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## Main Question or Discussion Point

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?

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?

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