2026
MATHEMATICS II
SUMMARY
KOEN HANEGREEFS
VUB
,Table of contents
Table of contents ..................................................................................................................................... 1
Chapter 1 - Linear Geometry ................................................................................................................... 10
1.1 Vectors in ℝ𝒏................................................................................................................................ 10
Intuition .......................................................................................................................................... 10
Definition ....................................................................................................................................... 10
Worked example ............................................................................................................................. 10
Note - length is defined in 1.3 .......................................................................................................... 10
1.2 Lines, planes, hyperplanes ............................................................................................................ 10
Definition ....................................................................................................................................... 10
Rule ............................................................................................................................................... 11
Worked example ............................................................................................................................. 11
Intuition - what "linearly independent" means here ........................................................................... 11
Rule - Procedure 1-C: is a point on a flat? ......................................................................................... 12
Worked example 1.C - point on a line in R^4 ..................................................................................... 12
Rule - Procedure 1-D: Cartesian equation to parametrisation............................................................ 13
Worked example 1.D - parametrising a plane in R^3 ......................................................................... 13
Rule - Procedure 1-E: parametrisation to Cartesian equation ............................................................ 14
Worked example 1.E - Cartesian equation and the origin test ............................................................ 14
Rule - Procedure 1-F: building a flat, and cutting it with a hyperplane................................................. 15
Worked example 1.F - a 3-flat in R^4 and its slice ............................................................................. 15
1.3 Dot product, length, angle ............................................................................................................. 16
Intuition .......................................................................................................................................... 16
Definition ....................................................................................................................................... 16
Rule ............................................................................................................................................... 16
Concept check ............................................................................................................................... 16
Rule - distance between points and from a point to a line .................................................................. 17
Chapter exercises .............................................................................................................................. 17
Chapter 2 - Vector Spaces ...................................................................................................................... 18
2.1 Vector space and subspace .......................................................................................................... 18
Intuition .......................................................................................................................................... 18
Koen Hanegreefs 1
, Definition ....................................................................................................................................... 18
Rule ............................................................................................................................................... 18
Worked example ............................................................................................................................. 18
Definition - vector spaces that are not R^n ....................................................................................... 19
Worked example 2.D - a subspace of polynomials ............................................................................ 19
Rule - checking the axioms for an exotic operation ........................................................................... 20
Worked example 2.E - finding the zero element and a failing axiom .................................................... 20
Concept check - intersection versus union of subspaces .................................................................. 20
2.2 Span, linear independence, basis .................................................................................................. 21
Intuition .......................................................................................................................................... 21
Definition ....................................................................................................................................... 21
Rule ............................................................................................................................................... 21
Worked example ............................................................................................................................. 21
Intuition - what dependence really says ........................................................................................... 21
Rule - Procedure 2-A: testing linear independence............................................................................ 22
Worked example 2.F - independence of three vectors in R^3 ............................................................. 23
Worked example 2.G - is a set of polynomials a basis of P_2? ............................................................ 24
Rule - span membership and describing a span ................................................................................ 24
Worked example 2.H - membership and Cartesian equation of a span .............................................. 25
Rule - trimming to a basis and extending to a basis ........................................................................... 25
Worked example 2.I - trimming four vectors to a basis ...................................................................... 25
2.3 Coordinate representation ............................................................................................................ 26
Definition ....................................................................................................................................... 26
Worked example ............................................................................................................................. 26
Chapter exercises .............................................................................................................................. 26
Chapter 3 - Maps Between Spaces.......................................................................................................... 27
3.1 Homomorphism and isomorphism ................................................................................................ 27
Intuition .......................................................................................................................................... 27
Definition ....................................................................................................................................... 27
Rule ............................................................................................................................................... 27
Worked example ............................................................................................................................. 27
Rule - what a linear map does to pictures ......................................................................................... 28
Koen Hanegreefs 2
, 3.2 Kernel, range, rank-nullity.............................................................................................................. 28
Definition ....................................................................................................................................... 28
