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Mathematics, Calculus and Linear Algebra Reference Card

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Math and Calculus Cheat Sheet

Overview

Tooling

Logic and Proof

Terminology:

1 Physical Units and SI Prefixes

Fundamental Units of S.I System

Physical Quantity Unit Symbol
Length metre \(m\)
Mass Kilogram \(Kg\)
Time Seconds \(s\)
Temperature Kelvin \(K\)
Planar Angle Radian \(\text{rad}\)

Main S.I Units

Physical Quantity Unit Symbol
Force metre \(m\)
Work Joule \(J\), \(N.m\)
Energy Joule \(J\), \(N.m\)
Power Watt \(W\)
Area square metre \(m^2\)
Volume cubic metre \(m^3\)
Pressure Pascal \(Pa\)
Stress Pascal \(Pa\)
Velocity/Speed meters per seconds \(m/s\)
Acceleration meters per square seconds \(m/s^2\)
Angular velocity radians per seconds \(rad/s\)
Angular acceleration radians per square seconds \(rad/s^2\)
Momentum (linear) Kilogram metre per second \(Kg m / s\)
Torque (Moment) Newton mere \(N.m\)
Density Kilogram/metre3 \(Kg/m^3\)

S.I Prefixes

Multiplication Factor Prefix Symbol
\(10^{12}\) Tera \(T\)
\(10^{9}\) Giga \(G\)
\(10^{6}\) Mega \(M\)
\(10^{3}\) kilo \(k\)
\(10^{2}\) hecto \(h\)
\(10^{1}\) deca \(da\)
\(10^{-1}\) deci \(d\)
\(10^{-2}\) centi \(c\)
\(10^{-3}\) milli \(m\)
\(10^{-6}\) micro \(\mu\)
\(10^{-9}\) nano \(n\)
\(10^{-12}\) pico \(p\)

2 Fundamental Curves

2.1 Line Function

A general line equation has the following format, where \(\alpha\) is the line slope or anglular coefficient and \(\beta\) is the intercept of the line and the Y axis when x = 0.

$$ y = f(x) = \alpha x + \beta $$

Given two points of this line equations \((x_1, \, y_1)\) and \((x_2, \, y_2)\), the line equation can be found by using,

$$ y - y_1 = \frac{y_2 - y_1}{x_2 - x_1}(x - x_1) $$

so, the line slope coefficient \(\alpha\) is given by,

$$ \alpha = \frac{y_2 - y_1}{x_2 - x_1} $$

and the coefficient \(\beta\) can be determined as:

$$ \beta = - \alpha x_1 + y_1 $$

The solution of the line equation \(f(x) = 0\), is the point x where this curve intercepts the \(X\) axis or \(y = 0\). It is observable that this point is:

$$ x = - \beta / \alpha $$

2.2 Quadratic Equation and Quadratic Curve

A quadratic cuve has the format \(\eqref{EqQuadratic}\), where \(a \in \mathbb{R}\), \(b \in \mathbb{R}\) and \(c \in \mathbb{R}\).

$$ \label{EqQuadratic} f(x) = a x^2 + b x + c $$

Solution of Quadratic Equation

The solution of the quadtratic equation \(f(x) = 0\) can be found as:

$$ \begin{cases} x &= \underbrace{ \dfrac{-b \pm \sqrt{\Delta}}{2a} }_{\text{real roots}} \in \mathbb{R} \quad &\text{if } \Delta \geq 0 \\ x &= \underbrace{ \dfrac{-b \pm \mathrm{j} \sqrt{-\Delta}}{2a} }_{\text{complex conjugate roots}} \in \mathbb{C} \quad &\text{if } \Delta < 0 \end{cases} $$

Where \(\Delta\) is given by:

$$ \Delta = b^2 - 4ac $$

Determine Quadratic Equation from Points

Given three distinct points \((x_1, \, y_1)\), \((x_2, \, y_2)\) and \((x_3, \, y_3)\) of a quadratic cuve \(f(x) = ax^2 + bx + c\), the coefficients of the quatratic curve that intercepts those points can be determined by solving the following linear system for the column vector \([a \, b \, c]^T\).

$$ \begin{bmatrix} x_1^2 & x_1 & 1 \\ x_2^2 & x_2 & 1 \\ x_3^2 & x_3 & 1 \end{bmatrix} \begin{bmatrix} a \\ b \\ c \end{bmatrix} = \begin{bmatrix} y_1 \\ y_2 \\ y_3 \end{bmatrix} $$

Solve the equation in Python's Sympy

Since this linear system is small, it is worth solving it in symbolic form in order to be able to implement it as software.

Import Sympy and enable pretty printing. 

import sympy as sp
sp.init_printing()

Create symbols and matrices. 

x1, x2, x3, y1, y2, y3, a, b, c = sp.symbols("x1 x2 x3 y1 y2 y3 a b c")

A = sp.Matrix([[x1**2, x1, 1], [x2**2, x2, 1], [x3**2, x3, 1]])

x = sp.Matrix([a, b, c])
y = sp.Matrix([y1, y2, y3])

Create the equation object. Solve A * x = y indirectly by converting this equation to sympy standard format eqrhs: A * x - y, that solves eqrhs == 0 or A * x - y == 0 for x, vector of variables a, b and c. 

eq = A * x - y
 
>>> eq
⎡    2                ⎤
⎢a⋅x₁  + b⋅x₁ + c - y₁⎥
⎢                     ⎥
⎢    2                ⎥
⎢a⋅x₂  + b⋅x₂ + c - y₂⎥
⎢                     ⎥
⎢    2                ⎥
⎣a⋅x₃  + b⋅x₃ + c - y₃⎦

Solve the equation system for a, b and c. 

sols = sp.solve([eq[0], eq[1], eq[2]], (a, b, c))

>>> sols[a]
  -x₁⋅y₂ + x₁⋅y₃ + x₂⋅y₁ - x₂⋅y₃ - x₃⋅y₁ + x₃⋅y₂   
───────────────────────────────────────────────────
  2        2           2        2     2           2
x₁ ⋅x₂ - x₁ ⋅x₃ - x₁⋅x₂  + x₁⋅x₃  + x₂ ⋅x₃ - x₂⋅x₃ 

