IA: 1P4: Mathematics Flashcards

Question

How can you determine if three planes all intersect at a point?

Answer 1

When none of the planes are parallel and solving the simultaneous equations gives a unique solution

Answer 2

f**'**(0) = 0

Answer 3

* **Limits**: What happens at x→±∞ * **Intercepts**: When does f(x) = 0, and likewise what is the value of f(0)? * **Symmetry**: is f(x) even or odd (or neither), or is there a point about which it is symmetric * **Turning points**: When f**'**(x) = 0 what sign is f**''**(x) (maxima, minima, inflection) * **Singularities**: Is there a value such that |f(x)|→±∞

Answer 4

## Footnote Note: odd function

Answer 5

## Footnote Note: even function

Answer 6

## Footnote Note: odd function

Answer 7

A taylor series for a function f(x) is a polynomial expansion about a point x = a such that:

Answer 8

## Footnote Where f⁽ⁿ⁾(a) is the n'th derivative of f(x) evaluated at a

Answer 9

It is the taylor series expanded about x = 0

Answer 10

The radius of convergence of a Taylor series is the distance from the center of the series (the expansion point) to the nearest point where the series fails to converge

Answer 11

A common approach is to let y = 1/x and then consider a series for which y is small, y→0.

Answer 12

For indeterminate cases where the limit produces 0/0 or ∞/∞

Answer 13

You can use a series expansion (such as taylor or maclaurin) to approximate a function in polynomial form and use this to approximate the limit

Answer 14

A circle of radius k centred at the complex number z₁

Answer 15

An ellipse

Answer 16

Let z = x + jy and then expand to form a cartesian equation

Answer 17

Example: Rather than using integration by parts, you can replace the sin with an exponential using Euler's formula and just take the imaginary component at the end of the workings

Answer 18

Let: x + jy = arcsin(2) Solve using compound angle formula and hyperbolic functions

Answer 19

A differential equation where the dependent variable is only a function of the single independent variable

Answer 20

A differential equation which has more than one independent variable

Answer 21

* Distinct Integration * Separable Equations * Integrating Factor method * Equations reducable to the separable form (scaling)

Answer 22

y = x arctan(Ln|x| + c)

Answer 23

An ordinary differential equation in which the dependent variable (such as y) and its derivatives only appear as a linear combination.

Answer 24

An ordinary differential equation which has no functions of the independent variable appearing on its own (or simply the right hand side of the equation is zero)

Answer 25

1. Real and distinct roots 2. Complex conjugate roots 3. Repeated real roots

Answer 26

1. Find the general solution of the homogenous equation (known as the complementary function) 2. Determine the particular integral 3. The general solution is the complementary function + particular integral

Answer 27

Trial and error, however there is a table of common particular integrals in the data booklet:

Answer 28

**When the right hand side of the differential equation has the same form as part of the complementary function**. In these cases an alternative PI should be used. These also appear in a table in the data booklet:

Answer 29

1. Draw a diagram of an element 2. Balance forces/moments or energy fluxes to get an ODE 3. Take δx → 0 to obtain an ODE 4. Solve the ODE

Answer 30

A type of recurrence relation that relates the values of a sequence at different points, with the relationship being linear

Answer 31

**Substitute yₙ = Aλⁿ:** This will form a quadratic equation that functions in the same way as an auxiliary equation. Therefore you can then continue by solving in a similar way to a second order linear ODE: ## Footnote We are looking for a solution in the form **yₙ = Aλⁿ**

Answer 32

1. Find the "complementary function" by solving with the RHS as 0 2. Find a "particular solution" this is analagous to a particular integral 3. General solution = complementary function + particular solution 4. Determine constants from intial conditions

Answer 33

For partial differentiation with respect to x, **y is treated as a constant** and you are slicing through the surface parallel with the x axis. Therefore, when differentiating, you do it completely as normal whilst treating y as a constant.