Rule ............................................................................................................................................... 28
Worked example ............................................................................................................................. 28
Rule - Procedure 3-A: kernel and image on an abstract space ........................................................... 29
Worked example 3.D - kernel and image of a map from P_2 to R^2 .................................................... 29
Rule - reading dimensions, injective and surjective ........................................................................... 30
3.3 Matrix representation .................................................................................................................... 30
Definition ....................................................................................................................................... 30
Worked example ............................................................................................................................. 30
Reading the identity Rep_D(h(v)) = Rep_{B,D}(h) Rep_B(v) .................................................................. 30
Rule - Procedure 3-E: from values on a non-standard basis to the standard matrix ............................. 31
Worked example 3.E - standard matrix from values on a non-standard basis ..................................... 31
Rule - Procedure 3-F: the matrix of the identity map is a change of basis ............................................ 31
Worked example 3.F - both change of basis matrices for a basis of R^2 ............................................. 32
Rule - matrix of a composition in non-standard bases ....................................................................... 32
Worked example 3.G - composition with a non-standard middle basis .............................................. 33
Chapter exercises .............................................................................................................................. 33
Chapter 4 - Matrix Operations ................................................................................................................. 34
4.1 Addition, scalar multiple, product .................................................................................................. 34
Definition ....................................................................................................................................... 34
Rule ............................................................................................................................................... 34
Worked example ............................................................................................................................. 34
Definition - transpose and identity matrix ......................................................................................... 34
Definition - elementary matrices ...................................................................................................... 35
Worked example 4.C - a matrix as a product of elementary matrices ................................................. 35
4.2 Inverse matrix ............................................................................................................................... 35
Definition ....................................................................................................................................... 35
Rule ............................................................................................................................................... 36
Worked example ............................................................................................................................. 36
Rule - Procedure 4-A: inverting by Gauss-Jordan on [A | I] .................................................................. 36
Worked example 4.D - inverting a 3x3 by Gauss-Jordan ..................................................................... 37
Koen Hanegreefs 3
, Chapter exercises .............................................................................................................................. 37
Chapter 5 - Linear Systems ..................................................................................................................... 38
5.1 Gauss elimination ......................................................................................................................... 38
Intuition .......................................................................................................................................... 38
Definition ....................................................................................................................................... 38
Rule ............................................................................................................................................... 38
Worked example 5.A - Full Gauss workflow ...................................................................................... 38
Rule - Gauss-Jordan versus plain Gauss........................................................................................... 39
Worked example 5.B - a 3x3 system by Gauss-Jordan ....................................................................... 39
Rule - computing the rank by row reduction ...................................................................................... 40
5.2 Solution-set structure ................................................................................................................... 40
Rule ............................................................................................................................................... 40
Classification ................................................................................................................................. 40
Concept check ............................................................................................................................... 40
Rule - Procedure 5-A: parametrising a solution set with free variables ............................................... 41
Worked example 5.C - 2 equations in 4 unknowns ............................................................................ 41
Chapter exercises .............................................................................................................................. 42
Chapter 6 - Determinants ....................................................................................................................... 43
6.1 Definition and small cases ............................................................................................................ 43
Definition ....................................................................................................................................... 43
Rule ............................................................................................................................................... 43
Worked example ............................................................................................................................. 43
Definition - minors, cofactors and the sign pattern ........................................................................... 44
Rule - Laplace expansion along any row or column ........................................................................... 44
Worked example 6.C - Laplace expansion using zeros ...................................................................... 45
Rule - shortcuts for large sparse determinants ................................................................................. 45
6.2 Properties and use ........................................................................................................................ 46
Rule ............................................................................................................................................... 46
Worked example ............................................................................................................................. 46
Definition - cofactor matrix, adjugate and the inverse formula ........................................................... 46
Rule - Cramer's rule ........................................................................................................................ 47
Worked example 6.D - solving a 2x2 system by Cramer ..................................................................... 47
Koen Hanegreefs 4
, Intuition - geometric meaning of the determinant ............................................................................. 48
Rule - the invertible matrix theorem ................................................................................................. 48
Chapter exercises .............................................................................................................................. 49
Chapter 7 - Eigenvalues, Eigenvectors, Diagonalisation ........................................................................... 50