>>> sols[b]
  2        2        2        2        2        2   
x₁ ⋅y₂ - x₁ ⋅y₃ - x₂ ⋅y₁ + x₂ ⋅y₃ + x₃ ⋅y₁ - x₃ ⋅y₂
───────────────────────────────────────────────────
  2        2           2        2     2           2
x₁ ⋅x₂ - x₁ ⋅x₃ - x₁⋅x₂  + x₁⋅x₃  + x₂ ⋅x₃ - x₂⋅x₃ 
 
>>> sols[c]
  2           2              2           2        2              2   
x₁ ⋅x₂⋅y₃ - x₁ ⋅x₃⋅y₂ - x₁⋅x₂ ⋅y₃ + x₁⋅x₃ ⋅y₂ + x₂ ⋅x₃⋅y₁ - x₂⋅x₃ ⋅y₁
─────────────────────────────────────────────────────────────────────
           2        2           2        2     2           2         
         x₁ ⋅x₂ - x₁ ⋅x₃ - x₁⋅x₂  + x₁⋅x₃  + x₂ ⋅x₃ - x₂⋅x₃

Generate the LaTeX expression for the values of a, b and c. 

>>> print(sp.latex(sols[a]))
\frac{- x_{1} y_{2} + x_{1} y_{3} + x_{2} y_{1} - x_{2} y_{3} - x_{3} y_{1} + x_{3} y_{2}}{x_{1}^{2} x_{2} - x_{1}^{2} x_{3} - x_{1} x_{2}^{2} + x_{1} x_{3}^{2} + x_{2}^{2} x_{3} - x_{2} x_{3}^{2}}

>>> print(sp.latex(sols[b]))
\frac{x_{1}^{2} y_{2} - x_{1}^{2} y_{3} - x_{2}^{2} y_{1} + x_{2}^{2} y_{3} + x_{3}^{2} y_{1} - x_{3}^{2} y_{2}}{x_{1}^{2} x_{2} - x_{1}^{2} x_{3} - x_{1} x_{2}^{2} + x_{1} x_{3}^{2} + x_{2}^{2} x_{3} - x_{2} x_{3}^{2}}


>>> print(sp.latex(sols[c]))
\frac{x_{1}^{2} x_{2} y_{3} - x_{1}^{2} x_{3} y_{2} - x_{1} x_{2}^{2} y_{3} + x_{1} x_{3}^{2} y_{2} + x_{2}^{2} x_{3} y_{1} - x_{2} x_{3}^{2} y_{1}}{x_{1}^{2} x_{2} - x_{1}^{2} x_{3} - x_{1} x_{2}^{2} + x_{1} x_{3}^{2} + x_{2}^{2} x_{3} - x_{2} x_{3}^{2}}

Solution:

$$ \begin{split} a &= \frac{- x_{1} y_{2} + x_{1} y_{3} + x_{2} y_{1} - x_{2} y_{3} - x_{3} y_{1} + x_{3} y_{2}}{x_{1}^{2} x_{2} - x_{1}^{2} x_{3} - x_{1} x_{2}^{2} + x_{1} x_{3}^{2} + x_{2}^{2} x_{3} - x_{2} x_{3}^{2}} \\ b &= \frac{x_{1}^{2} y_{2} - x_{1}^{2} y_{3} - x_{2}^{2} y_{1} + x_{2}^{2} y_{3} + x_{3}^{2} y_{1} - x_{3}^{2} y_{2}}{x_{1}^{2} x_{2} - x_{1}^{2} x_{3} - x_{1} x_{2}^{2} + x_{1} x_{3}^{2} + x_{2}^{2} x_{3} - x_{2} x_{3}^{2}} \\ c &= \frac{x_{1}^{2} x_{2} y_{3} - x_{1}^{2} x_{3} y_{2} - x_{1} x_{2}^{2} y_{3} + x_{1} x_{3}^{2} y_{2} + x_{2}^{2} x_{3} y_{1} - x_{2} x_{3}^{2} y_{1}}{x_{1}^{2} x_{2} - x_{1}^{2} x_{3} - x_{1} x_{2}^{2} + x_{1} x_{3}^{2} + x_{2}^{2} x_{3} - x_{2} x_{3}^{2}} \end{split} $$

Code Generation using CSE - Common Subexpression Elimination: 

>>> result = sp.Matrix([sols[a], sols[b], sols[c]])

>>> u = sp.numbered_symbols("u")
>>> r = sp.cse(result, symbols = u)

# --- Intermediate Variables -------#
for var, expr in r[0]: print(f" {var} := {expr}")

 u0 := x3**2
 u1 := u0*x1
 u2 := x1**2
 u3 := u2*x2
 u4 := x2**2
 u5 := u4*x3
 u6 := u4*x1
 u7 := u2*x3
 u8 := u0*x2
 u9 := 1/(u1 + u3 + u5 - u6 - u7 - u8)
   
eqq = r[1][0]

# --------- Final Variables -------#
>> print(f" a := {eqq[0]}")
 a := u9*(-x1*y2 + x1*y3 + x2*y1 - x2*y3 - x3*y1 + x3*y2)

>> print(f" b := {eqq[1]}")
 b := u9*(u0*y1 - u0*y2 + u2*y2 - u2*y3 - u4*y1 + u4*y3)

>>> print(f" c := {eqq[2]}")
 c := u9*(u1*y2 + u3*y3 + u5*y1 - u6*y3 - u7*y2 - u8*y1)

2.3 Circle Equation

The equation of circle with center at point \(\mathbf{r}_o = (x_o, y_o)\) and radius \(\rho\) can be stated in vector form as as

$$ \| \mathbf{r} - \mathbf{r}_o \| = \rho $$

in algebraic form this equation is

$$ (x - x_o)^2 + (y - y_o)^2 = \rho^2 $$
$$ \notag \| \mathbf{r} - \mathbf{r_o} \|^2 = \rho^2 $$
$$ \notag \| (x, y) - (x_o, y_o) \|^2 = \rho^2 $$
$$ \notag \| (x - x_o, y - y_o) \|^2 = \rho^2 $$

Hence,

$$ \notag (x - x_o)^2 + (y - y_o)^2 = \rho^2 $$

or in less concise form

$$ x^2 + y^2 + Dx + Ey + F = 0 $$

Where,

Finding the circle that passes through three points

Given the points \(\mathbf{r}_1 = (x_1, y_1)\), \(\mathbf{r}_2 = (x_2, y_2)\), and \(\mathbf{r}_3 = (x_3, y_3)\), it is possible to find the equation of the circle that passes through those points in the form \(x^2 + y^2 + Dx + Ey + F = 0\) by solving the following system of equations for the parameters \(D\), \(E\) and \(F\).

$$ \begin{bmatrix} x_1 & y_1 & 1 \\ x_2 & y_2 & 1 \\ x_3 & y_3 & 1 \end{bmatrix} \begin{bmatrix} D \\ E \\ F \end{bmatrix} = \begin{bmatrix} -(x_1^2 + y_1^2) \\ -(x_2^2 + y_2^2) \\ -(x_3^2 + y_3^2) \end{bmatrix} $$

or in terse form

$$ \begin{bmatrix} | & | & | \\ \mathbf{x} & \mathbf{y} & \mathbf{1}_3 \\ | & | & | \end{bmatrix} \begin{bmatrix} D \\ E \\ F \end{bmatrix} = - (\mathbf{x}.^2 + \, \mathbf{y}.^2) $$

where

The position of circle center \(\mathbf{r}_o = (x_o, y_o)\) and the radius \(\rho\) can be found as

$$ x_o = - \frac{1}{2} D \quad y_o = -\frac{1}{2} E \quad \rho^2 = x_o^2 + y_o^2 - F $$

Example (qd-edu-hk)

Find the equation of the circle passing through the pointsP(2,1), Q(0,5), R(-1,2).

Solution in Python's numpy

Import numpy 

import numpy as np
import linalg as nl

Define the input data. 

x = np.array([2, 0, -1])
y = np.array([1, 5,  2])

Create coefficients array of the linear system. 

A = np.c_[x, y, np.ones(3)]

>> A
array([[ 2.,  1.,  1.],
       [ 0.,  5.,  1.],
       [-1.,  2.,  1.]])

Define vector of constants. NOTE: the method call .reshape(-1, 1) is used for ensure that \(\mathbf{b}\) is a column vector with shape (2, 1). 

b = -(x**2 + y**2).reshape(-1, 1)

>> b
array([[ -5],
       [-25],
       [ -5]])

Solve the system of equations \(A \mathbf{x} = \mathbf{b}\). 

coeffs = nl.solve(A, b)

>> coeffs
array([[-2.],
       [-6.],
       [ 5.]])

Find the parameters, \(D\), \(E\) and \(F\). 

D, E, F = coeffs

>> D, E, F
(np.float64(-2.0), np.float64(-6.0), np.float64(5.0))

Determine \(x_o\), \(y_o\), the position of the circle center and its radius \(\rho\). 

xo = -1/2 * D
yo = -1/2 * E
rho_square = xo**2 + yo**2 - F

>> np.array([xo, yo, rho_square])
array([1., 3., 5.])

Circle radius \(\rho\). 

rho = np.sqrt(rho_square)

>> rho
np.float64(2.23606797749979)

So, for this set of points

  • \(D = -2\)
  • \(E = -6\)
  • \(F = 5\)
  • \(x_o = 1\)
  • \(y_o = 3\)
  • \(\rho^2 = 5\)
  • \(\rho = \sqrt{5}\)

This result can be confirmed by defining the function 

f = lambda x, y: (x - xo)**2 + (y - yo)**2 - rho_square

and applying it to \(\mathbf{x}\) and \(\mathbf{y}\). 

>> f(x, y)
array([0., 0., 0.])

Hence, the equation of the circle passing through the given three points is

$$ \notag x^2 + y^2 - 2x - 6y + 5 = 0 $$

or

$$ \notag (x - 1)^2 + (y - 3)^2 = 5 $$

Solution 2 in Python's Numpy.

Import numpy 

import numpy as np
import numpy.linalg as nl

Define helper function. 

def find_circle_params(r1, r2, r3):
    x1, y1 = r1
    x2, y2 = r2
    x3, y3 = r3
    x = np.array([x1, x2, x3])
    y = np.array([y1, y2, y3])
    A = np.c_[x, y, np.ones(3)]
    b = -(x**2 + y**2)
    coeffs = nl.solve(A, b)
    D = float(coeffs[0])
    E = float(coeffs[1])
    F = float(coeffs[2])
    xo = -D/2
    yo = -E/2
    radius = np.sqrt(xo**2 + yo**2 - F)
    radius = float(radius)
    return xo, yo, radius

Input parameters 

r1 = (2, 1)
r2 = (0, 5)
r3 = (-1, 2)

Find the circle that fits the points. 

xo, yo, radius = find_circle_params(r1, r2, r3)
xo, yo, radius**2
(1.0, 3.0, 5.000000000000001)

3 Trigonometric Functions and Identities

3.1 Trigonometric Functions

Basic Trigonometric Identities

$$ \cos^2 \alpha + \sin^2 \alpha = 1 $$

Euler's Identity

$$ e^{j \theta} = \cos\theta + j \sin \theta $$

Note: \(j = \sqrt{-1}\) imaginary unit and \(\theta\) is the angle in radians.

Half Angle Trigonometric Identities

$$ \begin{split} \sin \frac{\alpha}{2} &= \pm \sqrt{ \frac{1 - \cos \alpha}{2} } \\ \cos \frac{\alpha}{2} &= \pm \sqrt{ \frac{1 + \cos \alpha}{2}} \\ \tan \frac{\alpha}{2} &= \frac{1 - \cos \alpha}{\sin \alpha} = \frac{\sin \alpha}{ 1 + \cos \alpha} \end{split} $$

Double Angle Formulas Trigonometric Identities

$$ \begin{split} \sin 2 \alpha &= 2 \sin \alpha \cos \alpha \\ \cos 2 \alpha &= \cos^2 \alpha - \sin^2 \alpha \\ \cos 2 \alpha &= 1 - 2 \sin^2 \alpha \\ \tan 2 \alpha &= \frac{ 2 \tan \alpha }{ 1 - \tan^2 \alpha } \end{split} $$

Properties

Properties
Symmetry Properties
Cosine is an even function (pt: função par) \(\cos (-x) = \cos (x)\)
Sine is a odd function (pt: função impar) \(\sin (-x) = - \sin(x)\)
\(\sin (\pi/2 - x) = \sin(90 - x) = \cos x\)
\(\cos (\pi/2 - x) = \cos(90 - x) = \sin x\)
\(\cos (\pi/2 - x) = \cos(90 - x) = \sin x\)
Arc Operations
Sin of the sum of angles \(\sin(x + y) = \sin x . \cos y + \cos x . \sin y\)
Cos of the sum of angles \(\cos(x + y) = \cos x . \cos y - \sin x . \sin y\)
Identities
Pythagorean identity \((\sin x)^2 + (\cos x)^2 = 1\)

