Theory question about Exact Eqtns and Calculus

[flyingpig]

Homework Statement

I've noticed that exact equations take the same form as vector field dot <dx,dy> form, are there any relationships? Because I ended up writing an arrow on top of my F(x,y) everytime I do these exact equation problems

Is there a secret my book is not telling me?

 

[lanedance]

not sure if i understand the question, but if you have a differential
$$dQ = A(x,y)dx + B(x,y)dy$$

and they satisfy
$$\frac{A(x,y)}{dy} = \frac{B(x,y)}{dx}$$

then there exists a scalar function Q(x,y), such that dQ
$$dQ = A(x,y)dx + B(x,y)dy$$

similarly if you consider the vector field
$$F = \begin{pmatrix} A(x,y) \\ B(x,y)\end{pmatrix}$$
a line integral will independent of path, depending only on endpoints and equal to $Q(\vec{x}_f) - Q(\vec{x}_i)$, and you could write:
$$dQ = F \bullet \vec{dx}$$