7.1 Eigenvalues and eigenvectors ........................................................................................................ 50
Intuition .......................................................................................................................................... 50
Definition ....................................................................................................................................... 50
Rule ............................................................................................................................................... 51
Definition - algebraic and geometric multiplicity ............................................................................... 51
Rule - Procedure 7-A: finding eigenvectors ....................................................................................... 51
Worked example 7.B - when the two multiplicities differ ................................................................... 52
Rule - three extra eigenvalue situations ............................................................................................ 52
Worked example 7.C - a matrix with complex eigenvalues................................................................. 53
Rule - long run behaviour of powers of a matrix ................................................................................. 53
7.2 Diagonalisation ............................................................................................................................. 53
Definition ....................................................................................................................................... 53
Rule ............................................................................................................................................... 54
Worked example ............................................................................................................................. 54
Concept check ............................................................................................................................... 54
Rule - diagonalisability criterion with multiplicities ........................................................................... 55
Worked example 7.D - diagonalising and computing a high power ..................................................... 55
7.3 Change of basis and transition matrices ........................................................................................ 56
Intuition .......................................................................................................................................... 56
Definition ....................................................................................................................................... 56
Rule - Procedure 7-D: change of coordinates.................................................................................... 56
Rule - Procedure 7-E: change of basis for a linear map ...................................................................... 56
Worked example ............................................................................................................................. 56
Chapter exercises .............................................................................................................................. 57
Chapter 8 - Multivariable Calculus .......................................................................................................... 58
8.1 Partial derivatives and gradient ...................................................................................................... 58
Definition ....................................................................................................................................... 58
Rule ............................................................................................................................................... 58
Koen Hanegreefs 5
, Rule ............................................................................................................................................... 59
Worked example ............................................................................................................................. 59
Note - forward pointers from this section ......................................................................................... 59
8.2 The Hessian .................................................................................................................................. 59
Definition ....................................................................................................................................... 59
Worked example ............................................................................................................................. 59
Note - what the Hessian is used for .................................................................................................. 59
8.3 Homogeneity ................................................................................................................................ 60
Definition ....................................................................................................................................... 60
Rule - testing homogeneity .............................................................................................................. 60
Worked example 8.D - testing three functions .................................................................................. 60
Rule - Euler's theorem ..................................................................................................................... 61
8.4 Directional derivatives................................................................................................................... 61
Definition ....................................................................................................................................... 61
Rule ............................................................................................................................................... 61
Worked example ............................................................................................................................. 62
Concept check ............................................................................................................................... 62
Try it ............................................................................................................................................... 62
8.5 Limits, contour plots and the chain rule ......................................................................................... 62
Definition - multivariable limit .......................................................................................................... 62
Definition - contour plot (level curves) .............................................................................................. 62
Rule - chain rule in R^n .................................................................................................................... 63
Worked example ............................................................................................................................. 63
Note - the Jacobian matrix ............................................................................................................... 63
Chapter exercises .............................................................................................................................. 63
Chapter 9 - Optimisation ........................................................................................................................ 64
9.1 Unconstrained extrema ................................................................................................................. 64
Rule ............................................................................................................................................... 64
Worked example ............................................................................................................................. 64
Definition - leading principal minors and the n-variable test .............................................................. 65
Worked example 9.C - classifying a critical point in three variables.................................................... 65
9.2 Lagrange multipliers...................................................................................................................... 66
Koen Hanegreefs 6
, Definition ....................................................................................................................................... 66
Rule ............................................................................................................................................... 66
Worked example ............................................................................................................................. 66
