PH 316 Resources - Mostly math from text chapter 1.

Gradient of a scalar function (turns a scalar function into a vector function)
 

  • grad f = direction and magnitude of maximum rate of change of scalar function f
  • df = (grad f) dot dr
  • when dr is parallel to grad f , the largest change in f occurs per unit of travel
  • when dr is perpendicular to grad f , no change in f occurs in the direction of travel

    •  

     

    Dot product (produces a scalar from two vectors)
     
  • A dot B = A (projection of B on A) = B (projection of A on B)
  • i dot j = j dot k = k dot i = 0.
  • (Ax i + Ay j + Az k) dot (Bx i + By j + Bz k) = AxBx + AyBy + AzBz
  • dot product is an outstanding way to obtain the angle between two vectors

    •  

     

    Taylor series

    1-D: f(x) = f(a)+ (x-a) df/dx|a + higher terms.

    Since f(x)-f(a) = integral from a to x of f'(t) dt, we can integrate by parts again and again to get closer and closer.

    The result is f(x)-f(a) = sum on n from 1 to infinity of (x-a)^n d^n f/dx^n /n!

    This is the Taylor series. (The MacLaurin series, when a=0, is a special case of Taylor series).

    We will be using Taylor series when we examine Laplace's equation


    Binomial series

    We all know  (a+b)^2 = a^2 + 2ab + b^2, & maybe that (a+b)^3 = a^3+3 a^2b +3ab^2 + b^3.
    This is the stuff of Pascal's triangle. (121   1331  14641, etc.)

    The formula for (a+b)^n is sum on m from 0 to m of {a^(n-m) b^m Cnm}, where Cnm =n!/[m!(n-m)!].
     

    Letting a=1 and b=x, we have (1+x)^n = 1+nx +n(n-1)x^2/2! + ...

    If m is a positive integer, the series has only n+1 terms, but if m is anything else, the series is infinite.

    As an example of using the series, we may approximate g =sqrt(123) = (121+2)^2.
    We fish out 121 and get g = 11(1+2/121)^(1/2). With x=2/121 we may use only the leading term and one other to obtain an appproximation, namely g ~= 11(1+1/2 2/121) = 11 + 11/121, around 11.1.

    Likewise you can show the cube root of 65 is close to 4+4/(3*64), or about 4.02. Naturally, the cube root of 63 would be around 3.98.

    Divergence of a vector