Notable angles

Degrees Randians Sin Cos Tan
0 0 0 1 0
30 \(\pi/6\) 1/2 \(\sqrt{3}/2\) \(\sqrt{3}/3\)
45 \(\pi/4\) \(\sqrt{2}/2\) \(\sqrt{2}/2\) 1
60 \(\pi/3\) \(\sqrt{3}/2\) 1/2 \(\sqrt{3}\)
90 \(\pi/2\) 1 0 \(\infty\)
180 \(\pi\) 0 -1 0
270 \((3/2) \pi\) -1 0 \(-\infty\)
360 \(2\pi\) 0 1 0

3.2 Trigonometric Identities

Sine and Cosine Relation

$$ (\sin \theta)^2 + (\cos \theta)^2 = 1 $$

Euler's Equation and Complex Numbers

Let \(\mathrm{j} = \sqrt{-1}\) be the imaginary unit.

Complex Exponential

$$ e^{\mathrm{j} \theta} = \cos \theta + \mathrm{j} \sin \theta $$

and,

$$ e^{-\mathrm{j \theta}} = \cos \theta - \mathrm{j} \sin \theta $$

Cosine of angle in radians expressed as linear combination of complex exponential

$$ \cos \theta = \frac{1}{2} (e^{\mathrm{j} \theta} + e^{ - \mathrm{j} \theta} ) $$

Sine of angle in radians expressed as linear combination of complex exponential

$$ \cos \theta = \frac{1}{2 \mathrm{j}} (e^{\mathrm{j} \theta} + e^{ - \mathrm{j} \theta} ) $$

Derivative of complex exponential with respect to angle in radians.

$$ \frac{d}{d \theta} e^{j \theta} = j \cdot e^{j \theta} $$

Functions Symmetry

$$ \cos(-x) = \cos(x) $$
$$ \sin(-x) = -\sin(x) $$

Trigonometric expansions

$$ \cos (a + b) = \cos a \cdot \cos b - \sin a \sin b $$
$$ \sin (a + b) = \sin a \cdot \cos b - \sin a \cos b $$
$$ \tan(b - a) = \dfrac{ \tan y - \tan x }{\tan x \tan y + 1 } $$

4 Derivatives

4.1 Derivative Rules

Function Derivative Description
f(x) \(\dot{f}(x) = \dfrac{df}{dx}\) Derivative of generic function
\(r(x) = c\) 0 Derivative of constant
\(c f(x)\) \(c \dfrac{df}{dx}\) Function multiplied by constant
\(c + f(x)\) \(\dfrac{df}{dx}\) Function added to constant
\(f(x) + g(x)\) \(\dfrac{df}{dx} + \dfrac{dg}{dx}\) Sum of functions
\(f(x) g(x)\) \(f \dfrac{f}{dx} + g \dfrac{df}{dx}\) Product of functions
\(f(u(x))\) \(\left[\dfrac{d}{du} f(u) \right] \dfrac{du}{dx} = \dfrac{df}{du} \dfrac{du}{dx}\) Chain Rule
\(\dfrac{1}{f(x)}\) \(-\dfrac{1}{f^2} \dfrac{df}{dx}\) Dertivative of function
\(\dfrac{g}{f}\) \(\dfrac{f \dot{g} - g \dot{f}}{f^2}\) Derivative of ratio
\(f^{-1}(x)\) \(\left. \left[ \dfrac{d}{dy} f(y) \right]^{-1} \right\vert_{y = f^{-1}(x)}\) Derivative of inverse function
\(\left. \dfrac{d}{dx} f^{-1}(x) \right\vert_{x = a}\) \(\left. \left[ \dfrac{d}{dy} f(y) \right]^{-1} \right\vert_{y = f^{-1}(a)}\) Derivative of inverse function evaluated at x = a

4.2 Basic Derivatives

Function f(x) Derivative
\(x\) 1
\(x^2\) \(2x\)
\(x^3\) \(3 x^2\)
\(x^p\) \(p x^{p - 1}\) \(p \in \mathbb{R}\)
\(1/x\) \(-1/x^2\)
\(\sqrt{x}\) \(\dfrac{1}{2 \sqrt{x}}\)
\(e^x\) \(e^x\)
\(e^{a x}\) \(a e^{a x}\)
\(a^x\) \(a^x \ln a\)
\(e^{f(x)}\) \(e^{f(x)} f(x) \dot{f}(x)\)
\(\ln x\) \(1/x\) Derivative of natural logarithm base \(\mathrm{e}\)
\(\log_a x\) \(\dfrac{1}{x \ln a}\) Derivative of logarithm of base a
\(\sin x\) \(\cos x\)
\(\cos x\) \(- \sin x\)
\(\tan x\) \(\sec^2 x\)
\(\sec x\) \(\sec x \tan x\)
\(\sin^{-1}(x)\) \(\dfrac{1}{\sqrt{1 - x^2}}\) arc sine
\(\cos^{-1}(x)\) \(-\dfrac{1}{\sqrt{1 - x^2}}\) arc cosine
\(\tan^{-1}(x)\) \(\dfrac{1}{1 + x^2}\) arc tangent or arctan

4.3 Derivative of Inverse Function

Derivative of inverse function \(f^{-1}(x)\) of \(f(x)\) at \(x = a\).

$$ \notag \left. \dfrac{d}{dx} f^{-1}(x) \right\vert_{\displaystyle x = a} = \left. \left[ \dfrac{df}{dy} \right]^{-1} \right\vert_{\displaystyle y = f^{-1}(a)} $$

Procedure for Determining the derivative of the inverse

STEP 1: Given a, solve the following equation for y, in order to find \(y_a\), the solution of this equation.

$$ \notag a = f(y) $$

STEP 2: Compute the derivative of \(f(y)\) with respect to y at \(y = y_a\).

$$ \notag w = \left. \dfrac{d}{dy} f(y) \right\vert_{\displaystyle y = y_a} $$

STEP 3: Compute the value of the derivative of the inverse function of f(x) by taking the inverse of the previous result w.