Concept check ............................................................................................................................... 67
Rule - turning a word problem into a constrained optimisation .......................................................... 67
Worked example 9.D - cheapest closed crate ................................................................................... 67
Concept check - constrained optimisation ....................................................................................... 68
Chapter exercises ........................................................................................................................... 68
Chapter 10 - Multiple Integrals ................................................................................................................ 69
10.1 Iterated integrals ......................................................................................................................... 69
Definition ....................................................................................................................................... 69
Worked example ............................................................................................................................. 69
10.2 Change of variables and Jacobian ................................................................................................ 69
Definition ....................................................................................................................................... 70
Rule ............................................................................................................................................... 70
Worked example ............................................................................................................................. 70
Concept check ............................................................................................................................... 70
Rule - when the new variables are given as functions of x and y ......................................................... 70
Worked example 10.C - affine substitution over a parallelogram ....................................................... 71
Rule - polar bounds for discs, annuli and wedges ............................................................................. 71
Worked example 10.D - integrating over an annulus.......................................................................... 72
Rule - transforming and sketching a region under a substitution ........................................................ 72
10.3 Choosing the integration order..................................................................................................... 72
Rule - when to swap dx dy and dy dx ................................................................................................. 72
Worked example ............................................................................................................................. 73
Chapter exercises .............................................................................................................................. 73
Chapter 11 - Ordinary Differential Equations ........................................................................................... 74
11.0 Classification reference .............................................................................................................. 74
The four classification axes ............................................................................................................. 74
Compact reference table ................................................................................................................ 74
Worked example ............................................................................................................................. 75
Concept check ............................................................................................................................... 75
Koen Hanegreefs 7
, Try it - 4 classification exercises ....................................................................................................... 75
11.1 First-order ODEs ......................................................................................................................... 76
Rule ............................................................................................................................................... 76
Rule ............................................................................................................................................... 76
Rule ............................................................................................................................................... 76
Rule ............................................................................................................................................... 76
Worked example ............................................................................................................................. 76
Rule - first-order linear as homogeneous plus particular ................................................................... 76
11.2 Second-order linear with constant coefficients............................................................................. 77
Rule ............................................................................................................................................... 77
Rule ............................................................................................................................................... 77
Worked example ............................................................................................................................. 77
Rule - undetermined coefficients ..................................................................................................... 77
Worked example 11.C - undetermined coefficients .......................................................................... 78
Concept check - existence and uniqueness ..................................................................................... 78
11.3 Modelling economic problems with ODEs .................................................................................... 78
Intuition .......................................................................................................................................... 78
Rule - common templates ............................................................................................................... 79
Worked example - retirement fund with withdrawals ........................................................................ 79
11.4 Classification cheat-sheet........................................................................................................... 79
Chapter exercises ........................................................................................................................... 79
Chapter 12 - Cheat Sheet and Canonical Proofs ...................................................................................... 80
12.1 One-page formula sheet .............................................................................................................. 80
12.2 Canonical proofs (exam-relevant) ................................................................................................ 80
Proof 1 - Zero-scalar identity in a vector space ................................................................................. 80
Proof 2 - Span of a subset is a subspace........................................................................................... 80
Proof 3 - Kernel of a linear map is a subspace ................................................................................... 80
Proof 4 - Eigenspace is a non-trivial subspace .................................................................................. 81
Proof 5 - Solution set of 𝑨𝒙 = 𝟎 is a subspace .................................................................................. 81
Proof 6 - Distinct eigenvalues give linearly independent eigenvectors ................................................ 81
Proof 7 - rank-nullity theorem .......................................................................................................... 81
Proof 8 - adding a vector outside the span preserves independence .................................................. 81
Koen Hanegreefs 8
, Proof 9 - every n-dimensional space is isomorphic to R^n ................................................................. 82
Proof 10 - any two bases have the same number of elements ............................................................ 82
Proof 11 - when a translated subspace is a subspace ....................................................................... 82
Chapter exercises .............................................................................................................................. 83
Koen Hanegreefs 9
