IB maths ALL

dxdy in polar

rdrd(theta)

What IS the Jacobean

A scale factor of area when changing variables and it is in the databook pg9

How to find equations of field lines

dy/dx equals the j bit over the i bit and then integrate

What IS divergence

Flux out of an area

How to calculate divergence

Dot product of d/dx d/dy etc with vector

What does the curl vector define

The axis of rotation

What generates curl

Local shear and global rotation

What does zero curl mean

Conservative field, path irrelevant

What does zero divergence mean

Closed surfaces have no net flux

Can you divide by a vector

No

What is the other name for the divergence theorem

Gauss theorem

What does gauss theorem do

Tells you literally divergence of a volume equals flux out of its surface

What does stokes theorem do

Lets you find a curl rather than a circulation

Where do you find the diffusion equations

In the materials databook

How do you solve d2X/dx2 + d2Y/dy2 = 0

Separation of variables, both sides equal a const 1/X d2X/dx2 = a² decide if positive or negative by considering if it should be sinosoidal or exponential in time etc

What is an alternative to exponentials in your differential eqn solutions

Ae^-ax + Be^ax OR Ccoshax + Dsinhax

How do you solve a PDE of the form d2f/dx2 + d2f/dt2 = 0

Assume form f=func(x)func(t).

1/X d2X/dx2 = 1/T d2T/dt2 = const

Decide whether the separation constant is positive or negative based on whether you expect it to be sinusoidal in time or distance, solve each side and use boundary conditions

How do you solve a PDE where the separation constant is zero

Solutions for X and T are both linear

How do you solve a PDE of the form d2f/dx2 = 1/a df/dt

1/X d2X/dx2 = 1/aT dT/dt (don’t forget the 1/T) = const. For the first order equation you just get one exponential

How do you solve a PDE of the form d2f/dx2 = 1/c² d2f/dt2

You start with 1/X d2X/dx2 = 1/c²T d2T/dt2 = const

Decide whether const is positive or negative, solve with boundary conditions.

As this is the wave equation, your solution might have an n (because boundary condition periodic)

When do you expect a self-similar solution to a PDE

When the boundary conditions are steady in time and there was no inherent length scale

How do you solve self similar PDEs

Using a characteristic diffusion length scale, L=c root(alpha t).

(T-To)/(delta)T = f(y/ root(alpha t))

There is only one question, just practice it ep4q8)

What is a sample space

A set of all possible outcomes

What is the entropy of a function

The number of states and their relative likelyhood. H= -(sum of all) P(x) x log2P(x) (probability function)

Is expectation linear

Yes

How do you calculate expectation

Sum/integral over all possible x of xp(x)

How else to write E(XY)

E(X)E(Y)

How do you do questions which need you to find the standard deviation

Trial and error

How do you calculate the expectation

Sum of all xP(x)

What is the central second moment more commonly called

The variance

What gives a Bernoulli distribution

Single trial with binary output

What gives a geometric distribution

How many trials until first success

What gives a binomial distribution

How many successes occur in n trials

What gives a poisson distribution

How many events occur in a given interval at rate lambda

How are binomial and Poisson distributions linked

For n going to infinity, B(n, L/n) = Pois(L)

What is a CDF F(x)

Cumulative distribution function. Always increasing, difference between two values = probability of event in that region

What is a PDF f(x)

Probability density function. It is the gradient of the CDF

How are PDF CDF and probability linked

(Integral from a to b)f(x) = F(b) - F(a) = P(a<x<b)

What gives exponential density

What is the time/distance between two successive lambda-rate successes?

What gives beta density

What is the PDF of the trial probability if we observe a-1 successes and b-1 fails?

