Difference between revisions of "Calc3.2"
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=Vectors, and Vector Functions= | =Vectors, and Vector Functions= | ||
| + | ==Vectors== | ||
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. | 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. | ||
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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>. | 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>. | ||
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| + | We can, if we wish, talk about just the length of the vector - this is its absolute value, or <math>\left\| \vec{v} \right\| \ </math>. For a vector written in Cartesian coordinates, the Pythagorean theorem tells us this is just <math>\sqrt{x^2 + y^2} \ </math>. | ||
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| + | Vectors can be added and subtracted, just like numbers. See the illustration below. | ||
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| + | In Cartesian coordinates, this is particularly easy: <math>\begin{pmatrix}x\\y\end{pmatrix} + \begin{pmatrix}a\\b\end{pmatrix} = \begin{pmatrix}x+a\\y+b\end{pmatrix} \ </math> and <math>\begin{pmatrix}x\\y\end{pmatrix} - \begin{pmatrix}a\\b\end{pmatrix} = \begin{pmatrix}x-a\\y-b\end{pmatrix} \ </math>. | ||
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| + | Multiplication by a number is also simple: the vectors direction is unchanged, but its length is multiplied by that number. Again, in Cartesian coordinates, this becomes simple: <math>c\begin{pmatrix}a\\b\end{pmatrix} = \begin{pmatrix}ca\\cb\end{pmatrix} \ </math>. | ||
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| + | There are two ways to multiply a vector by another vector. In two dimensions, we can perform only one, which is called the '''dot product''' or in more advanced terminology, '''inner product.''' Given two vectors, this dot product is how far one vector goes in the direction of the other, using the other as the "unit" of length. It is defined <math>\vec{u} \cdot \vec{v} = \left\|\vec{u}\right\| \left\|\vec{v}\right\| \cos(\theta) \ </math>, where <math>\theta \ </math> is the angle between <math>\vec{u} \ </math> and <math>\vec{v} \ </math>. As before, this takes a particularly simple form in Cartesian coordinates: <math>\begin{pmatrix}x\\y\end{pmatrix} \cdot \begin{pmatrix}a\\b\end{pmatrix} =xa+by \ </math>. This expression is similar for the three dimensional case: <math>\begin{pmatrix}x\\y\\z\end{pmatrix} \cdot \begin{pmatrix}a\\b\\c\end{pmatrix} = xa+yb+zc \ </math>. | ||
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| + | ==Vector Functions== | ||
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| + | Before, we described scalar functions as taking a point in space and ascribing a number to it - for example, <math>f(x,y,z) = \ </math>. | ||
=Coordinate Systems= | =Coordinate Systems= | ||
Revision as of 23:17, December 31, 2009
Vectors, and Vector Functions
Vectors
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 Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \vec{v} \ } or Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf{v} \ } . 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 Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \vec{v} = \begin{pmatrix}x\\y\\z\end{pmatrix} \ } .
We can, if we wish, talk about just the length of the vector - this is its absolute value, or Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \left\| \vec{v} \right\| \ } . For a vector written in Cartesian coordinates, the Pythagorean theorem tells us this is just Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \sqrt{x^2 + y^2} \ } .
Vectors can be added and subtracted, just like numbers. See the illustration below.
In Cartesian coordinates, this is particularly easy: Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{pmatrix}x\\y\end{pmatrix} + \begin{pmatrix}a\\b\end{pmatrix} = \begin{pmatrix}x+a\\y+b\end{pmatrix} \ } and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{pmatrix}x\\y\end{pmatrix} - \begin{pmatrix}a\\b\end{pmatrix} = \begin{pmatrix}x-a\\y-b\end{pmatrix} \ } .
Multiplication by a number is also simple: the vectors direction is unchanged, but its length is multiplied by that number. Again, in Cartesian coordinates, this becomes simple: Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle c\begin{pmatrix}a\\b\end{pmatrix} = \begin{pmatrix}ca\\cb\end{pmatrix} \ } .
There are two ways to multiply a vector by another vector. In two dimensions, we can perform only one, which is called the dot product or in more advanced terminology, inner product. Given two vectors, this dot product is how far one vector goes in the direction of the other, using the other as the "unit" of length. It is defined Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \vec{u} \cdot \vec{v} = \left\|\vec{u}\right\| \left\|\vec{v}\right\| \cos(\theta) \ } , where Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \theta \ } is the angle between Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \vec{u} \ } and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \vec{v} \ } . As before, this takes a particularly simple form in Cartesian coordinates: Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{pmatrix}x\\y\end{pmatrix} \cdot \begin{pmatrix}a\\b\end{pmatrix} =xa+by \ } . This expression is similar for the three dimensional case: Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{pmatrix}x\\y\\z\end{pmatrix} \cdot \begin{pmatrix}a\\b\\c\end{pmatrix} = xa+yb+zc \ } .
Vector Functions
Before, we described scalar functions as taking a point in space and ascribing a number to it - for example, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f(x,y,z) = \ } .
Coordinate Systems
cartesian
polar
brief discussion about cylindrical
spherical
Limits of Multivariate Functions
Elementary Vector Calculus
Partial Derivatives
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