MATHEMATICS II
SUMMARY
KOEN HANEGREEFS
VUB
,Table of contents
Table of contents ..................................................................................................................................... 1
Chapter 1 - Linear Geometry ................................................................................................................... 10
1.1 Vectors in ℝ𝒏................................................................................................................................ 10
Intuition .......................................................................................................................................... 10
Definition ....................................................................................................................................... 10
Worked example ............................................................................................................................. 10
Note - length is defined in 1.3 .......................................................................................................... 10
1.2 Lines, planes, hyperplanes ............................................................................................................ 10
Definition ....................................................................................................................................... 10
Rule ............................................................................................................................................... 11
Worked example ............................................................................................................................. 11
Intuition - what "linearly independent" means here ........................................................................... 11
Rule - Procedure 1-C: is a point on a flat? ......................................................................................... 12
Worked example 1.C - point on a line in R^4 ..................................................................................... 12
Rule - Procedure 1-D: Cartesian equation to parametrisation............................................................ 13
Worked example 1.D - parametrising a plane in R^3 ......................................................................... 13
Rule - Procedure 1-E: parametrisation to Cartesian equation ............................................................ 14
Worked example 1.E - Cartesian equation and the origin test ............................................................ 14
Rule - Procedure 1-F: building a flat, and cutting it with a hyperplane................................................. 15
Worked example 1.F - a 3-flat in R^4 and its slice ............................................................................. 15
1.3 Dot product, length, angle ............................................................................................................. 16
Intuition .......................................................................................................................................... 16
Definition ....................................................................................................................................... 16
Rule ............................................................................................................................................... 16
Concept check ............................................................................................................................... 16
Rule - distance between points and from a point to a line .................................................................. 17
Chapter exercises .............................................................................................................................. 17
Chapter 2 - Vector Spaces ...................................................................................................................... 18
2.1 Vector space and subspace .......................................................................................................... 18
Intuition .......................................................................................................................................... 18
Koen Hanegreefs 1
, Definition ....................................................................................................................................... 18
Rule ............................................................................................................................................... 18
Worked example ............................................................................................................................. 18
Definition - vector spaces that are not R^n ....................................................................................... 19
Worked example 2.D - a subspace of polynomials ............................................................................ 19
Rule - checking the axioms for an exotic operation ........................................................................... 20
Worked example 2.E - finding the zero element and a failing axiom .................................................... 20
Concept check - intersection versus union of subspaces .................................................................. 20
2.2 Span, linear independence, basis .................................................................................................. 21
Intuition .......................................................................................................................................... 21
Definition ....................................................................................................................................... 21
Rule ............................................................................................................................................... 21
Worked example ............................................................................................................................. 21
Intuition - what dependence really says ........................................................................................... 21
Rule - Procedure 2-A: testing linear independence............................................................................ 22
Worked example 2.F - independence of three vectors in R^3 ............................................................. 23
Worked example 2.G - is a set of polynomials a basis of P_2? ............................................................ 24
Rule - span membership and describing a span ................................................................................ 24
Worked example 2.H - membership and Cartesian equation of a span .............................................. 25
Rule - trimming to a basis and extending to a basis ........................................................................... 25
Worked example 2.I - trimming four vectors to a basis ...................................................................... 25
2.3 Coordinate representation ............................................................................................................ 26
Definition ....................................................................................................................................... 26
Worked example ............................................................................................................................. 26
Chapter exercises .............................................................................................................................. 26
Chapter 3 - Maps Between Spaces.......................................................................................................... 27
3.1 Homomorphism and isomorphism ................................................................................................ 27
Intuition .......................................................................................................................................... 27
Definition ....................................................................................................................................... 27
Rule ............................................................................................................................................... 27
Worked example ............................................................................................................................. 27
Rule - what a linear map does to pictures ......................................................................................... 28
Koen Hanegreefs 2
, 3.2 Kernel, range, rank-nullity.............................................................................................................. 28
Definition ....................................................................................................................................... 28
