maths

homogenous equation

sub in u=y/x then dy/dx = u + xdu/dx then do separable equations

modulus rule complex numbers

|z₁ z₂ | = |z₁| |z₂|

modulus and arg of z¹⁰

1) find z

2) put into re^iarg form then re(cos(arg) + isin(arg))

3) reduce angle if needed

4) then simplify back to normal

all solutions of z⁶ = -8i

• convert all the way to polar

• put n in brackets

• then do for n = 0,1,2,3,4,5 number of n depends on power STARTS FROM 0

principal logs

ln(w) = ln|w| + iArg(w)

cos(x)

e^ix + e^-ix / 2

sin(x)

e^ix - e^-ix / 2i

linear equation

1. make standard equation dy/dx + f(x)y = g(x)

2. make integrating factor I(x) = exp(integral of the part with y but not y)

3. then multiply standard equation by I(x) to both sides

4. whole left hand side becomes d/dx (integrating factor * y)

sec(x)

1/cos(x)

bernoulli equatinos

• want to convert to linear

• its bernoulli when n ≠ 0 ≠ 1 (power on the y not x)

• multiply whole equation by (1-n)y^-n

• then substitute u = y^1-n

homogeneous equation form

linear equation form

bernoulli equation form

e^-ln(x)

1 / e^ln(x) = 1 / x

exact equation form

exact equation steps

1. check if they are exact by checking opposite partial derivatives Q/x P/y

2. integrate P respect to x

3. differentiate that respect to y

4. compare to Q thats you g(y)

5. write f(x,y) = integrated P + g(y)

6. set to constant A and solve for y

chain rule with parital derivatives

chain rule where f(x(u,v), y(u,v)

directional derivative in direction v

∇ f(p) ⋅ v / |v| = |∇ f (p)| cos(theta)

equation of plane that goes through point normal to vector n

(x - p) dot n = 0

equation of line through in direction v

x = p + tv

normal line to a surface S through point p where grad f is a normal vector to s

x = p + t ∇f(p)

normal line to a plane

x = p + tv

tangent plane to S at p

(x-p) dot (∇f(p)) = 0

• so ∇f(p) is a normal vector to the surface

plane equation

(x-p) dot n = 0

critical points

1. find fx and fy and solve for x and y

2. those r critical points

3. find fxx, fyy, fxy

4. solve for discriminant and compare to formula book

LU decomposition

Lu=b (first)
Ux = u (then use u)

• for L put opposite sign of what the row transformation is

at the end the original matrix A = LU

when does a matrix have an LU decomposition/can be factorised for LU

small matrix:

• all three principal minors are not zero (eg. determinants of first element, 4 elements, whole thing)

big matrix:

• strictly diagonally dominant = the absolute value of diagonal element in each row is greater than absolute value of sum of other elements

• if not doesnt mean it doesnt have just a way of checking

PTLU decomp

PA = LU

A = P^TLU

do LU decomp

P^T means transpose so rows become columns

• P is permutation matrix that rearranges rows - identity matrix and then swap the one to where row 2s one is etc.

jacobis method

• will converge if matrix of coefficients is strictly diagonally dominant

• make strictly diagonally dominant by swapping rows

• for second iteration use only the old values from the iteration before

• variable on left (k+1) variable on right is k

gauss-seidel

• uses newest available value (so in same iteration)

• will converge if positive definite or SDD

• first equation is LHS (k+1) RHS (k)

• next few equations are all (k+1)

• when they give u starting conditions put it straight into first equation fro x

positive definite

• symmetric so A = A^T (transpose)

• principal minors all positive

transpose of a matrix

rows become columns

SOR method

• rearrange for variable then add eg. (1-w)x + w(rearranged equation)

• for y = … y = (1-w)y + w(rearranged eq)

• same as gauss seidel immiediately use new updates