Welcome to NRICH.

 
Partial Differentiation
By Brad Rodgers (P1930) on Sunday, October 15, 2000 - 12:19 am:

Does anyone know of a book to learn Partial Differentiation from? This site's previous advice on books has been so good, I thought I'd ask again. It appears that this will be necessary to entirely understand relativity.

Thanks,

Brad

By Thomas Mooney (P3048) on Sunday, October 15, 2000 - 12:39 am:

Yeah I do Brad. It's called Elementary applied partial differential equations and it's by a guy called Richard Haberman. Don't let the Elementary confuse you, this book is some serious maths! but also theres another one called advanced Engineering maths and it's by Denis G Zill and Michael R Cullen. It's da bomb!. The latter is better.

By Sean Hartnoll (Sah40) on Sunday, October 15, 2000 - 11:20 am:

Actually, I think Partial Differential Equations (PDEs) is not what you want to know about, just partial differentiation. PDEs are fairly advanced and you probably need to study ordinary (non-partial) differential equations first. Partial differentiation is probably talked about in books with titles like Advanced Calculus or Mathematical Methods.

The concept is not a difficult one. If you have a function, say f(x,y,z) = xy2z3 from a point in space (x,y,z) to a real number f(x,y,z) then the partial derivatives with respect to x, y and z are just what you get when you treat the other variables as constants, so

f/x = y2z3
f/y = 2xyz3
f/z = 3xy2z2

And that's it!

it has important property that you can change the order (check this for the example above)

/x (/ y) f = /y (/ x) f

And if x,y,z are functions of t, so

f(x(t),y(t),z(t))

then the total derivative of f is

df/dt = f/ x dx/dt + f/ y dy/dt + f/ z dz/dt

As an exercise, let x(t) = t, y(t) = t2, z(t) = t+1 and use this in the previous example for f to calculate the total deriative df/dt.

Sean