Answer 34

The value of the small change of a function when the small changes in the independent variables tend to zero

Answer 35

A method of approximating a multivariable function near a given point using the first-order terms of its Taylor series expansion

Answer 36

The gradient along a particular direction: It is the rate of change of the function f along the direction **u** = (a,b), where **u** is a unit vector

Answer 37

∇f = ∇f(x,y) **u** = (a,b)

Answer 38

The gradient of a multivariable function f(x,y)

Answer 39

p' = transformed coordinates A = matrix p = original coordinates

Answer 40

An identity matrix, I, is defined such that for a square matrix A:

Answer 41

No, the order is important

Answer 42

The transpose of a matrix is formed by intercahnging the rows and columns of a matrix

Answer 43

For a square matrix A, the inverse A⁻¹ is defined such that: A⁻¹A = AA⁻¹ = I

Answer 44

1. Find the determinant of A, Det A 2. Form the matrix of minors, M 3. From the matrix of minors, form the matrix of cofactors, C 4. Find the Transpose of the matrix of cofactors, Cᵀ 5. A⁻¹ = (1/DetA) Cᵀ

Answer 45

The transpose of the cofactor matrix

Answer 46

When 2 rows (or columns) are exchanged

Answer 47

If two rows (or columns) are equal

Answer 48

You can add a multiple of one row to another row (or one column to another) as it will leave the determinant unchanged

Answer 49

They do not, they are the same

Answer 50

* Rotations * Reflections * Pure stretches/compressions

Answer 51

Consider the columns of the transformation vector (a1, a2, a3). These are the vectors that result from transforming the x- y- and z- unit basis vectors respectively. So to determine the matrix, all you must establish is where the three basis vectors map to under the transformation.

Answer 52

anticlockwise

Answer 53

A matrix with the property:

Answer 54

* A proper orthogonal matrix * A rotation matrix

Answer 55

Unit vectors that are mutually perpendicular

Answer 56

No, rotations do not commute in 3D **unless** they are about the same axis. Therefore rotations are commutable in 2 dimensions

Answer 57

It has the same effect as keeping the coordinate frame fixed, but rotating the vector in the opposite direction. Therefore: ## Footnote Rᵀ = R⁻¹ as it is a pure rotation

Answer 58

A' = Rᵀ A R

Answer 59

You can find the transformation in a convenient coordinate system that makes the shear transformation simple. You can then transform it back to the orignal coordinates. All you require is the orthognal matrix R that describes the rotation of the original coordinate system to the new coordinate system: e' = Re. A' = RᵀAR

Answer 60

**An eigenvector is a special vector associated with a square matrix (or a linear transformation) that remains aligned in the same direction after the transformation**, although it may be scaled (stretched or compressed). Where **A** is the transformation, **x** is a non-zero vector, and λ is a scalar: **x** is said to be an eigenvector if it satisfies the equation:

Answer 61

It is a scalar value known as an **eigenvalue**

Answer 62

The scale factor applied to the magnitude of the eigenvector following the transformation

Answer 63

Yes, the only important thing is that the eigen**vector** is non-zero

Answer 64

No, only the direction. If **x** is an eigenvector of **A**, then any scalar multiple of **x** is also an eigenvector of **A**. It is customary (but not obligatory) to normalise the eigenvectors so that they are unit vectors anyway.

Answer 65

A final (optional) step can then be to normalise the eigenvectors to make them unit length

Answer 66

For a nxn matrix, there will be exactly n eigenvalues.

Answer 67

* The eigenvalues are real * The eigenvectors are orthogonal

Answer 68

**A**ᵀ = -**A**

Answer 69

**A defective matrix is an nxn matrix which has fewer than n linearly independent eigenvectors**. This *may* occur if the matrix has a repeated eigenvalue

Answer 70

**A** = **U Λ U**⁻¹

Answer 71

For **U**⁻¹ to exist, det(**U**) ≠ 0. For a 3x3 matrix, this requires **u**₁, **u**₂, and **u**₃ to be linearly independent. Therefore defective matrices cannot be diagonalised as they do not have a full set of linear independent eigenvectors.

Answer 72

A symmetric matrix has orthogonal eigenvectors so they are linearly independent and so **U**⁻¹ always exists. Furthermore, **U**⁻¹ = **U**ᵀ and so for a symmetric matrix S:

Answer 73

It represents the same physical transformation as **S** but in a basis (coordinate system) aligned with the orthogonal eigenvectors. This is because:

Answer 74

For any symmetric matrix, the transformation it represents is a pure stretch (or compression) along the **mutually orthogonal** eigenvectors with the scale factor being the eigenvalues (which are always real). There is no rotation or skewing as they are scaling along orthogonal axes.

