Some fundamental algebraic formulas and identities Part 2

 

51. Volume Formulas:

• Cylinder: $�=�{�}^2ℎ$
• Sphere: $�=\frac{4}{3}�{�}^3$
• Cone: $�=\frac{1}{3}�{�}^2ℎ$

52. Matrix Determinant:

• For a 2x2 matrix $\left[\begin{array}{cc}�& �\\ �& �\end{array}\right]$, the determinant is $��-��$. For a 3x3 matrix $�$: $\text{det}(�)=�(��-�ℎ)-�(��-��)+�(�ℎ-��)$

53. Vandermonde Determinant:

• The determinant of a Vandermonde matrix $�$ with distinct values ${�}_1,{�}_2,\dots ,{�}_{�}$ is given by: $\text{det}(�)={\prod }_{1\le �<�\le �}({�}_{�}-{�}_{�})$

54. Laplace's Equation:

• In two dimensions, Laplace's equation ${\mathrm{\nabla }}^2�=0$ describes a harmonic function $�(�,�)$.

55. Conic Sections:

• Ellipse: $\frac{{�}^2}{{�}^2}+\frac{{�}^2}{{�}^2}=1$
• Hyperbola: $\frac{{�}^2}{{�}^2}-\frac{{�}^2}{{�}^2}=1$
• Parabola: ${�}^2=4��$

56. Cauchy-Schwarz Inequality:

• For vectors $�=({�}_1,{�}_2,\dots ,{�}_{�})$ and $�=({�}_1,{�}_2,\dots ,{�}_{�})$, the inequality is: ${\left({\sum }_{�=1}^{�}{�}_{�}{�}_{�}\right)}^2\le \left({\sum }_{�=1}^{�}{�}_{�}^2\right)\left({\sum }_{�=1}^{�}{�}_{�}^2\right)$

57. Euler's Line:

• In a triangle, Euler's line is a line passing through the centroid, circumcenter, and orthocenter.

58. Bézout's Identity:

• Bézout's identity states that for any integers $�$ and $�$, there exist integers $�$ and $�$ such that $��+��=\text{gcd}(�,�)$.

59. Riemann Sum:

• The Riemann sum approximates the definite integral of a function over an interval by dividing it into subintervals and summing the function values multiplied by the subinterval width.

60. Singular Value Decomposition (SVD):

• For a matrix $�$, SVD expresses it as $�=�\mathrm{\Sigma }{�}^{�}$, where $�$ and $�$ are orthogonal matrices, and $\mathrm{\Sigma }$ is a diagonal matrix of singular values.

61. Ptolemy's Theorem:

• For a cyclic quadrilateral (a quadrilateral inscribed in a circle), Ptolemy's theorem relates the sides and diagonals: $��+��=��+��$

62. Lagrange Interpolation:

• Given $�+1$ data points, Lagrange interpolation provides a polynomial of degree at most $�$ passing through these points.

63. Taylor Series:

• The Taylor series expansion of a function $�(�)$ about a point $�$ is given by: $�(�)=�(�)+{�}^{\mathrm{\prime }}(�)(�-�)+\frac{{�}^{\mathrm{\prime }\mathrm{\prime }}(�)}{2!}(�-�{)}^2+\dots$

64. Fourier Transform:

• The Fourier transform of a function $�(�)$ is given by: $\mathcal{�}\{�(�)\}={\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}�(�){�}^{-���}��$

65. Frobenius Norm:

• The Frobenius norm of a matrix $�$ is defined as $\mathrm{\parallel }�{\mathrm{\parallel }}_{�}=\sqrt{{\sum }_{�,�}\mathrm{\mid }{�}_{��}{\mathrm{\mid }}^2}$.

66. Fibonacci Sequence:

• The Fibonacci sequence is defined by the recurrence relation ${�}_{�}={�}_{�-1}+{�}_{�-2}$ with initial conditions ${�}_0=0$ and ${�}_1=1$.