$$ \left. \dfrac{d}{dx} f^{-1}(x) \right\vert_{\displaystyle x = a} = \frac{1}{w} $$

Examples

4.4 Chain Rule

Chain rule for function of one variable

This notation explicitly states that the in the right-hand side of the equation f is expressed as function of u and not as function of x.

$$ \dfrac{d}{dx} f(u(x)) = \left[ \dfrac{d}{du} f(u) \right] \dfrac{du}{dx} $$

Concise notation for chain rule for functions of a single variable

$$ \dfrac{d}{dx} f(u(x)) = \dfrac{df}{du} \dfrac{du}{dx} $$

Chain rule for function of multi-variable functions

$$ \dfrac{d}{dt} f(x(t), y(t)) = \dfrac{\partial f}{\partial x} \dfrac{dx}{dt} + \dfrac{\partial f}{\partial y} \dfrac{dy}{dt} $$

Chain rule for coordinate changing

Let f be function of \(x\) and \(y\). A new coordinate is introduced withcoordinates \(v\) and \(w\) in a way that \(x = x(v, w)\) and \(y = y(v, w)\). The partial derivative of f with respect to the new coordinate \(v\), \(w\) can be determined as:

$$ \begin{split} x &= x(v, w) \quad y = y(v, w) \\ \frac{\partial f}{\partial v} &= \frac{\partial f}{\partial x} \frac{\partial x}{\partial v} + \frac{\partial f}{\partial y} \frac{\partial y}{\partial v} \\ \frac{\partial f}{\partial w} &= \frac{\partial f}{\partial x} \frac{\partial x}{\partial w} + \frac{\partial f}{\partial y} \frac{\partial y}{\partial w} \\\end{split} $$

Examples

4.5 Tangent Line Equation

The derivative can be used for obtaining the tangent line equation \(\eqref{eqtanget_diff}\) to a curve represented by a function f(x) at a particular point, where \(\alpha \in \mathbb{R}\) and \(\beta \in \mathbb{R}\) are the coefficients of the line equation.

$$ y = \alpha x + \beta \label{eqtanget_diff} $$

The coefficient \(\alpha\), the line slope, can be computed as:

$$ \alpha = f'(x_a) = \left. \frac{df}{dx} \right\vert_{\displaystyle x = x_a} $$

Then, the coefficient \(\beta\) can be found by replacing the pair $x_a, : \(y_a = f(x_a)\) in \(\eqref{eqtanget_diff}\) and solving it for \(\beta\).

$$ \beta = y_a - \alpha x_a $$

so,

$$ \beta = f(x_a) - f'(x_a) x_a $$

The, the coefficients of the line equation tangent to the point \((x_a, y_a)\) are given by:

$$ \begin{cases} \alpha = f'(x_a) \\ \beta = f(x_a) - f'(x_a) x_a \end{cases} $$

5 Integrals

References:

General Rules

Linear combination of integrals:

$$ \int [a \cdot f(x) + b \cdot g(x)] dx = a \int f(x) dx + b \int g(x) dx $$

Power of a function:

$$ \int [g(x)]^n g'(x) dx = \frac{1}{n + 1} [g(x)]^{n + 1} \quad n \neq -1 $$

Ratio between function's derivate and itself:

$$ \int \frac{g'(x)}{g(x)} dx = \ln || g(x) || $$

Common Integrals

Constant:

$$ \int C \cdot dx = C \cdot x $$

Power:

$$ \int x^n \cdot dx = \frac{x^{n + 1}}{n + 1} \quad \neq -1 $$

Power:

$$ \int a^x \cdot dx = \frac{a^x}{\ln a} \quad a > 0 $$

Exponential:

$$ \int e^x \cdot dx = e^x $$

Logarithm:

$$ \int \frac{1}{x} \cdot dx = \ln || x || $$

Sine:

$$ \int \sin x = - \cos x $$

Cosine:

$$ \int \cos x \cdot dx = \sin x $$

Square sine:

$$ \int \sin^2 x \cdot dx = \frac{x}{2} - \frac{1}{4} \sin 2x $$

Square cosine:

$$ \int \cos^2 x \cdot dx = \frac{x}{2} + \frac{1}{4} \sin 2x $$

Sinh - hyperbolic sine:

$$ \int \text{sinh} x \cdot dx = \text{cosh} x $$

Cosh - hyperbolic consine:

$$ \int \text{cosh} x \cdot dx = \text{sinh} x $$

6 Sums of Series

$$ \sum_{k = 1}^{n} k = \frac{n(n+1)}{2} $$
$$ \sum_{k = 1}^{n} k^2 = \frac{n(n+1)(2n + 1)}{6} $$
$$ \sum_{k = 1}^{n} k^3 = \left(\frac{n(n+1)}{2} \right)^2 $$

7 Taylor Series

7.1 Taylor series expansion

The Taylor series is a expansion of a real function f(x) about a point x a. When a is equal to 0, the expansion is also known as Maclaurin series.

$$ \begin{split} f(x) &= \sum_{n = 0}^{\infty} \frac{f^{(n)}(a)}{n!}(x - a)^n \\ &= f(a) + f^{(1)}(a)(x - a) + \frac{f^{(2)}(a)}{2!}(x - a)^2 + \frac{f^{(3)}(a)}{3!}(x - a)^3 \cdots \end{split} $$

The Taylor series expansion is sometimes written as:

$$ \begin{split} f(x + a) &= \sum_{n = 0}^{\infty} \frac{f^{(n)}(a)}{n!}x^n \\ &= f(a) + f^{(1)}(a)x + \frac{f^{(2)}(a)}{2!} x^2 + \frac{f^{(3)}(a)}{3!} x^3 \cdots \end{split} $$

where:

7.2 Some Taylor series expansion

Exponential

$$ e^{x} = \sum_{n = 0}^{\infty} \frac{x^n}{n!} = 1 + x + \frac{1}{2!} x^2 + \frac{1}{3!} 3^2 + \frac{1}{4!} 4^2 + \cdots $$

Consine

$$ \cos x = \sum_{n = 0}^{\infty} \frac{(-1)^n}{(2n)!} x^{2n} = 1 - \frac{1}{2!}x^2 + \frac{1}{4!} x^4 - \frac{1}{6!} x^6 + \cdots $$