Rule ............................................................................................................................................... 28
Worked example ............................................................................................................................. 28
Rule - Procedure 3-A: kernel and image on an abstract space ........................................................... 29
Worked example 3.D - kernel and image of a map from P_2 to R^2 .................................................... 29
Rule - reading dimensions, injective and surjective ........................................................................... 30
3.3 Matrix representation .................................................................................................................... 30
Definition ....................................................................................................................................... 30
Worked example ............................................................................................................................. 30
Reading the identity Rep_D(h(v)) = Rep_{B,D}(h) Rep_B(v) .................................................................. 30
Rule - Procedure 3-E: from values on a non-standard basis to the standard matrix ............................. 31
Worked example 3.E - standard matrix from values on a non-standard basis ..................................... 31
Rule - Procedure 3-F: the matrix of the identity map is a change of basis ............................................ 31
Worked example 3.F - both change of basis matrices for a basis of R^2 ............................................. 32
Rule - matrix of a composition in non-standard bases ....................................................................... 32
Worked example 3.G - composition with a non-standard middle basis .............................................. 33
Chapter exercises .............................................................................................................................. 33
Chapter 4 - Matrix Operations ................................................................................................................. 34
4.1 Addition, scalar multiple, product .................................................................................................. 34
Definition ....................................................................................................................................... 34
Rule ............................................................................................................................................... 34
Worked example ............................................................................................................................. 34
Definition - transpose and identity matrix ......................................................................................... 34
Definition - elementary matrices ...................................................................................................... 35
Worked example 4.C - a matrix as a product of elementary matrices ................................................. 35
4.2 Inverse matrix ............................................................................................................................... 35
Definition ....................................................................................................................................... 35
Rule ............................................................................................................................................... 36
Worked example ............................................................................................................................. 36
Rule - Procedure 4-A: inverting by Gauss-Jordan on [A | I] .................................................................. 36
Worked example 4.D - inverting a 3x3 by Gauss-Jordan ..................................................................... 37
Koen Hanegreefs 3
, Chapter exercises .............................................................................................................................. 37
Chapter 5 - Linear Systems ..................................................................................................................... 38
5.1 Gauss elimination ......................................................................................................................... 38
Intuition .......................................................................................................................................... 38
Definition ....................................................................................................................................... 38
Rule ............................................................................................................................................... 38
Worked example 5.A - Full Gauss workflow ...................................................................................... 38
Rule - Gauss-Jordan versus plain Gauss........................................................................................... 39
Worked example 5.B - a 3x3 system by Gauss-Jordan ....................................................................... 39
Rule - computing the rank by row reduction ...................................................................................... 40
5.2 Solution-set structure ................................................................................................................... 40
Rule ............................................................................................................................................... 40
Classification ................................................................................................................................. 40
Concept check ............................................................................................................................... 40
Rule - Procedure 5-A: parametrising a solution set with free variables ............................................... 41
Worked example 5.C - 2 equations in 4 unknowns ............................................................................ 41
Chapter exercises .............................................................................................................................. 42
Chapter 6 - Determinants ....................................................................................................................... 43
6.1 Definition and small cases ............................................................................................................ 43
Definition ....................................................................................................................................... 43
Rule ............................................................................................................................................... 43
Worked example ............................................................................................................................. 43
Definition - minors, cofactors and the sign pattern ........................................................................... 44
Rule - Laplace expansion along any row or column ........................................................................... 44
Worked example 6.C - Laplace expansion using zeros ...................................................................... 45
Rule - shortcuts for large sparse determinants ................................................................................. 45
6.2 Properties and use ........................................................................................................................ 46
Rule ............................................................................................................................................... 46
Worked example ............................................................................................................................. 46
Definition - cofactor matrix, adjugate and the inverse formula ........................................................... 46
Rule - Cramer's rule ........................................................................................................................ 47
Worked example 6.D - solving a 2x2 system by Cramer ..................................................................... 47
Koen Hanegreefs 4
, Intuition - geometric meaning of the determinant ............................................................................. 48
Rule - the invertible matrix theorem ................................................................................................. 48
Chapter exercises .............................................................................................................................. 49
Chapter 7 - Eigenvalues, Eigenvectors, Diagonalisation ........................................................................... 50