67. Bayes' Rule for Probability:

• Bayes' rule relates conditional and marginal probabilities: $�(�\mathrm{\mid }�)=\frac{�(�\mathrm{\mid }�)\cdot �(�)}{�(�)}$

68. Gamma Function:

• The gamma function $\mathrm{\Gamma }(�)$ is an extension of the factorial function to complex numbers and real numbers, except for non-positive integers.

69. Cauchy Integral Formula:

• For a function $�(�)$ that is analytic on and within a simple closed contour $�$, the Cauchy Integral Formula is: $�(�)=\frac{1}{2��}{\oint }_{�}\frac{�(�)}{�-�}��$

70. Laplace Operator:

• The Laplace operator ${\mathrm{\nabla }}^2$ is defined as the divergence of the gradient of a scalar function. In Cartesian coordinates, ${\mathrm{\nabla }}^2�=\frac{{\mathrm{\partial }}^2�}{\mathrm{\partial }{�}^2}+\frac{{\mathrm{\partial }}^2�}{\mathrm{\partial }{�}^2}+\frac{{\mathrm{\partial }}^2�}{\mathrm{\partial }{�}^2}$.

71. Beta Function:

• The beta function $�(�,�)$ is defined as $�(�,�)={\int }_0^1{�}^{�-1}(1-�{)}^{�-1}��$ and is related to the gamma function.

72. Logistic Map:

• The logistic map is a mathematical model of population growth, defined by the recurrence relation ${�}_{�+1}=�{�}_{�}(1-{�}_{�})$.

73. Bessel Functions:

• Bessel functions, denoted by ${�}_{�}(�)$, are solutions to Bessel's differential equation and have applications in various areas, including wave propagation and heat conduction.

74. Hessian Matrix:

• The Hessian matrix is a square matrix of second-order partial derivatives of a scalar function. For a function $�(�,�)$, the Hessian matrix is: $\text{Hess}(�)=\left[\begin{array}{cc}{�}_{��}& {�}_{��}\\ {�}_{��}& {�}_{��}\end{array}\right]$

75. Legendre Polynomials:

• Legendre polynomials ${�}_{�}(�)$ are solutions to Legendre's differential equation and have applications in spherical harmonics and potential theory.

76. Möbius Transformation:

• A Möbius transformation is a fractional linear transformation of the form $�(�)=\frac{��+�}{��+�}$, where $�,�,�,�$ are complex numbers.

77. Heat Equation:

• The heat equation describes the distribution of heat in a given region over time and is given by $\frac{\mathrm{\partial }�}{\mathrm{\partial }�}=�{\mathrm{\nabla }}^2�$.

78. Inverse Laplace Transform:

• The inverse Laplace transform of a function $�(�)$ is denoted as ${\mathcal{�}}^{-1}\{�(�)\}$ and represents the original function in the time domain.

79. Chebyshev Polynomials:

• Chebyshev polynomials ${�}_{�}(�)$ are solutions to Chebyshev's differential equation and are often used in approximation theory and signal processing.

80. Routh-Hurwitz Criterion:

• The Routh-Hurwitz criterion is a mathematical test to determine the stability of a linear time-invariant system based on the coefficients of its characteristic equation.

81. Euler's Differential Equation:

• Euler's differential equation is a type of linear homogeneous second-order differential equation with constant coefficients and is given by $�{�}^{\mathrm{\prime }\mathrm{\prime }}+�{�}^{\mathrm{\prime }}+��=0$, where $�$, $�$, and $�$ are constants.

82. Lagrangian Mechanics:

• Lagrangian mechanics is a reformulation of classical mechanics that replaces Newton's laws with the principle of stationary action, expressed through the Lagrangian function.

83. Laplace's Equation in Spherical Coordinates:

• In spherical coordinates $(�,�,�)$, Laplace's equation is ${\mathrm{\nabla }}^2�=\frac{1}{{�}^2}\frac{\mathrm{\partial }}{\mathrm{\partial }�}\left({�}^2\frac{\mathrm{\partial }�}{\mathrm{\partial }�}\right)+\frac{1}{{�}^2\sin �}\frac{\mathrm{\partial }}{\mathrm{\partial }�}(\sin �\frac{\mathrm{\partial }�}{\mathrm{\partial }�})+\frac{1}{{�}^2{\sin }^2�}\frac{{\mathrm{\partial }}^2�}{\mathrm{\partial }{�}^2}=0$.

