Vector Functions (Vector Fields)
Vector Functions
To understand Maxwell's Equations, we need to understand 3dimensional vector functions (also known as vector fields). Fortunately, these aren't too complicated. Vector functions are basically 3 stacked functions  the first describes the xcomponent of the vector, the second describes the ycomponent and the 3rd function describes the zcomponent of the vector. Equation 1 gives an example of a vector function (these type of functions are written in bold), and we'll call it A:
The vector function A is a function of 3spatial variables: (x,y,z). These are just the coordinates of a 3dimensional standard Cartesian space. We can write the vector A in terms of 3 nonvector ("scalar") functions, , and . is the magnitude of the vector A in the x direction. Similarly, is the magnitude of A in the y direction and is the magnitude in the z direction. The vector field A can also be written as:
In Equation [2], is a unit vector (a vector with magnitude equal to 1) in the xdirection, is a unit vector in the ydirection, and is a unit vector in the zdirection. Vector functions are relatively simple to evaluate. For instance, if you want to know what the value of A is at location (1,2,3)  you simply substitute this in and evaluate:
Equation [3] states that =3, =5 and =27 when A is evaluated at location (x,y,z)=(1,2,3). For Maxwell's Equations, we will only need 3dimensional vectors (vector functions with x, y and z components). To make things slightly more complicated, we will usually have a time coordinate  but it will still be a 3dimensional vector function. As an example, look at the vector function B in Equation 4:
The vector function B still only has 3 components, , and . However, we've added a variable t, which is simply the time component. Now, the vector function B varies with position (x,y,z) and time (t). To make it clear, if we want to evaluate B at (x,y,z,t)=(1,2,3,4), it is still simple:
