Accelerometry, 81 Adjacency Matrix, 129, 153 Adjoint Matrix, 141 Arabic

Index

A D Accelerometry, 81 Deﬁnite integrals, 203–209 Adjacency matrix, 129, 153 Denominator, 3 Adjoint matrix, 141 Dense matrix, 138 Arabic numerals, 2 Derivatives, 174–176 Arithmetic rules for limits, 169, 170 differential and total, 183, 184 Arrays, 129, 130 higher order, 180, 181 Asymptotic discontinuity, 170, 173 partial, 181–183 practical use, 184–191 and rules for differentiation, 177–180 B Diagonal matrix, 137, 138, 147, 148, 153, Basis vector principle, 155 154, 156 BEDMAS (Brackets-Exponent-Division- Diffusion tensor imaging (DTI), 182 Multiplication-Addition-Subtraction), 7 Domains, 167 Double integral, 209 2D sinusoidal waves, 84 C Dynamic causal modeling, 190–191 Classes of numbers, 3–16 arithmetic with exponents and logarithms, 8–10 E arithmetic with fractions, 5–7 Eigendecomposition, 156 numeral systems, 10–16 Eigenvalues, 145, 146 Column-matrices, 130 Eigenvectors, 145, 146 Common denominator, 5 Electromyography, 81 Complex numbers, 4, 15, 16 Equations, 27, 34–39 Conjugate transpose, 137, 139, 153, general deﬁnitions, 29 154, 156 solving linear equations, 29–34 Cramer’s rule, 39, 57, 142, 143, 158 by elimination, 36–38 Cross product, 109, 110, 112, 113 solving graphically, 38, 39

Equations (cont.) L by substitution, 34–36 Least squares fitting, 187–189 using Cramer’s rule, 39 Leslie matrix, 129, 153 types, 29 Limits, 163–165 Euclidean length, 107 application, 172, 173 Euler’s formula, 72–74, 76, 79, 90 arithmetic rules, 169, 170 Exponent, 8 determining graphically, 167–169 at infinity, 170–172 intuitive definition, 166, 167 F special limits, 173, 174 Factor multiplication rule, 58 Logarithmic equations, 48, 49 Finite decimal numbers, 3 Logical matrix, 137, 138, 153 Lower-triangular matrix, 138

G Gram-Schmidt (or Schmidt-Gram) M orthogonalization, 116 Mathematical symbols and formulas, 16–19 conventions for writing, 17 Latin and Greek letters, 17, 18 H reading formulas, 17–19 Hadamard product, 136, 153–155 Matrices, 129 Hermitian matrix, 137, 138, 156 Matrices as transformations, 133–137 Higher order derivatives, 180, 181 Matrix operations, 139–152 addition, 131, 132 alternative multiplication, 136, 137 I applications, 139–152 Identity matrix, 137, 138 diagonalization, 147, 148 Imaginary axis, 13 Eigenvectors and eigenvalues, 145, 146 fi – Inde nite integrals, 199 203, 206 inverse and determinant, 139–144 – Inequations, 50 54 SVD, 148–152 fi In nite decimal numbers, 3 multiplication, 131–135 – Integrals, 199 200 alternative, 136–137 fi – de nite, 203 209 subtraction, 131, 132 fi – inde nite, 200 203 Moment arm, 112 – multiple, 208 209 Multiple integrals, 208–209 Integration techniques, 209–219 integration by parts, 209–212 integration by reverse chain rule, N 215–217 Normal matrix, 137, 138, 156 integration by substitution, 212–215 Null vector, 103 Intuitive definition of limit, 166, 167 Numbers and mathematical symbols, 1, 2 Irrational, 3 Numerator, 3

K O Kronecker product, 136, 137, 153, 156 Orthogonalize vector, 115, 116, 118, 119 Index 233

P logical matrix, 137 Partial derivatives, 181–183 normal matrix, 137 Pythagoras’ theorem, 65 square matrix, 137 triangular matrix, 137 unit matrix, 137 Q unitary matrix, 137 Quadratic equation rule, 58 Square matrix, 131, 135, 137, 140, 143, 145, – Quadratic equations, solving, 39 46 147–150, 153, 156 – by factoring, 43 46 Substitution, 34 – graphically, 41 42 Symmetric matrix, 138 using quadratic equation rule, 42, 43 Quadratic function, 184 Quadratic inequations, solving, 52–54 T Transcendental equations, 47–49 Triangular matrix, 137, 154 R Trigonometry, 61, 62 Rational equations, 46 degrees and radians, 66, 67 Real axis, 13 Euler’s formula and trigonometric formulas, Real numbers, 3 72–74 Roman numeral system, 10 Fourier analysis, 74–87 Rotation matrix, 155 functions and definitions, 68–74 Row-matrices, 130 trigonometric ratios and angles, 63–67 Triple integral, 209 S Scientific notation, 12 U Shearing matrix, 155 Unit matrix, 137, 154 fi Signi cant digits, 12 Unitary matrix, 137, 138, 156 – Singular value decomposition (SVD), 148 152 Upper-triangular matrix, 138 Skew-symmetric matrix, 138 Smooth function, 184 Solving linear equations, 29–34 V by elimination, 36–38 Vector, 99–120 solving graphically, 38, 39 covariance, 114 by substitution, 34–36 cross product, 109–113 using Cramer’s rule, 39 linearly combination, 113 Sparse, 137, 143, 144, 154, 158 linearly dependent, 113 Sparse matrix, 138 multiplication, 105–113 Special matrices, 137–139 operations, 101–113 diagonal matrix, 137 projection and orthogonalization, 115–119 Hermitian matrix, 137 Vertical axis, 13 identity matrix, 137