. Then we will continue to study vector-valued functions, partial derivatives, the gradient vector, Lagrange multipliers, double and triple integrals and line integrals, culminating with the fundamental theorems of Green, Stokes, and Gauss. We will also apply these ideas to a wide range of problems that include motion in space, optimization, arc length, surface area, volumes, and centers of mass, time permitting.
The students should be able to interpret the concepts of Calculus algebraically, graphically and verbally. More generally, the students will improve their ability to think critically, to analyze a problem and solve it using a wide array of tools. These skills will be invaluable to them in whatever path they choose to follow, be it as a mathematics major or in pursuit of a career in one of the other sciences.
The textbook is