Answer 75

The determinant of the matrix, and subsequently the scale factor of the transformation

Answer 76

## Footnote Note: if λ₂ was the largest eigenvalue, it would be α₂ and **u**₂ etc etc

Answer 77

An eigenplane of a matrix or linear transformation is a two-dimensional subspace (a plane) on which any vector gets mapped to another point on this plane following the linear transformation. An example of where an eigenplane may occur is a 3D rotation

Answer 78

* When there is a complex conjugate pair of eigenvalues * When two of the eigenvectors are complex

Answer 79

In a linear system the output is computed as some linear combination of the inputs

Answer 80

1. Linear systems satisfy the principle of superposition (see image) 2. A linear **time invariant** system have the property that a sine wave at the input leads to a sine wave at the output, with the same frequency (However the amplitude and/or phase can change)

Answer 81

A system is time invariant if delaying an input results in the same output, just delayed (by the same amount): if f(t) → y(t), then f(t - τ) → y(t - τ) for any τ

Answer 82

A step function is a piecewise constant function that changes values only at specific points, creating a series of flat segments or "steps"

Answer 83

The unit step function where the step is at t = 0

Answer 84

The delta function is a spike with unit area, it is infinitely tall but infinitely thin. δ(x-a) is zero everywhere apart from at a. The definite integral of δ(x-a), where the bounds pass over x = a will equal 1, if the bounds do not pass over x = a it will equal zero.

Answer 85

* The integral of the delta function is the step function * The derivative of the step function is the delta function

Answer 86

When you integrate a function multiplied by a delta function, the result is simply the function’s value at the point where the delta function is centered. ## Footnote If "b" is not within the integration range, then the output is zero.

Answer 87

Using convolution

Answer 88

Set the input f(t) = H(t). However since this cannot be done directly we must:

Answer 89

step response

Answer 90

impulse response

Answer 91

Solve the differential equation to find the impulse response (by differentiating the step response) and then use superposition to find the corresponding output:

Answer 92

Considering the input f(t) to be made up of a sequence of strips of width Δτ, each strip can be formed from a scaled and delayed delta function. Therefore the output is the sum of these delated, scaled impulse responses. As Δτ approaches zero, this sum turns into an integral called the convolution integral.

Answer 93

* Treat t as a constant, τ is the integration variable * t is the time as it relates to the output of the system y(t) * τ is the time as it relates to the input of the system f(t)

Answer 94

It allows you to find the output of a differential equation from the input f(t) once you have determined the impulse response

Answer 95

**You may have to split up the integrals if there is a piecewise function for an input (i.e a series of step functions)**. However, when splitting them up you must remember that the original convolution integral starts from -∞ and so you must include all previous stages aswell. By including the previous stages over their entire ranges (rather than up to time t) it gives a constant as the start point for the stage! For example: ## Footnote Note: For the second and third stages the stages previous have still been included over their entire range (rather than up to t) as to give a constant!!!

Answer 96

If the expression for g(t) is very complicated, it may be easier to compute the integral if it has a factor in the form g(τ) rather than g(t-τ). Therefore you can swap the arguments to the functions in the convolution integral to make it easier to compute. It does not matter which way round the arguments to the functions in the convolution integral are, **so long as both functions are zero for t < 0** ## Footnote Only valid when both functions are zero for t < 0!!! (This is almost always the case for time-varying systems)

Answer 97

These are systems which have a temporal input (time), they have no output before the input that causes it and so g(t) = 0 for t < 0

Answer 98

These are systems which respond to spatial inputs, the inputs can affect the output on either side and therefore **g(x) can be non-zero for any x (including x < 0)**

Answer 99

Only "past" inputs contribute to the output at t

Answer 100

Inputs to both the left and right of x contribute to the output at x

Answer 101

Note: upper bound is not x

Answer 102

Systems for which g(t) = 0 for all t < 0

Answer 103

Systems with time-varying inputs and outputs

Answer 104

If you can determine the spatial impulse response, you can compute the complete displacement under a continuous load K using the spatial convolution integral. To find this impulse response you first have to determine the maximum displacement for a point load, F = 1, at position a. This lets you express the impulse response as two piecewise functions, corresponding to the two straight-line segments of the response. You can then use the convolution integral (split into a sum for the 2 sections) to find the overall displacement as a function of x.

Answer 105

A Fourier series represents a periodic function as an infinite sum of sines and cosines with different frequencies and amplitudes.

