I'm getting very confused while learning about flow lines and the flow of a vector field. This may be because I've never formally seen differential equations. This is in my textbook:
I understand a flow line to be the path of an object whose velocity at any point is exactly equal to the vector field the object is in. I can see how $\hat F$ and $\hat r(t)$ are resolved into components, but I don't understand the next line that states $x'(t) =F_1(x(t), y(t))$ and $y'(t) =F_2(x(t), y(t))$
How can a component of the $\hat F$ function take two parameters, which themselves are functions...? And where are $\hat i$ and $\hat j$ in the equations?
My lack of understanding seems huge: I'd be more than happy to do whatever reading I should, but I don't know what I don't know. I know linear algebra well, so that's not where the gap is. Any help is greatly appreciated.
P.S. Do I have to understand differential equations to understand things like line and flow integrals, divergence, curl, etc.? Just wondering if should learn them ASAP for my vector calculus class.
$\hat{i}$ and $\hat{j}$ are unit vectors in the $x$- and $y$-directions, respectively. This means that $$ x \hat{i} + y \hat{j} = (x,y) \text{,} $$ where the expression on the left-hand side is a vector written as a linear combination of other vectors and the expression on the right-hand side is the same vector expressed as a pair of coordinates. So $$ x'(t) = F_1(x(t), y(t)) $$ has the same meaning (after a slight abuse of notation, described below) as $$ x'(t) = F_1(x(t) \hat{i} + y(t) \hat{j}) \text{.} $$ If one were being very careful about notation, one would write \begin{align*} \vec{F} &= \vec{F}(\vec{r}(t)) \\ &= F_1(\vec{r}(t)) \hat{i} + F_2(\vec{r}(t)) \hat{j} \\ &= F_1(x(t)\hat{i} + y(t) \hat{j}) \hat{i} + F_2(x(t)\hat{i} + y(t) \hat{j}) \hat{j} \\ &= F_1( \, \underbrace{(x(t), y(t))}_{\text{coordinate pair}} \, ) \hat{i} + F_2((x(t), y(t))) \hat{j} \end{align*} Here, $F_1$ and $F_2$ are always functions receiving a vector as an argument. However, it is very common to abuse notation slighty to replace $((\dots))$ with $(\dots)$. This means we really have two functions called "$F_1$". One receives a vector as its single argument. The other receives the components of a vector as its two arguments. They both do the same thing to that vector. The only difference is the details of how that vector is expressed as arguments to $F_1$. Similarly for $F_2$. So it is common to finish with $$ {}= F_1(x(t), y(t)) \hat{i} + F_2(x(t), y(t)) \hat{j} \text{.} $$
The only differential equations content in these lines should be familiar from calculus. "$x'(t) = F_1(x(t),y(t))$" just says "the rate of change of the $x$-coordinate with respect to time at the time $t$ is the first component of $\vec{F}$ evaluated at the point/position $(x(t), y(t))$."
In comments, OP asks why $F_1$ and $F_2$ have to take the entire position vector as inputs and not just the $\hat{i}$ and $\hat{j}$ components, respectively.
The simplest reason is because $\vec{F}$ does. $\vec{F}$ receives the position vector $\vec{r}$ as input, so all of its components receive the same thing as input.
But also, because if the $x$-coordinate of $\vec{r}$ could only alter the $\hat{i}$ component of $\vec{F}$ and likewise, the $y$-component, the $\hat{j}$ component, the only vector fields you could have would factor into two independent fields. Some fields do this. Consider the radially outward field $\vec{F} = (x,y) = (r_1, r_2)$.
Notice the $x$-component of each vector is the same in each column, i.e., is the same for each choice of $x$-coordinate of position. Likewise, the $y$-components of the vectors are the same in each row, i.e., the same for each fixed $y$-coordinate.
But not all flow fields do this. Consider the vortex $\vec{F} = (-y,x) = (-r_2, r_1)$.
Here's an even more complicated field, where both components of the flow field vectors depend on both components of the position. $\vec{F} = \left( \frac{\ln(1+|x|)}{y^2+1}, \sin(x) + y \right)$.
We want to be able to deal with arbitrary flow fields (although usually we require continuity, or even more smoothness), so we allow each component of the flow field vectors to depend on all of the position coordinates.