Sine

$$ \sin x = \sum_{n = 0}^{\infty} \frac{(-1)^n}{(2n + 1)!} x^{2n + 1} = x - \frac{1}{3!}x^3 + \frac{1}{5!}x^5 - \frac{1}{7!}x^7 + \cdots $$

Inverse tan or arctan

$$ \arctan x = \sum_{n = 0}^{\infty} \frac{(-1)^n}{2n + 1} x^{2n + 1} = x - \frac{1}{3} x^3 + \frac{1}{5} x^5 - \frac{1}{7!} x^7 + \cdots $$

Natural Logarithm

$$ \ln(1 + x) = \sum_{n = 1}^{\infty} \frac{x^n}{n} (-1)^{n+1} = x - \frac{1}{2} x^2 + \frac{1}{3}x^3 - \frac{1}{4} x^4 + \cdots $$

Geometric Power Series

$$ \frac{1}{1 + x} = 1 + x + x^2 + x^3 + \cdots $$

7.3 Approximations for small values of x

Misc Approximation

$$ \begin{split} e^x & \approx 1 + x \\ \ln (1 + x) & \approx x \\ \ln (1 - x) & \approx 1 - x \\ \frac{1}{1 + x} & \approx 1 - x \\ \frac{1}{1 - x} & \approx 1 + x \\ (1 - x)^n & \approx 1 - n x + n(n-1) x^2 / 2 \end{split} $$

Approximations for small angles

$$ \begin{split} \theta^2 &\approx 0 \\ \sin \theta & \approx \theta \\ \cos \theta & \approx 1 - \frac{\theta^2}{2} \approx 0 \\ \tan \theta & \approx \theta \\ \csc \theta & \approx 1 / \theta + \theta / 6 \\ \sec \theta & \approx 1 + \theta^2 / 2 \approx 0 \\ \cot \theta & \approx 1 / \theta - \theta* 3 \end{split} $$

7.4 References

8 Multivariate Calculus

8.1 Gradient Vector

Consider a scalar-valued function \(f\) of two variables \(f = f(x, y) \in \mathbb{R}\), the gradient of this function is the vector

$$ \nabla f = \frac{\partial f}{\partial \mathbf{r}} = \dfrac{\partial f}{\partial x} \hat{\mathbf{x}} + \dfrac{\partial f}{\partial y} \hat{\mathbf{y}} = \begin{bmatrix} \dfrac{\partial f}{\partial x} \\ \dfrac{\partial f}{\partial y} \end{bmatrix} $$

Where \(\hat{\mathbf{x}}\) is the unit vector of \(x\) axis, \(\hat{\mathbf{y}}\) is the unit vector of the \(y\) axis and \(\nabla\) is the operator

$$ \nabla = \dfrac{\partial}{\partial x} \hat{\mathbf{x}} + \dfrac{\partial}{\partial y} \hat{\mathbf{y}} $$

or in matrix form,

$$ \nabla = \begin{bmatrix} \dfrac{\partial f}{\partial x} \\ \dfrac{\partial f}{\partial y} \end{bmatrix} $$

In some text books, the \(x\) axis unit vector is referred as \(i\) or \(\mathbf{i}\) and the \(y\) axis unit vector is referred as \(j\) or \(\mathbf{j}\). Those notations are avoided due to the possible confusion since \(i\) is sometimes used to referrer to either electrical current or the imaginary unit \(i = \sqrt{-1}\) and the letter \(j\) is also used in electrical engineering for denoting the imaginary unit \(j = \sqrt{-1}\).

8.2 Directional Derivative

The directional derivative $ D_{\mathbf{u}} f$ in the direction \(\mathbf{u}\) of a function \(z = f(x, y)\) of two variables, which represents a surface in space, is given by

$$ D_{\mathbf{u}} f = \nabla f \cdot \hat{\mathbf{u}} = \frac{\partial f}{\partial x} u_x + \frac{\partial f}{\partial y} u_y $$

or

$$ D_{\mathbf{u}} f = \| \nabla f \| \| \mathbf{u} \| \cos \theta $$

Where \(\mathbf{u} = (u_x, u_y)\) and \(\theta\) is the angle betwen \(\mathbf{u}\) and the gradient vector \(\nabla f\). The directional derivative has the highest positive value when \(\theta = 0\) in the direction of the gradient vector. The direction of the highest decrease of the directional derivative is in the opposite direction of the gradient vector when \(\theta = \pi\) (180 degrees).

8.3 Change of Variables and Jacobian Matrix in a Plane

Let \((x, y)\) be the cartesian coordinates of a plance, a 2-dimensional space, where \(x = x(u, v)\) and \(y = y(u, v)\).

The jacobian matrix of this variable transformation is defined as

$$ J = \begin{bmatrix} \dfrac{ \partial \mathbf{r} }{\partial u} & \dfrac{ \partial \mathbf{r} }{\partial v} \end{bmatrix} = \begin{bmatrix} \nabla_{(u, v)}^T x \\ \nabla_{(u, v)}^T y \end{bmatrix} = \begin{bmatrix} \dfrac{ \partial{x} }{ \partial u} & \dfrac{ \partial{x} }{ \partial v} \\ \dfrac{ \partial{y} }{ \partial u} & \dfrac{ \partial{y} }{ \partial v} \end{bmatrix} $$

Where

Area Double Integral

The area in double integral is given by

$$ A = \iint dA = \iint dx \, dy = \iint | \det J | \, du \, dv $$

Area Differential

The area differential is given by

$$ dA = dx \, dy = | \det J | \, du \, dv $$

where \(| \det J |\) is absolute value of the determinant of the jacobian matrix.

