Changes

Normally, we don't allow vectors (like <math>\vec{i} \ </math>) to be elements of a matrix or a determinant, but in this case, we make an exception because the formula is so useful and easy to remember.  We will encounter a similar exception to the rules when we give an easy way to remember the formula for '''curl.'''

Normally, we don't allow vectors (like <math>\vec{i} \ </math>) to be elements of a matrix or a determinant, but in this case, we make an exception because the formula is so useful and easy to remember.  We will encounter a similar exception to the rules when we give an easy way to remember the formula for '''curl.'''

==Vector Functions==

==Vector Functions and Vector Fields==

[[Image:VectorField.png|right|thumb|200px|Vector functions are often visualized as fields, with the output vectors drawn with their base at the point that generated that output.]]

[[Image:VectorField.png|right|thumb|200px|An illustration of a vector field.  The arrows show the magnitude and direction of the vector at the location of the arrow's base.]]

Before, we described scalar functions as taking a point in space and ascribing a number to it - for example, <math>f(x,y,z) = x^2 + y^2 +z^2 \ </math>.  With our new notation, we can describe this as <math>f(\vec{r}) = \vec{r} \cdot \vec{r} \ </math>, which is a scalar function written with vectors.   

Before, we described scalar functions as taking a point in space and ascribing a number to it - for example, <math>f(x,y,z) = x^2 + y^2 +z^2 \ </math>.  With our new notation, we can describe this as <math>f(\vec{r}) = \vec{r} \cdot \vec{r} \ </math>, which is a scalar function written with vectors.   

There is another class of functions, though, which return not numbers but vectors.  These are vector functions, and they are useful for describing many phenomenon, most notably flows, winds, and heat dispersion.  For example, at every point in the ocean, the water is moving with both a speed and a direction.  If we have some vector <math>\vec{r} \ </math> which can be used to desrcibe points in the ocean, we can write a function <math>f(\vec{r}) = \vec{v} \ </math>, which will give us the velocity vector of the water at <math>\vec{r} \ </math>.  Very often, we use <math>\vec{r} \ </math> as a variable vector describing a position or location, <math>\vec{v} \ </math> for velocity, and later we'll encounter <math>\vec{a} \ </math> for acceleration.

There is another class of functions, though, which return not numbers but vectors.  Mathematicians define a function as anything that maps some set (the "domain") to some set (the "range".)  For the functions we have seen so far, the range is the real numbers (scalars.)

 

When the range is a vector space, it can be called a vector function.  But the most common types of vector functions are those for which the domain is physical space.  In that case, the function is called a ''[[vector field]]''.

 

Vector fields are useful for describing many phenomena, most notably flows, winds, heat dispersion, and electric and magnetic forces.  For example, at every point in the ocean, the water is moving with both a speed and a direction.  If we have some vector <math>\vec{r} \ </math> which can be used to describe points in the ocean, we can write a function <math>f(\vec{r}) = \vec{v} \ </math>, which will give us the velocity vector of the water at <math>\vec{r} \ </math>.  Very often, we use <math>\vec{r} \ </math> as a variable vector describing a position or location, <math>\vec{v} \ </math> for velocity, and later we'll encounter <math>\vec{a} \ </math> for acceleration.

=Coordinate Systems=

=Coordinate Systems=