7.1 Eigenvalues and eigenvectors ........................................................................................................ 50
Intuition .......................................................................................................................................... 50
Definition ....................................................................................................................................... 50
Rule ............................................................................................................................................... 51
Definition - algebraic and geometric multiplicity ............................................................................... 51
Rule - Procedure 7-A: finding eigenvectors ....................................................................................... 51
Worked example 7.B - when the two multiplicities differ ................................................................... 52
Rule - three extra eigenvalue situations ............................................................................................ 52
Worked example 7.C - a matrix with complex eigenvalues................................................................. 53
Rule - long run behaviour of powers of a matrix ................................................................................. 53
7.2 Diagonalisation ............................................................................................................................. 53
Definition ....................................................................................................................................... 53
Rule ............................................................................................................................................... 54
Worked example ............................................................................................................................. 54
Concept check ............................................................................................................................... 54
Rule - diagonalisability criterion with multiplicities ........................................................................... 55
Worked example 7.D - diagonalising and computing a high power ..................................................... 55
7.3 Change of basis and transition matrices ........................................................................................ 56
Intuition .......................................................................................................................................... 56
Definition ....................................................................................................................................... 56
Rule - Procedure 7-D: change of coordinates.................................................................................... 56
Rule - Procedure 7-E: change of basis for a linear map ...................................................................... 56
Worked example ............................................................................................................................. 56
Chapter exercises .............................................................................................................................. 57
Chapter 8 - Multivariable Calculus .......................................................................................................... 58
8.1 Partial derivatives and gradient ...................................................................................................... 58
Definition ....................................................................................................................................... 58
Rule ............................................................................................................................................... 58
Koen Hanegreefs 5
, Rule ............................................................................................................................................... 59
Worked example ............................................................................................................................. 59
Note - forward pointers from this section ......................................................................................... 59
8.2 The Hessian .................................................................................................................................. 59
Definition ....................................................................................................................................... 59
Worked example ............................................................................................................................. 59
Note - what the Hessian is used for .................................................................................................. 59
8.3 Homogeneity ................................................................................................................................ 60
Definition ....................................................................................................................................... 60
Rule - testing homogeneity .............................................................................................................. 60
Worked example 8.D - testing three functions .................................................................................. 60
Rule - Euler's theorem ..................................................................................................................... 61
8.4 Directional derivatives................................................................................................................... 61
Definition ....................................................................................................................................... 61
Rule ............................................................................................................................................... 61
Worked example ............................................................................................................................. 62
Concept check ............................................................................................................................... 62
Try it ............................................................................................................................................... 62
8.5 Limits, contour plots and the chain rule ......................................................................................... 62
Definition - multivariable limit .......................................................................................................... 62
Definition - contour plot (level curves) .............................................................................................. 62
Rule - chain rule in R^n .................................................................................................................... 63
Worked example ............................................................................................................................. 63
Note - the Jacobian matrix ............................................................................................................... 63
Chapter exercises .............................................................................................................................. 63
Chapter 9 - Optimisation ........................................................................................................................ 64
9.1 Unconstrained extrema ................................................................................................................. 64
Rule ............................................................................................................................................... 64
Worked example ............................................................................................................................. 64
Definition - leading principal minors and the n-variable test .............................................................. 65
Worked example 9.C - classifying a critical point in three variables.................................................... 65
9.2 Lagrange multipliers...................................................................................................................... 66
Koen Hanegreefs 6
, Definition ....................................................................................................................................... 66
Rule ............................................................................................................................................... 66
Worked example ............................................................................................................................. 66
