Fast Proofs
A couple of neat proofs, because they're just uber-cool.
The maths ones first:
Product rule - we want to show that if f(x)=u(x)v(x), then f'(x)=u'v +uv' where the ' represents a derivative wrt x. Here's the quick way: we use a somewhat clunky notation to let f be a multivariate function:
f(x) = f( u(x) , v(x) )
Then just take a derivative using the chain rule (f,u means partial derivative, because I can't format the backwards 6)
f'(x) =f,u u'(x) + f,v v'(x)
f'(x) =vu' + uv'
Which is approximately a million times faster than the proof using the definition of the derivative and limits.
Asin(x)+Bcos(x) - we want a good way to remember that this is Rcos(x-phi) where R^2=a^2+b^2 and phi is some angle we can get using trig formulas and equating co-efficients (it's arctan(B/A) I think...)
1st method is to use vectors - Asin(x) + Bcos(x) = <A,B> . <sinx , cosx> where the . means a scalar product. We now use the geometric definition, noting that the latter is simply a unit vector making an angle of x with the axis:
Asin(x) + Bcos(x) = sqrt(A^2+B^2)(1)cos(x - some angle)
If you draw out the two vectors, then you can work out the angle between them using geometry, no trig required.
2nd method is to use complex numbers - acos(x) + bsin(x) is the real part of (a-bi)(cos(x) + isin(x))
Which is just the real part of sqrt(a^2 + b^2)e^(-i some angle) times e^(ix) using polar notation.
Multiply out, take the real part and your done. Again, the complex perspective provides us with a picture of the two complex numbers from which to work out the correct angles and lengths.
Physics Proofs:
Potential energy for a dipole in a field -
You have two charges, +q and -q seperated by a distance d (make it a vector displacement, actually) in an electric field E(r) with a potential V(r).
The potential for the +ve charge is +qV(r+d), and for the negative charge it's -qV(r). When you add them together you get q[V(r+d)-V(r)]~qd.grad(V)
Using qd=p (dipole moment) and -grad(V)=E, we get the total potential as -p.E We're not done yet though, beacause as above the . means a scalar product. We use the same trick and get the total energy, U as:
U = -|P||E| cos(theta)
This shows that the minimum energy for a dipole in an external field is aligned with it, because -cos(theta) is minimal at theta=0. This fact finds applications in e.g. an NMR scanner (additionally, we can use the formula above to note that we should find an unstable equilibrium anti-parallel to the field i.e. at theta=pi. This is also put to use in NMR scanners as compounds absorb radiation and flip between the two equilibria.)
