COURSE DESCRIPTION: |
In this course we learn how to extend the ideas of calculus to two and three dimensions.
The concepts of 1-variable calculus arise in studying the motion of a particle along a line. For a particle moving through space, not just along a line, the position, velocity and acceleration at each moment are described by vectors, not just by single real numbers. Many other physical quantities, such as force and angular velocity are also modelled mathematically as vectors.
We begin by studying the algebra of vectors (linear algebra), which allows us to describe the relationships between vector quantities in physics and also forms the basis of analytic geometry in 3-dimensional space. Vector-valued functions of a single real variable (time) are used to represent, for example, the velocity of a moving particle and also to study the geometry of space curves. We learn how to generalize the concepts of derivative and integral to vector-valued functions.
A real-valued function of 2 variables can be used to model quantities such as the temperature on the surface of the earth, which varies from one location to another. The graph of such a function is a surface in space. At a point of such a graph, one has a tangent plane, not just a tangent line. We learn how to describe the tangent plane in terms of ideas of calculus, and learn how the concepts of derivative and integral generalize to functions of several variables.
In the last part of the course we learn the 2-dimensional version of the Fundamental Theorem of Calculus, Green's Theorem. This is the mathematics behind the physical notions of work and potential energy, and is a big step toward understanding electric and magnetic fields.
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