Area in Polar Coordinates

Planar cartesian coordinates \((x, y)\) can be turned into polar coordinates as \((\theta, r)\), where

Compute the partial derivative of \(\mathbf{r} = (x, y)\) vector with respec to \(\theta\).

$$ \frac{\partial \mathbf{r} }{\partial \theta} = \frac{\partial}{\partial \theta} \begin{bmatrix} r \cos \theta \\ r \sin \theta \end{bmatrix} = \begin{bmatrix} - r \sin \theta \\ r \cos \theta \end{bmatrix} $$

Compute the partial derivative of \(\mathbf{r} = (x, y)\) vector with respec to \(\theta\).

$$ \frac{\partial \mathbf{r} }{\partial r} = \frac{\partial}{\partial r} \begin{bmatrix} r \cos \theta \\ r \sin \theta \end{bmatrix} = \begin{bmatrix} \cos \theta \\ \sin \theta \end{bmatrix} $$

Determine the jacobian matrix

$$ J = \begin{bmatrix} \dfrac{\partial \mathbf{r} }{\partial \theta} & \dfrac{\partial \mathbf{r} }{\partial r} \end{bmatrix} $$
$$ J = \begin{bmatrix} - r \sin \theta & \cos \theta \\ r \cos \theta & \sin \theta \end{bmatrix} $$

Compute the determinant of the jacobian matrix.

$$ \det J = (- r \sin \theta) (\sin \theta) - (r \cos \theta) (\cos \theta) $$
$$ \det J = - r \sin^2 \theta - r \cos^2 \theta $$
$$ \det J = -r (\sin^2 \theta + \cos^2 \theta) $$
$$ \det J = -r $$

The area differential in polar coordinates is given by

$$ dA = | \det J | \, d\theta \, dr $$
$$ dA = | - r | \, d\theta \, dr $$
$$ dA = r \, \d\theta \, dr $$

Therefore, the area of a region in polar coordinates is

$$ \boxed{ A = \iint dA = \iint r \, \d\theta \, dr } $$

8.4 Change of Variables and Jacobian Matrix in a Space

8.5 Derivative Chain Rule

8.6 See also

9 Statistical Formulas

Source:

Notation:

Expected Value Identities

Expected value of constant 'a':

$$ E(a) = a $$

Expected value of a linear combination of random variables, 'a' and 'b' are constants.

$$ E( a \cdot \pmb{X} + b \cdot \pmb{Y} ) = a \cdot \pmb{X} + b \cdot \pmb{Y} $$

Covariance, Variance, Standard Deviation Identities

$$ \text{Var}( \pmb{X} ) E[(X - \mu)^2] E(X^2) - \mu^2 = \sigma_x^2 $$
$$ \text{Var}(c \cdot \pmb{X} + d) = c^2 \cdot \text{Var}( \pmb{X} ) $$
$$ \text{Cov}( \pmb{X}, \pmb{Y} ) = \text{Cov}( \pmb{Y}, \pmb{X} ) $$
$$ \text{Cov}(c \cdot \pmb{X}, \pmb{Y} ) = \text{Cov}( \pmb{X}, c \cdot \pmb{Y} ) = c \cdot \text{Cov}( \pmb{X}, \pmb{Y} ) $$
$$ \text{Cov}( \pmb{X} + \pmb{Y}, \pmb{Z} ) = \text{Cov}( \pmb{X}, \pmb{Y} ) + \text{Cov}( \pmb{X}, \pmb{Z} ) $$
$$ \text{Var}( \pmb{X} + \pmb{Y}) = \text{Var}( \pmb{X} ) + \text{Var}( \pmb{Y} ) + 2 \text{Cov}( \pmb{X},\pmb{Y}) $$

10 Linear algebra

10.1 Notation

General:

Column vector:

$$ \text{Vcol}_{(n x 1)} = \begin{bmatrix} v_1 \\ v_2 \\ \cdots \\ v_n \end{bmatrix} $$

Row vector:

$$ \text{Vrow}_{(1 x n)} = \begin{bmatrix} v_1 & v_2 & \cdots & v_n \end{bmatrix} $$

10.2 Special matrices

Zero matrix:

$$ 0_{3 x 4} = \begin{bmatrix} 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ \end{bmatrix} $$ 
julia> zeros(3, 3)
  3×3 Array{Float64,2}:
   0.000  0.000  0.000
   0.000  0.000  0.000
   0.000  0.000  0.000


  julia> zeros(3, 1)
  3×1 Array{Float64,2}:
   0.000
   0.000
   0.000

Identity Matrix:

$$ I_{2} = \begin{bmatrix} 1 & 0 \\ 0 & 1 \\ \end{bmatrix} $$
$$ I_{3} = \begin{bmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \\ \end{bmatrix} $$ 
using LinearAlgebra 


  julia> eye(n) = zeros(n, n) + I


  julia> eye(3)
  3×3 Array{Float64,2}:
   1.000  0.000  0.000
   0.000  1.000  0.000
   0.000  0.000  1.000


  julia> eye(4)
  4×4 Array{Float64,2}:
   1.000  0.000  0.000  0.000
   0.000  1.000  0.000  0.000
   0.000  0.000  1.000  0.000
   0.000  0.000  0.000  1.000

Ones-based matrices:

\(1_{n, n} = 1 \quad \text{for every i, j}\) 

julia> ones(2, 2)
  2×2 Array{Float64,2}:
   1.00000E+00  1.00000E+00
   1.00000E+00  1.00000E+00


  julia> ones(3, 2)
  3×2 Array{Float64,2}:
   1.00000E+00  1.00000E+00
   1.00000E+00  1.00000E+00
   1.00000E+00  1.00000E+00


  julia> ones(3, 1)
  3×1 Array{Float64,2}:
   1.00000E+00
   1.00000E+00
   1.00000E+00

11 Types of matrices

Symmetric Matrix

Any matrix which equals to its transpose:

Aij = Aji 

# -------- Symmetric matrix  ---------


  julia> A = [1 7 3; 7 4 -5; 3 -5 6]
  3×3 Array{Int64,2}:
   1   7   3
   7   4  -5
   3  -5   6


  julia> A'
  3×3 LinearAlgebra.Adjoint{Int64,Array{Int64,2}}:
   1   7   3
   7   4  -5
   3  -5   6


  julia> A'  A
  true

Non Symmetric Matrix: 

julia> B = [3 8 7; 9 5 6; 1 2 3]
  3×3 Array{Int64,2}:
   3  8  7
   9  5  6
   1  2  3


  julia> B'
  3×3 LinearAlgebra.Adjoint{Int64,Array{Int64,2}}:
   3  9  1
   8  5  2
   7  6  3


  julia> B  B'
  false

11.1 Matrix Operations Rules

Matrix multiplication is not symmetric

$$ A B \neq B A $$ 
julia> A = [ 4 5; 6 7]
  2×2 Array{Int64,2}:
   4  5
   6  7


  julia> B = [ 8 2; 1 5]
  2×2 Array{Int64,2}:
   8  2
   1  5


  julia> A * B
  2×2 Array{Int64,2}:
   37  33
   55  47


  julia> B * A
  2×2 Array{Int64,2}:
   44  54
   34  40


  julia> A * B  B * A
  false

Associativity

$$ A B C = (A B) C = A (B C) $$

Transpose matrix

Tranpose of product:

$$ (A B)^T = B^T A^T $$
$$ (A B C)^T = C^T (A B)^T = (B C)^T A = C^T B^T A^T $$

Transpose of sum:

$$ (A + B)^T = A^T + B^T $$

11.2 Column vector properties

Let a and b be two (n x 1) column vectors:

$$ a_{(n x 1)} = \begin{bmatrix} a_1 \\ a_2 \\ \cdots \\ a_n \end{bmatrix} $$
$$ b_{(n x 1)} = \begin{bmatrix} b_1 \\ b_2 \\ \cdots \\ b_n \end{bmatrix} $$

