Understanding Vector Calculus: A Brief Summary

[deltabourne]

Can anyone give me a very brief summary of what vector calculus means? I know this sounds like a "specify what you mean" type question, but I hope it isn't. Let me explain further. I know all the equations, how to find line integrals, what Green's theorem is, etc. but I don't exactly know what they mean (I have an idea but I'm just not as set with it as I am with single/multivariable calculus). When I find an integral with Green's theorem, what am I finding (generally)? What about line integrals (mass if you have a density if I recall correctly)?

Also I have about 5-6 ways of writing what seems to be finding the same thing, ie the integral of F(r(t))*r'(t) = the integral of F*T ds (where * is dot product), etc. Are they just the same thing? Is independence of path just a special case of sorts for finding line integrals in vector fields?

Any help is appreciated

 

[chroot]

When people refer to just "calculus," they mean calculus on a single real function. When people talk about multi-variable calculus, they mean calculus on a single real function with several dependent variables. When people talk about vector calculus, they mean calculus on systems of equations described by vectors.

For example, $\vec F = m \vec a$ is a vector equation, in which every vector is composed of three components. The vector equation is exactly equivalent to three independent real equations, $F_x = m a_x, F_y = m a_y, F_z = m a_z$.

Vector calculus is calculus applied to vector equations, which are just systems of multiple real equations.

- Warren

 

[Theelectricchild]

Leading up to three higher dimensional versions of the fundamental thm of calc!

Greens, Stokes and Divergence Thms ---