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21-259 Calculus in Three Dimensions

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Calculus in Three Dimensions is the third course in the Calculus sequence. In the first course, we learn about the derivative and its applications. In the second course, we learn integration techniques, and we make full use of the Fundamental Theorem of Calculus. In this course we will learn about differentiation and integration of functions of more than one variable, and tie these topics together with the study of vector calculus.

First we will spend some time with the basic elements that we'll need to understand: the geometry of three dimensions, lines and planes, functions of 2 (or more) variables, limits and continuity for these functions, and so on. Partial derivatives, which allows us to determine rates of change of a function with respect to each independent variable, will then be introduced. Other differentiation concepts will be studied as well, including the total derivative of a function, the gradient of a function, and the divergence and curl of a field. Various applications of derivatives will again be considered, and most importantly the optimization problems.

Integration will be introduced, and the various types of integrals will depend on the function and domain considered. Keep in mind that the integral is a tool, an extremely flexible and powerful tool to measure and analyze functions. We'll encounter double integrals, triple integrals, line integrals along curves, and surface integrals. Also, we'll integrate vector fields over both oriented curves and surfaces. Fortunately, we'll be able to reduce the calculation of these integrals to one-variable integrals.

Finally, the course reaches its beautiful and natural conclusion with the higher dimensional versions of the Fundamental Theorem of Calculus.

Learning Objectives

After completing this course, you will have

  1. Developed your ability to understand and visualize three dimensional space.
  2. Extended your intuition for the calculus concpets of differentiation and integration to a three dimensional environment.
  3. Expanded your ability to perform calculations related to three dimensional geometric objects
  4. Enhanced your capacity to think precisely and express yourself clearly.

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The description gives a brief overview of the topics we will discuss this semester. The learning objectives give an itemized list of the skills you should be developing. The list of learning objectives may give you some useful direction in terms of studying for exams.

Course Information

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