If S=X+Y what is the expectation and variance of S

E(S) = E(X)+E(Y)

Var(S) = E(XY) - E(X)E(Y)

What is an ML estimator

Choosing a parameter that makes the observed data most probable

Estimating mean- minimising sum of all

(Xi - theta)²

What is an MAP estimator

Choosing the parameter that is most probable after observing the data

How do you do hypothesis testing

Make a null hypothesis (no effect)

Compute how likely data is if null hypothesis true

P value is probability of outcome being what it is or more extreme

Compare to significance level

Halve significance level for two tailed test

What is the column space of a matrix A

For Ax=b it is the set of all possible b

What is the null space of a matrix A

The solution of Ax=0

What is the row space of a matrix A

All x such that Ax ≠ 0

What is the left null space of a matrix A

All b so that Ax=b has no solution

What is the dimension of a vector space

The smallest number of basis vectors that could span it

What are basis vectors

The smallest set of spanning vectors that are linearly independent

What does it mean for vectors to be linearly independent

One can not be written as a multiple of the others

Do the dimensions of a space and the dimensions of a vector in it have to match

No

What is the rank of a matrix

The dimension of the vector space spanned by its columns

What sort of matrices have an equal inverse and transpose

Orthonormal ones

How do you do LU factorisation

Write out the LH column and top row of a matrix, starting with a 1 on the left top make a multiplication grid that gives these values, fill in rest, write out remainder. Repeat.

L = side columns, U = top rows

L is lower triangular U is upper echelon

How do you solve Ax=b if you know A=LU

Solve Lc=b then Ux=c

If no unique answer, set free variables to zero for particular solution then set them to 1 one at a time for homogeneous solutions and solution has form x0. + ax1 +bx2 etc

What is the dimension of a matrix

How big it is (ie how many rows and columns)

How do you find a least squares solution to Ax=b

Solve A(t)Ax=A(t)b

What is Gram Schmidt

It takes vectors describing a space and gives a set of orthonormal vectors describing the same space

How do you do Gram Schmidt

q1 = normalised a1

q2 = normalised a2 - (a2 dot q1) q1

q3 = normalised a3 - (a3 dot q1) q1 - (a3 dot q2) q2

Etc

a’s are columns of A

q’s are columns of Q

How do you do QR factorisation

Get Q from Gram-Schmidt

The columns of R are the coefficients of the equations for a in terms of q’s

R should be upper echelon

If you know the eigenvectors what is the column space

The eigenvectors with non zero eigenvalues

If you know the eigenvectors what is the null space

Any eigenvector with a zero eigenvalue (this is also the left null space)

How do you diagonalise a matrix

A=QLQ(t)

Where the columns of Q are the eigenvectors and L has the eigenvalues down the diagonal

What is the method for singular value decomposition

A = Q_1 S Q_2(t)

Find AA(t), Q1 is its eigenvectors and S is its eigenvalues

Find A(t)A, Q2 is its eigenvectors (eigenvalues should be the same)

Make sure you use the same columns for eigenvectors corresponding to the same eigenvalues

How do you find the eigenvalues of a triangular matrix

The eigenvalues are the values on the diagonal.

How do you find flux through a surface

Double integral F.dA, turn to a volume integral with Gauss theorem

Are you scared of spherical polar coordinates

No!

How do you do a path integral in a conservative vector field

Find the scalar potential, it is the difference between the scalar potentials at the two end points

How do you show that one matrix is the inverse if another

Show that their multiple is the identity matrix

How do you do Gaussian elimination and what is it for

To solve Ax=b without finding inverse of A by making A upper echelon

Append b to RHS of A

Subtract multiples of row 1 from lower rows to get zeros below 1,1

Repeat for all rows until upper echelon

Peel off b and solve by back substitution

How do you diagonalise a matrix

P^(-1)AP = L P has columns that are A’s eigenvectors and L is the eigenvalues down the diagonal

If you give an answer with letters in what is it important to say

Values for which there is no answer (eg 1/x for x≠0)

If you have found a least squares solution v to Ax=b, what vector p is the projection of b onto A’s column space

p=A v

What is the equation for a projection matrix p=Pb where p is b projected onto A

P = A (A(t)A)^-1 A(t)

What is in your integration toolkit

By parts, substitution, the cases given in the databook, guess and adjust