Undergrad Different types of vector fields?

[Hacca]

Vector fields confuses me. What are the differences between ($t$ could be any variable, not just time):

1. If the position vector don't have an argument, $\mathbf{r}=x\mathbf{\hat e}_x+y\mathbf{\hat e}_y+z\mathbf{\hat e}_z=(x,y,z)$ so
$\mathbf{E}(\mathbf{r},t)=E_x(\mathbf{r},t)\mathbf{\hat e}_x+E_y(\mathbf{r},t)\mathbf{\hat e}_y+E_z(\mathbf{r},t)\mathbf{\hat e}_z$

2. The position vector have an argument $t$, $\mathbf{r}(t)=x(t)\mathbf{\hat e}_x+y(t)\mathbf{\hat e}_y+z(t)\mathbf{\hat e}_z=(x(t),y(t),z(t))$ so
$\mathbf{E}(\mathbf{r}(t),t)=E_x(\mathbf{r}(t),t)\mathbf{\hat e}_x+E_y(\mathbf{r}(t),t)\mathbf{\hat e}_y+ E_z(\mathbf{r}(t),t)\mathbf{\hat e}_z$

3. The position vector have a different argument, say $u$ and $u\neq t$, $\mathbf{r}(u)=x(u)\mathbf{\hat e}_x+y(u)\mathbf{\hat e}_y+z(u)\mathbf{\hat e}_z=(x(u),y(u),z(u))$ so
$\mathbf{E}(\mathbf{r}(u),t)=E_x(\mathbf{r}(u),t)\mathbf{\hat e}_x+ E_y(\mathbf{r}(u),t)\mathbf{\hat e}_y+E_z(\mathbf{r}(u),t)\mathbf{\hat e}_z$

Are all vector fields? Are all $\mathbb{R}^4 \rightarrow \mathbb{R}^3$?

Also, in the context of Maxwell's equations, the fields are denoted without an argument, just $\mathbf{E}$, $\mathbf{B}$ etc. Is it just an abbreviation for any of the above?

 

[jedishrfu]

When the position vector has an argument like t it means its describing a path with t as the parameter

In the case if the E field r can take on any position and can take on any time meaning the E field varies with position and with time.

 

[andrewkirk]

Equation 1 tells us the vector field at an arbitrary point in space and time (spacetime). The arguments $\mathbf r$ and $t$ specify that point.

Equation 2 tells us the vector field experienced at time $t$ by a particle that is moving through space in such a way that its location at time $t$ is $\mathbf r(t)$. The equation gives less information about the vector field than Equation 1, but more information about the location of the particle.

To understand Equation 3 you need to understand the parametrisation of curves. This equation applies when we are interested in the field at points on a curve that is parametrised by the function $\mathbf r:\mathbb R\to \mathbf R^3$ that gives us a position in space $\mathbf r(u)$ for each parameter value $u$. The situations where this would be used are somewhat more complex so it's best not to worry about this until it's covered in your course and you're given problems involving it. But if you have a particular example, feel free to ask about it. Note that if the curve is the path of a particle and we parametrise it by the time variable, this type of equation turns into type 2. In other words, Equation 2 is a special case of Equation 3.

 

[fresh_42]

What would the same question be, if formulated with the velocity vector fields $\mathbf{v}\, , \,\mathbf{v}(x,t) \, , \,\mathbf{v}(x(t),t)\, , \,\mathbf{v}(x(t),s)$ instead of $\mathbf{E}$ and $\mathbf{B}$?