Answer 106

**For a length of 2π:**

Answer 107

Note: the bounds do not have to be +-π, they can be any combindation with a length of 2π. f(t) is only modelled within this range and so will repeat with a period of 2π outside this range and thus will only be useful over this range (unless f(t) is periodic itself)

Answer 108

Note: the bounds do not have to be +-π, they can be any combindation with a length of 2π. f(t) is only modelled within this range and so will repeat with a period of 2π outside this range and thus will only be useful over this range (unless f(t) is periodic itself)

Answer 109

Note: the bounds do not have to be +-π, they can be any combindation with a length of 2π. f(t) is only modelled within this range and so will repeat with a period of 2π outside this range and thus will only be useful over this range (unless f(t) is periodic itself).

Answer 110

**bₙ = 0** The aₙ terms model the even component and the bₙ terms model the off component in the function. The d term models the mean value of the function.

Answer 111

**aₙ = 0** The aₙ terms model the even component and the bₙ terms model the off component in the function. The d term models the mean value of the function.

Answer 112

**d = 0** The aₙ terms model the even component and the bₙ terms model the off component in the function. The d term models the mean value of the function.

Answer 113

the fraction 2π/L, often written as ω₀. It appears in the expression for a fourier series:

Answer 114

1. Use the general range Fourier formulae directly 2. Differentiate the waveform twice to get a sequence of delta functions. Find a Fourier series for the delta functions, and then integrate the series twice to get the Fourier series of the triangular wave. 3. Look up the Fourier series of a similar waveform in the Maths Data book and use a substitution of variables to find the series for the waveform we require

Answer 115

You must choose your bounds carefully as to ensure the limit does not lie on a delta function as it is uncertain what to do in that scenario. In the below example rather than choosing bounds of 0 and L, you may choose -L/4 and 3L/4. The fourier series will still have a period of L, but the integral becomes much clearer.

Answer 116

The convergence of a Fourier series describes how closely the series approximates the original function as more terms are added, i.e. a square wave converges with 1/n and a triangular wave converges with 1/n²

Answer 117

It does NOT converge

Answer 118

Converges as 1/n

Answer 119

Converges as 1/n²

Answer 120

Converges as 1/n³

Answer 121

If you want to model a signal f(x) in the range 0 to T, you can use the fourier formulae for a general series to generate a variery of different serieses. They will all be the same in the range 0 to T, but since this is the only part we care about they may differ outside this range. Different serieses converge at different rates and therefore may be suitable for different scenarios.

Answer 122

* Some series may converge faster than others * Some series are easier to calculate (i.e. some of aₙ, bₙ, or d are zero) * You may be limited by the demands of the questions (i.e. it must be even or odd or have a given convergence)

Answer 123

These can all be represented by: **eʲⁿᵗ** ## Footnote j = i

Answer 124

Note: Lower bound is -∞, not 1

Answer 125

Note: Lower bound is -∞, not 1

Answer 126

Note: bounds do not have to be between 0 and 2π, they can be any combination as long as the length is 2π

Answer 127

Cₙ may have an undefined value for a given n, therefore you must take the limit for this value. Notice in the below example the fourier series has been split into 3 sections: -∞ to -1, 0, and 1 to ∞. This ensures that the undefined value is accounted for ## Footnote Remember to look out for this!

Answer 128

You can determine the real coefficients aₙ, bₙ, and d from the complex coefficient cₙ: * **aₙ = 2 Re(Cₙ)** * **bₙ = -2 Im(Cₙ)** * **d = c₀** These are then the coefficients you can use for the normal Fourier series

Answer 129

You can determine the complex coefficient Cₙ from the real coefficients aₙ and bₙ:

Answer 130

n! (n factorial)

Answer 131

A probability distribution where each event can only carry certain integer values for the probability

Answer 132

Where each event has the same probability

Answer 133

A sample mean is the mean average of a sample, it is also a random variable as many samples can be taken from the population

Answer 134

The normal distribution

Answer 135

## Footnote μ = mean σ = standard deviation

Answer 136

Regardless of the original distribution of a population, the distribution of the sample means (or sums) approaches a normal (Gaussian) distribution as the sample size increases, provided the samples are independent and identically distributed

Answer 137

* 50% of the data falls within 0.67σ of μ * 68% of the data falls within 1σ of μ * 95% of the data falls within 2σ of μ * 99.73% of the data falls within 3σ of μ