domingo, 8 de junho de 2014

18-02sc-multivariable-calculus-fall-2010

This unit covers the basic concepts and language we will use throughout the course. Just like every other topic we cover, we can view vectors and matrices algebraically and geometrically. It is important that you learn both viewpoints and the relationship between them.


 In this unit we will learn about derivatives of functions of several variables. Conceptually these derivatives are similar to those for functions of a single variable.
  1. They measure rates of change.
  2. They are used in approximation formulas.
  3. They help identify local maxima and minima.
As you learn about partial derivatives you should keep the first point, that all derivatives measure rates of change, firmly in mind. Said differently, derivatives are limits of ratios. For example,
Of course, we’ll explain what the pieces of each of these ratios represent.
Although conceptually similar to derivatives of a single variable, the uses, rules and equations for multivariable derivatives can be more complicated. To help us understand and organize everything our two main tools will be the tangent approximation formula and the gradient vector.
Our main application in this unit will be solving optimization problems, that is, solving problems about finding maxima and minima. We will do this in both unconstrained and constrained settings.
This unit starts our study of integration of functions of several variables. To keep the visualization difficulties to a minimum we will only look at functions of two variables. (We will look at functions of three variables in the next unit.)
Our main objects of study will be two types of integrals:
  1. Double integrals, which are integrals over planar regions.
  2. Line or path integrals, which are integrals over curves.
All integrals can be thought of as a sum, technically a limit of Riemann sums, and these will be no exception. If you make sure you master this simple idea then you will find the applications and proofs involving these integrals to be straightforward.
We will conclude the unit by learning Green's theorem which relates the two types of integrals and is a generalization of the Fundamental Theorem of Calculus. Along the way we will introduce the concepts of work and two dimensional flux and also two types of derivatives of vector valued functions of two variables, the curl and the divergence.
» Exam

In our last unit we move up from two to three dimensions. Now we will have three main objects of study:
  1. Triple integrals over solid regions of space.
  2. Surface integrals over a 2D surface in space.
  3. Line integrals over a curve in space.
As before, the integrals can be thought of as sums and we will use this idea in applications and proofs.
We'll see that there are analogs for both forms of Green's theorem. The work form will become Stokes' theorem and the flux form will become the divergence theorem (also known as Gauss' theorem). To state these theorems we will need to learn the 3D versions of div and curl.
» Session 98: Maxwell's Equations

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