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Here is a comprehensive list of fundamental mathematical equations across various branches of mathematics: ### Arithmetic and Algebra 1. **Addition**: \( a + b = c \) 2. **Subtraction**: \( a - b = c \) 3. **Multiplication**: \( a \times b = c \) 4. **Division**: \( \frac{a}{b} = c \) 5. **Quadratic Equation**: \( ax^2 + bx + c = 0 \) 6. **Linear Equation**: \( ax + b = 0 \) 7. **Slope of a Line**: \( m = \frac{y_2 - y_1}{x_2 - x_1} \) 8. **Point-Slope Form**: \( y - y_1 = m(x - x_1) \) 9. **Slope-Intercept Form**: \( y = mx + b \) 10. **Polynomial Equation**: \( P(x) = a_n x^n + a_{n-1} x^{n-1} + \ldots + a_1 x + a_0 \) 11. **Exponential Growth/Decay**: \( y = y_0 e^{kt} \) ### Geometry 1. **Pythagorean Theorem**: \( a^2 + b^2 = c^2 \) 2. **Area of a Circle**: \( A = \pi r^2 \) 3. **Circumference of a Circle**: \( C = 2 \pi r \) 4. **Area of a Triangle**: \( A = \frac{1}{2} b h \) 5. **Volume of a Sphere**: \( V = \frac{4}{3} \pi r^3 \) 6. **Volume of a Cylinder**: \( V = \pi r^2 h \) 7. **Surface Area of a Sphere**: \( A = 4 \pi r^2 \) 8. **Surface Area of a Cylinder**: \( A = 2 \pi r (r + h) \) ### Trigonometry 1. **Sine Function**: \( \sin(\theta) = \frac{\text{opposite}}{\text{hypotenuse}} \) 2. **Cosine Function**: \( \cos(\theta) = \frac{\text{adjacent}}{\text{hypotenuse}} \) 3. **Tangent Function**: \( \tan(\theta) = \frac{\text{opposite}}{\text{adjacent}} \) 4. **Sine Rule**: \( \frac{a}{\sin(A)} = \frac{b}{\sin(B)} = \frac{c}{\sin(C)} \) 5. **Cosine Rule**: \( c^2 = a^2 + b^2 - 2ab \cos(C) \) ### Calculus 1. **Derivative Definition**: \( f'(x) = \lim_{h \to 0} \frac{f(x+h) - f(x)}{h} \) 2. **Integral Definition**: \( \int f(x) \, dx = F(x) + C \) 3. **Fundamental Theorem of Calculus**: \( \int_a^b f(x) \, dx = F(b) - F(a) \) 4. **Chain Rule**: \( \frac{d}{dx} f(g(x)) = f'(g(x)) g'(x) \) 5. **Product Rule**: \( \frac{d}{dx} [u(x)v(x)] = u'(x)v(x) + u(x)v'(x) \) 6. **Quotient Rule**: \( \frac{d}{dx} \left( \frac{u(x)}{v(x)} \right) = \frac{u'(x)v(x) - u(x)v'(x)}{[v(x)]^2} \) 7. **Partial Derivative**: \( \frac{\partial f}{\partial x} \) ### Differential Equations 1. **First Order Differential Equation**: \( \frac{dy}{dx} + P(x)y = Q(x) \) 2. **Second Order Differential Equation**: \( \frac{d^2y}{dx^2} + P(x)\frac{dy}{dx} + Q(x)y = R(x) \) 3. **Laplace Transform**: \( \mathcal{L}\{f(t)\} = \int_0^\infty e^{-st} f(t) \, dt \) ### Linear Algebra 1. **Matrix Multiplication**: \( C = AB \) 2. **Determinant of a 2x2 Matrix**: \( \det(A) = ad - bc \) for \( A = \begin{pmatrix} a & b \\ c & d \end{pmatrix} \) 3. **Inverse of a Matrix**: \( A^{-1} \) such that \( AA^{-1} = I \) 4. **Eigenvalue Equation**: \( A \mathbf{v} = \lambda \mathbf{v} \) ### Probability and Statistics 1. **Bayes' Theorem**: \( P(A|B) = \frac{P(B|A)P(A)}{P(B)} \) 2. **Expected Value**: \( E(X) = \sum x_i P(x_i) \) 3. **Variance**: \( \text{Var}(X) = E[(X - \mu)^2] \) 4. **Standard Deviation**: \( \sigma = \sqrt{\text{Var}(X)} \) 5. **Normal Distribution**: \( f(x) = \frac{1}{\sqrt{2\pi\sigma^2}} e^{-\frac{(x-\mu)^2}{2\sigma^2}} \) ### Complex Numbers 1. **Euler's Formula**: \( e^{i\theta} = \cos(\theta) + i\sin(\theta) \) 2. **Complex Conjugate**: \( \overline{z} = a - bi \) for \( z = a + bi \) 3. **Magnitude of a Complex Number**: \( |z| = \sqrt{a^2 + b^2} \) ### Miscellaneous 1. **Binomial Theorem**: \( (a + b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^k \) 2. **Fibonacci Sequence**: \( F(n) = F(n-1) + F(n-2) \) with \( F(0) = 0 \) and \( F(1) = 1 \) This list covers a wide range of fundamental mathematical equations across different areas of mathematics. Let me know if you need more details on any specific equations or additional equations in a particular field. Here's a list of some fundamental mathematical equations across various domains: **Basic Arithmetic** * **Commutative Laws:** * a + b = b + a (Addition) * a * b = b * a (Multiplication) * **Associative Laws:** * (a + b) + c = a + (b + c) (Addition) * (a * b) * c = a * (b * c) (Multiplication) * **Distributive Law:** * a * (b + c) = a * b + a * c * **Identity Elements:** * a + 0 = a (Addition) * a * 1 = a (Multiplication) * **Inverse Elements:** * a + (-a) = 0 (Addition) * a * (1/a) = 1 (Multiplication, for a ≠ 0) **Algebra** * **Quadratic Formula:** * x = (-b ± √(b² - 4ac)) / 2a * **Slope-Intercept Form of a Line:** * y = mx + b * **Distance Formula:** * d = √((x₂ - x₁)² + (y₂ - y₁)² ) **Geometry** * **Pythagorean Theorem:** * a² + b² = c² * **Area of a Circle:** * A = πr² * **Circumference of a Circle:** * C = 2πr **Trigonometry** * **SOH CAH TOA:** * Sin(θ) = Opposite/Hypotenuse * Cos(θ) = Adjacent/Hypotenuse * Tan(θ) = Opposite/Adjacent **Calculus** * **Definition of the Derivative:** * f'(x) = lim (h→0) (f(x + h) - f(x)) / h * **Fundamental Theorem of Calculus:** * ∫ (from a to b) f'(x) dx = f(b) - f(a) **Other Important Equations** * **Euler's Identity:** e^(iπ) + 1 = 0 * **Einstein's Mass-Energy Equivalence:** E=mc² **Important Note:** This is just a small sample of fundamental mathematical equations. Mathematics is vast and diverse, with many other essential equations in areas like statistics, probability, linear algebra, differential equations, and more. Here is a list of fundamental mathematical equations from various branches of mathematics: 1. Algebra: - Quadratic formula: $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$ - Pythagorean theorem: $a^2 + b^2 = c^2$ 2. Calculus: - Derivative of a function: $\frac{d}{dx}f(x) = \lim_{h \to 0} \frac{f(x+h) - f(x)}{h}$ - Integral of a function: $\int_a^b f(x)dx = F(b) - F(a)$, where $F(x)$ is an antiderivative of $f(x)$ - Fundamental theorem of calculus: $\int_a^b f(x)dx = F(b) - F(a)$, where $F'(x) = f(x)$ 3. Trigonometry: - Sine rule: $\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}$ - Cosine rule: $c^2 = a^2 + b^2 - 2ab \cos C$ 4. Exponential and Logarithmic Functions: - Exponential function: $f(x) = a^x$, where $a > 0$ and $a \neq 1$ - Natural logarithm: $\ln e = 1$ - Logarithmic identity: $\log_a(xy) = \log_a(x) + \log_a(y)$ 5. Complex Numbers: - Euler's identity: $e^{i\pi} + 1 = 0$ - De Moivre's formula: $(\cos \theta + i \sin \theta)^n = \cos(n\theta) + i \sin(n\theta)$ 6. Differential Equations: - First-order linear differential equation: $\frac{dy}{dx} + P(x)y = Q(x)$ - Second-order linear differential equation: $a\frac{d^2y}{dx^2} + b\frac{dy}{dx} + cy = f(x)$ 7. Statistics and Probability: - Bayes' theorem: $P(A|B) = \frac{P(B|A)P(A)}{P(B)}$ - Normal distribution: $f(x) = \frac{1}{\sigma\sqrt{2\pi}}e^{-\frac{(x-\mu)^2}{2\sigma^2}}$ 8. Physics: - Newton's second law of motion: $F = ma$ - Einstein's mass-energy equivalence: $E = mc^2$ 9. Other notable equations: - Euler's formula: $V - E + F = 2$ (for polyhedra) - Cauchy-Schwarz inequality: $|\langle x,y\rangle| \leq ||x|| \cdot ||y||$ - Stirling's approximation: $n! \sim \sqrt{2\pi n}\left(\frac{n}{e}\right)^n$ These equations represent a small subset of the vast number of fundamental equations in mathematics and its applications. Here are more fundamental mathematical equations from various branches of mathematics: 10. Number Theory: - Fermat's little theorem: $a^p \equiv a \pmod{p}$ (for prime $p$) - Euler's totient function: $\phi(n) = n \prod_{p|n} (1 - \frac{1}{p})$ 11. Linear Algebra: - Matrix multiplication: $(AB)_{ij} = \sum_k A_{ik}B_{kj}$ - Determinant of a 2x2 matrix: $\det \begin{pmatrix} a & b \\ c & d \end{pmatrix} = ad - bc$ - Gaussian elimination: $A\vec{x} = \vec{b}$ 12. Fourier Analysis: - Fourier series: $f(x) = \frac{a_0}{2} + \sum_{n=1}^\infty \left(a_n \cos\frac{n\pi x}{L} + b_n \sin\frac{n\pi x}{L}\right)$ - Fourier transform: $\hat{f}(\xi) = \int_{-\infty}^\infty f(x)e^{-2\pi i x\xi}dx$ 13. Partial Differential Equations: - Heat equation: $\frac{\partial u}{\partial t} = \alpha \nabla^2 u$ - Wave equation: $\frac{\partial^2 u}{\partial t^2} = c^2 \nabla^2 u$ - Laplace's equation: $\nabla^2 u = 0$ 14. Combinatorics: - Binomial theorem: $(x + y)^n = \sum_{k=0}^n \binom{n}{k} x^k y^{n-k}$ - Permutations: $P(n,r) = \frac{n!}{(n-r)!}$ - Combinations: $C(n,r) = \binom{n}{r} = \frac{n!}{r!(n-r)!}$ 15. Topology: - Euler characteristic: $\chi = V - E + F$ - Fundamental group: $\pi_1(X)$ 16. Information Theory: - Shannon's entropy: $H(X) = -\sum_{i=1}^n p(x_i) \log_2 p(x_i)$ - Mutual information: $I(X;Y) = \sum_{x \in X} \sum_{y \in Y} p(x,y) \log \frac{p(x,y)}{p(x)p(y)}$ 17. Numerical Analysis: - Newton's method: $x_{n+1} = x_n - \frac{f(x_n)}{f'(x_n)}$ - Runge-Kutta methods: $y_{n+1} = y_n + h\sum_{i=1}^s b_i k_i$ 18. Graph Theory: - Euler's formula for planar graphs: $V - E + F = 2$ - Dijkstra's shortest path algorithm: $dist(u) + w(u,v) \geq dist(v)$ 19. Geometry: - Volume of a sphere: $V = \frac{4}{3}\pi r^3$ - Heron's formula for triangle area: $A = \sqrt{s(s-a)(s-b)(s-c)}$, where $s = \frac{a+b+c}{2}$ These additional equations further demonstrate the breadth and depth of mathematics across various subdisciplines. However, this list still represents only a fraction of the many important equations in mathematics."