Concept check ............................................................................................................................... 67
Rule - turning a word problem into a constrained optimisation .......................................................... 67
Worked example 9.D - cheapest closed crate ................................................................................... 67
Concept check - constrained optimisation ....................................................................................... 68
Chapter exercises ........................................................................................................................... 68
Chapter 10 - Multiple Integrals ................................................................................................................ 69
10.1 Iterated integrals ......................................................................................................................... 69
Definition ....................................................................................................................................... 69
Worked example ............................................................................................................................. 69
10.2 Change of variables and Jacobian ................................................................................................ 69
Definition ....................................................................................................................................... 70
Rule ............................................................................................................................................... 70
Worked example ............................................................................................................................. 70
Concept check ............................................................................................................................... 70
Rule - when the new variables are given as functions of x and y ......................................................... 70
Worked example 10.C - affine substitution over a parallelogram ....................................................... 71
Rule - polar bounds for discs, annuli and wedges ............................................................................. 71
Worked example 10.D - integrating over an annulus.......................................................................... 72
Rule - transforming and sketching a region under a substitution ........................................................ 72
10.3 Choosing the integration order..................................................................................................... 72
Rule - when to swap dx dy and dy dx ................................................................................................. 72
Worked example ............................................................................................................................. 73
Chapter exercises .............................................................................................................................. 73
Chapter 11 - Ordinary Differential Equations ........................................................................................... 74
11.0 Classification reference .............................................................................................................. 74
The four classification axes ............................................................................................................. 74
Compact reference table ................................................................................................................ 74
Worked example ............................................................................................................................. 75
Concept check ............................................................................................................................... 75
Koen Hanegreefs 7
, Try it - 4 classification exercises ....................................................................................................... 75
11.1 First-order ODEs ......................................................................................................................... 76
Rule ............................................................................................................................................... 76
Rule ............................................................................................................................................... 76
Rule ............................................................................................................................................... 76
Rule ............................................................................................................................................... 76
Worked example ............................................................................................................................. 76
Rule - first-order linear as homogeneous plus particular ................................................................... 76
11.2 Second-order linear with constant coefficients............................................................................. 77
Rule ............................................................................................................................................... 77
Rule ............................................................................................................................................... 77
Worked example ............................................................................................................................. 77
Rule - undetermined coefficients ..................................................................................................... 77
Worked example 11.C - undetermined coefficients .......................................................................... 78
Concept check - existence and uniqueness ..................................................................................... 78
11.3 Modelling economic problems with ODEs .................................................................................... 78
Intuition .......................................................................................................................................... 78
Rule - common templates ............................................................................................................... 79
Worked example - retirement fund with withdrawals ........................................................................ 79
11.4 Classification cheat-sheet........................................................................................................... 79
Chapter exercises ........................................................................................................................... 79
Chapter 12 - Cheat Sheet and Canonical Proofs ...................................................................................... 80
12.1 One-page formula sheet .............................................................................................................. 80
12.2 Canonical proofs (exam-relevant) ................................................................................................ 80
Proof 1 - Zero-scalar identity in a vector space ................................................................................. 80
Proof 2 - Span of a subset is a subspace........................................................................................... 80
Proof 3 - Kernel of a linear map is a subspace ................................................................................... 80
Proof 4 - Eigenspace is a non-trivial subspace .................................................................................. 81
Proof 5 - Solution set of 𝑨𝒙 = 𝟎 is a subspace .................................................................................. 81
Proof 6 - Distinct eigenvalues give linearly independent eigenvectors ................................................ 81
Proof 7 - rank-nullity theorem .......................................................................................................... 81
Proof 8 - adding a vector outside the span preserves independence .................................................. 81
Koen Hanegreefs 8
, Proof 9 - every n-dimensional space is isomorphic to R^n ................................................................. 82
Proof 10 - any two bases have the same number of elements ............................................................ 82
Proof 11 - when a translated subspace is a subspace ....................................................................... 82
Chapter exercises .............................................................................................................................. 83
Koen Hanegreefs 9
