Vector Functions (Vector Fields)

Maxwell's Equations (Home)

Vector Functions

To understand Maxwell's Equations, we need to understand 3-dimensional vector functions (also known as vector fields). Fortunately, these aren't too complicated. Vector functions are basically 3 stacked functions - the first describes the x-component of the vector, the second describes the y-component and the 3rd function describes the z-component 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:

example vector function
[Equation 1]

The vector function A is a function of 3-spatial variables: (x,y,z). These are just the coordinates of a 3-dimensional standard Cartesian space. We can write the vector A in terms of 3 non-vector ("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:

vector function in terms of unit vectors
[Equation 2]

In Equation [2], is a unit vector (a vector with magnitude equal to 1) in the x-direction, is a unit vector in the y-direction, and is a unit vector in the z-direction.

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:

evaluation of vector function
[Equation 3]

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 3-dimensional 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 3-dimensional vector function. As an example, look at the vector function B in Equation 4:

example of vector function with x,y,z,t
[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:

evaluation of vector function with x,y,z,t
[Equation 5]

Maxwell's Equations

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