Difference between revisions of "Calc3.2"
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Vectors are the bread and butter of multivariable calculus because they pack multiple variables into a single, easily manipulated quantity. Let's look at some more specific examples. | Vectors are the bread and butter of multivariable calculus because they pack multiple variables into a single, easily manipulated quantity. Let's look at some more specific examples. | ||
| − | In our first descriptions, we used a coordinate system you should be familiar with call "polar coordinates." If you aren't familiar with this system, we will review them shortly. When using Cartesian coordinates, vectors are very, very familiar. The point <math>(x,y) \ </math> has position vector, or put more simply, ''is'' the vector written <math>\begin{pmatrix}x\\y\end{pmatrix} \ </math> | + | In our first descriptions, we used a coordinate system you should be familiar with call "polar coordinates." If you aren't familiar with this system, we will review them shortly. When using Cartesian coordinates, vectors are very, very familiar. The point <math>(x,y) \ </math> has position vector, or put more simply, ''is'' the vector written <math>\begin{pmatrix}x\\y\end{pmatrix} \ </math>. See illustration at right. Unlike variables which may be numbers, like <math>a, x, \ </math> etc., vectors are usually written <math>\vec{v} \ </math> or <math>\mathbf{v} \ </math>. While the latter is common in print and online, we will use the former since it is more easily duplicated in handwriting. |
| + | In three dimensional space we use three variables to write down a vector. In Cartesian coordinates, this would look like <math>\vec{v} = \begin{pmatrix}x\\y\\z\end{pmatrix} \ </math>. | ||
=Coordinate Systems= | =Coordinate Systems= | ||
Revision as of 22:13, December 31, 2009
Vectors, and Vector Functions
A vector is a mathematical object which has magnitude like a number, but also direction. For example, if I wanted to describe how a person is driving at any particular time, I would need two numbers: their speed, and some measure of their direction (the angle it makes with true north, for example). These two quantities together constitute the motion vector of that person.
We also use vectors to describe position: for example, to locate an object, it suffices to know how far it is from you, and in what direction.
Vectors are the bread and butter of multivariable calculus because they pack multiple variables into a single, easily manipulated quantity. Let's look at some more specific examples.
In our first descriptions, we used a coordinate system you should be familiar with call "polar coordinates." If you aren't familiar with this system, we will review them shortly. When using Cartesian coordinates, vectors are very, very familiar. The point has position vector, or put more simply, is the vector written . See illustration at right. Unlike variables which may be numbers, like etc., vectors are usually written or . While the latter is common in print and online, we will use the former since it is more easily duplicated in handwriting.
In three dimensional space we use three variables to write down a vector. In Cartesian coordinates, this would look like .
Coordinate Systems
cartesian
polar
brief discussion about cylindrical
spherical
Limits of Multivariate Functions
Elementary Vector Calculus
Partial Derivatives
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