84. Discrete Fourier Transform (DFT):

• The discrete Fourier transform is a mathematical technique that transforms a sequence of complex numbers ${�}_0,{�}_1,\dots ,{�}_{�-1}$ into another sequence of complex numbers ${�}_0,{�}_1,\dots ,{�}_{�-1}$ defined by ${�}_{�}={\sum }_{�=0}^{�-1}{�}_{�}{�}^{-�2���\mathrm{/}�}$.

85. Dirac Delta Function:

• The Dirac delta function, denoted by $�(�)$, is a mathematical function that models an idealized distribution of unit amplitude at a single point and is used in distribution theory and signal processing.

86. Runge-Kutta Methods:

• Runge-Kutta methods are a family of numerical techniques for solving ordinary differential equations. The most common form is the fourth-order Runge-Kutta method.

87. Principal Component Analysis (PCA):

• Principal Component Analysis is a dimensionality reduction technique used in linear algebra and statistics to transform high-dimensional data into a lower-dimensional space while retaining as much variance as possible.

88. Laplace Transform of Derivatives:

• The Laplace transform of a derivative is given by $\mathcal{�}\{{�}^{\mathrm{\prime }}(�)\}=�\mathcal{�}\{�(�)\}-�(0^{+})$, where $�$ is the complex variable.

89. Leibniz Rule (Product Rule for Derivatives):

• Leibniz rule states that if $�(�)$ and $�(�)$ are differentiable functions, then the derivative of their product is given by $(��{)}^{\mathrm{\prime }}={�}^{\mathrm{\prime }}�+�{�}^{\mathrm{\prime }}$.

90. Poisson Distribution:

• The Poisson distribution models the number of events occurring in a fixed interval of time or space and is characterized by a single parameter $�$, representing the average rate of occurrence.

91. Legendre's Three-Square Theorem:

• Legendre's Three-Square Theorem states that a natural number can be expressed as the sum of three squares if and only if it is not of the form $4^{�}(8�+7)$ for integers $�$ and $�$.

92. Cantor Set:

• The Cantor set is a perfect set with interesting properties, obtained by recursively removing the middle third of a line segment.

93. Pigeonhole Principle:

• The Pigeonhole Principle states that if $�$ items are placed into $�$ containers and $�>�$, then at least one container must contain more than one item.

94. Taylor's Theorem:

• Taylor's theorem provides an expansion of a function about a point in terms of its derivatives. The nth-degree Taylor polynomial is given by: ${�}_{�}(�)=�(�)+{�}^{\mathrm{\prime }}(�)(�-�)+\frac{{�}^{\mathrm{\prime }\mathrm{\prime }}(�)}{2!}(�-�{)}^2+\dots +\frac{{�}^{�}(�)}{�!}(�-�{)}^{�}$

95. Prime Number Theorem:

• The Prime Number Theorem describes the asymptotic distribution of prime numbers and states that the number of primes less than $�$ is asymptotically $�\mathrm{/}\log (�)$.

96. Cayley-Hamilton Theorem:

• The Cayley-Hamilton theorem states that every square matrix satisfies its own characteristic equation.

97. Brouwer Fixed-Point Theorem:

• Brouwer's Fixed-Point Theorem states that every continuous function from a closed ball in Euclidean space to itself has a fixed point.

98. Nash Equilibrium:

• In game theory, a Nash equilibrium is a solution where no player has an incentive to deviate unilaterally from their chosen strategy.

99. Bézier Curves:

• Bézier curves are defined by control points and are widely used in computer graphics and design for creating smooth curves.

100. Quaternions:

• Quaternions are a number system that extends complex numbers and are used in computer graphics, control theory, and quantum mechanics.
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