| Line 19: |
Line 19: |
| | =Introduction and Prerequisites= | | =Introduction and Prerequisites= |
| | | | |
| − | Welcome to Multivariable Calculus! This course will move quickly, and cover a lot of ground. For this reason, the instructors personal email and phone number will be available to students. We will begin with a discussion of functions of multiple variables, including differentiaion and integration, and move on to more complicated matters associated with these functions, such as optimization with constraints, surface integrals, and the basics of differential geometry. It will cover the same material as a University-level Calculus 3 course, but we will begin with a review of concepts from Calculus 2 that the student should already be familiar with. | + | Welcome to Multivariable Calculus! This course will move quickly, and cover a lot of ground. For this reason, the instructors personal email and phone number will be available to students. We will begin with a discussion of functions of multiple variables, including differentiation and integration, and move on to more complicated matters associated with these functions, such as optimization with constraints, surface integrals, and the basics of differential geometry. It will cover the same material as a University-level Calculus 3 course, but we will begin with a review of concepts from Calculus 2 that the student should already be familiar with. |
| | | | |
| | It is essential that the student have firm grasp of concepts from Calculus 1. This includes, but is not limited to, | | It is essential that the student have firm grasp of concepts from Calculus 1. This includes, but is not limited to, |
| | | | |
| − | *Limits of functions of a single variable, including L'Hôpitals rule, | + | *Limits of functions of a single variable, including [[L'Hopital's Rule|L'Hôpitals rule]], |
| − | *Differention of a function of a single variable, | + | *Differentiation of a function of a single variable, |
| | *Integration of a function of a single variable, including integration by parts, | | *Integration of a function of a single variable, including integration by parts, |
| | *The fundamental theorem of calculus | | *The fundamental theorem of calculus |
| Line 142: |
Line 142: |
| | We begin with a review of functions. Functions, in the most general sense, are rules which associate one thing with another. For example, there is a function which takes a place on Earth and returns its temperature - in that case, the "domain" of the function is the Earth, and the "range" are temperatures. In elementary calculus, the domain and the range tend to be subsets of the real numbers. In this course, the domain and range may be real numbers, or sets of vectors in higher dimensions, or matrix sets. | | We begin with a review of functions. Functions, in the most general sense, are rules which associate one thing with another. For example, there is a function which takes a place on Earth and returns its temperature - in that case, the "domain" of the function is the Earth, and the "range" are temperatures. In elementary calculus, the domain and the range tend to be subsets of the real numbers. In this course, the domain and range may be real numbers, or sets of vectors in higher dimensions, or matrix sets. |
| | | | |
| − | Functions which take a real numbers, and output a real number, are said to be '''continuous''' if, basically, they are connected. At right are illustrations of continuous and discontinuous functions. A strict definition of continuity is usually saved for a course in [[|real analysis|Real Analysis]]. | + | Functions which take a real numbers, and output a real number, are said to be '''continuous''' if, basically, they are connected. At right are illustrations of continuous and discontinuous functions. A strict definition of continuity is usually saved for a course in [[real analysis|Real Analysis]]. |
| | | | |
| | The '''limit''' of a function is defined so that <math>\lim_{x\rightarrow x_0} f(x) = L \ </math> if, for any <math>\epsilon>0 \ </math>, there is an <math>\delta>0 \ </math> such that every number within <math>\delta \ </math> of <math>x_0 \ </math> is taken by the function to a number within <math>\epsilon \ </math> of <math>L \ </math>. | | The '''limit''' of a function is defined so that <math>\lim_{x\rightarrow x_0} f(x) = L \ </math> if, for any <math>\epsilon>0 \ </math>, there is an <math>\delta>0 \ </math> such that every number within <math>\delta \ </math> of <math>x_0 \ </math> is taken by the function to a number within <math>\epsilon \ </math> of <math>L \ </math>. |