The scalar product (dot product) between a and b is:

$$ \mathbf{a} \cdot \mathbf{b} = \sum_{i = 1}^n a_i b_i = \mathbf{a}^T \mathbf{b} = \mathbf{b}^T \mathbf{a} $$

The L-2 norm or euclidian norm of a vector is given by:

$$ \text{norm}(\mathbf{a}) = \| \mathbf{a} \| = \sqrt{ \sum_{i = 1}^n a_i^2 } = \sqrt{ \mathbf{a}^T \cdot \mathbf{a}} $$

12 Matrix Calculus

12.1 Definitions

Let x be a (n x 1) column vector and y be a (m x 1) column vector.

$$ x_{(n x 1)} = \begin{bmatrix} x_1 \\ x_2 \\ \cdots \\ x_n \end{bmatrix} $$
$$ y_{(m x 1)} = \begin{bmatrix} y_1 \\ y_2 \\ \cdots \\ y_n \end{bmatrix} $$

12.2 Derivate of a scalar with respect to a vector

If \(F(X): R^N \rightarrow R\) - F(X) = F(x1, x2, ... xn) is a multivariate function, the derivate of the scalar function F(X) with respect to the vector X is:

$$ \frac{\partial F}{\partial x} = \nabla F = \begin{bmatrix} \frac{\partial F}{\partial x_1} \\ \frac{\partial F}{\partial x_2} \\ \cdots \\ \frac{\partial F}{\partial x_n} \\ \end{bmatrix} $$

See:

12.3 Derivate of a vector with respect to a scalar

If the vector Y is a function (Y = Y(t)) of the scalar t, the derivate of Y with respect to t is:

$$ \frac{\partial Y(t)}{\partial t} = \begin{bmatrix} \frac{\partial y_1(t)}{\partial t} \\ \frac{\partial y_2(t)}{\partial t} \\ \frac{\partial y_3(t)}{\partial t} \\ \cdots \\ \frac{\partial y_m(t)}{\partial t} \\ \end{bmatrix} $$

12.4 Derivate of a vector with respect to a vector

If the vector Y (m x 1) is a function of the vector X (n x 1), then the derivate of Y with respect to Y is:

$$ \frac{\partial Y(X)}{\partial X} = J[Y(X)] = \begin{bmatrix} \frac{\partial y_1}{\partial x_1} & \frac{\partial y_2}{\partial x_1} & \cdots & \frac{\partial y_m}{\partial x_1} \\ \frac{\partial y_1}{\partial x_2} & \frac{\partial y_2}{\partial x_2} & \cdots & \frac{\partial y_m}{\partial x_2} \\ \cdots & \cdots & \cdots & \cdots \\ \frac{\partial y_1}{\partial x_2} & \frac{\partial y_2}{\partial x_2} & \cdots & \frac{\partial y_m}{\partial x_2} \\ \end{bmatrix} $$

12.5 Derivate rules for matrices and vectors

References:

Let X be n x 1 vector where \(x \in R^n\)

$$ x_{(n x 1)} = \begin{bmatrix} x_1 \\ x_2 \\ \cdots \\ x_n \end{bmatrix} $$

Let C be matrix of (n x n) constant elements where \(C \in R^{n x n}\)

$$ C = \begin{bmatrix} c_{11} & c_{12} & c_{13} \cdots c_{1n} \\ c_{21} & c_{22} & c_{23} \cdots c_{2n} \\ c_{31} & c_{32} & c_{33} \cdots c_{3n} \\ \cdots & \cdots & \cdots & \cdots \\ c_{n1} & c_{n2} & c_{n3} \cdots c_{nn} \\ \end{bmatrix} $$

The following identities are valid:

$$ \frac{\partial }{\partial X} (C . X) = \frac{\partial (C . x)}{\partial X} = C^T $$
$$ \frac{\partial }{\partial X} (X^T . C) = \frac{\partial (X^T C)}{\partial X} = C $$
$$ \frac{\partial }{\partial X} (X^T . X) = \frac{\partial (X^T . X)}{\partial X} = 2 X $$
$$ \frac{\partial }{\partial X} (y^T x) = \frac{\partial }{\partial X} (x^T y) = y^T $$
$$ \frac{\partial }{\partial X} (X^T . C . X) = \frac{\partial (X^T . C. X)}{\partial X} = (C + C^T) . x = x^T (C + C^T) $$
$$ \frac{\partial \alpha}{\partial X} = \frac{\partial}{\partial X} \left( y^T . A . x \right) = y^T . A $$
$$ \frac{\partial \alpha(x)}{\partial x} = \frac{\partial}{\partial x} (v^t . v) = 2 v^T \frac{\partial v}{\partial x} $$

12.6 Useful gradients

Reference: Imperial College of London and (Petersen and Pederson, 2012)

List of useful and pervasive identities for computing gradient in machine learning.

$$ \displaystyle \begin{cases} \dfrac{\partial f(X)^T}{\partial X} &=& \left[ \dfrac{\partial f(X)}{\partial X} \right]^T \\ \dfrac{\partial}{\partial x} tr(f(X)) &=& tr \left( \dfrac{\partial f(X)}{\partial X} \right) \\ \dfrac{\partial}{\partial X} \det f(X) &=& det X \; tr \left(X^{-1} \dfrac{\partial f(X)}{\partial X} \right) \\ \dfrac{\partial}{\partial x} (x^t \cdot a) &=& a^t \\ \dfrac{\partial}{\partial x} (a^t \cdot X \cdot b) &=& a \cdot b^t \\ \dfrac{\partial}{\partial x} (x^t \cdot B \cdot x) &=& x^t (B + B^t) \end{cases} $$

Where,