📐 Integration of sin²(x) Using the identity $\\sin^2(x) = \\frac{1 - \\cos(2x)}{2}$, we have $\\int \\sin^2(x) \\, dx = \\frac{x}{2} - \\frac{\\sin(2x)}{4} + C$. From: Beginner Calculus Learn more:

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Calculus Made Easy - An interactive introduction to calculus

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💡 Equilateral Triangle Construction Given segment $AB$, an equilateral triangle can be constructed with $AB$ as one side. Proof: Draw circle centered at $A$ with radius $AB$. Draw circle centered at $B$ with radius $BA$. Let $C$ be an intersection point of these circles. Then $AC = AB$ (radius of first circle) and $BC = BA$ (radius of second). Therefore $AB = BC = CA$, so triangle $ABC$ is equilateral. From: numbers-geometry Learn more:

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📐 Rank-Nullity Theorem For a linear transformation $T: V \\to W$ where $V$ is finite-dimensional: $\\dim(V) = \\text{rank}(T) + \\text{nullity}(T)$. Proof: Let $\\dim(V) = n$ and $\\dim(\\ker(T)) = k$. Let $\\{v_1, \\ldots, v_k\\}$ be a basis for $\\ker(T)$. Extend this to a basis $\\{v_1, \\ldots, v_k, u_1, \\ldots, u_{n-k}\\}$ of $V$. **Claim:** $\\{T(u_1), \\ldots, T(u_{n-k})\\}$ is a basis for $\\text{im}(T)$. **Spanning:** Any $w \\in \\text... From: Advanced Linear Algebra Learn more:

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A rigorous treatment of linear algebra based on Hoffman & Kunze, covering vector spaces, linear transformations, and canonical forms.

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📐 Theorem 11.9.1 (First Fundamental Theorem of Calculus) Let $f : [a, b] \\to \\mathbb{R}$ be Riemann integrable, and define $F(x) := \\int_a^x f$. Then $F$ is continuous on $[a, b]$. Moreover, if $f$ is continuous at $x_0 \\in (a, b)$, then $F$ is differentiable at $x_0$ with $F\ Proof: **Continuity:** For $x < y$, $|F(y) - F(x)| = |\\int_x^y f| \\leq M(y - x)$ where $M$ bounds $|f|$. As $y \\to x$, $F(y) \\to F(x)$. **Differentiability:** Fix $x_0$ where $f$ is continuous. For $h > 0$: $\\frac{F(x_0 + h) - F(x_0)}{h} = \\frac{1}{h} \\int_{x_0}^{x_0 + h} f$ Since $f$ is conti... From: tao-analysis-1 Learn more:

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🎮 Interactive: Polynomial Grapher Plot polynomials and find their roots. See how the degree determines the shape of the curve. From: Basic Algebra Try it:

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📐 Projection Theorem If $S$ is a closed subspace of an inner product space $V$ and $v \\in V$, there exists a unique $s \\in S$ such that $\\|v - s\\| = \\inf\\{\\|v - w\\| : w \\in S\\}$. Moreover, $v - s \\perp S$. Proof: The infimum is achieved by completeness of $S$. Uniqueness follows from the parallelogram law. The orthogonality condition characterizes the projection. From: adv_linalg Learn more:

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📐 Cycle Decomposition Every permutation can be written as a product of disjoint cycles, unique up to order. From: aa-pinter Learn more:

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📖 Axiom 3.1 (Sets are Objects) If $A$ is a set, then $A$ is also an object. In particular, given two sets $A$ and $B$, it is meaningful to ask whether $A$ is also an element of $B$. From: tao-analysis-1 Learn more:

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📐 First Isomorphism Theorem for Modules If $\\phi: M \\to N$ is an $R$-module homomorphism, then $M/\\ker(\\phi) \\cong \\text{Im}(\\phi)$. From: df-course Learn more:

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📖 Cyclic Shift A cyclic shift of $(c_0, c_1, \\ldots, c_{n-1})$ is $(c_{n-1}, c_0, \\ldots, c_{n-2})$. From: intro-discrete Learn more:

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📖 Order of an Element The \\textbf{order} of an element $a \\in G$, denoted $|a|$ or $\\text{ord}(a)$, is the smallest positive integer $n$ such that $a^n = e$. If no such $n$ exists, $a$ has \\textbf{infinite order}. From: df-course Learn more:

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📐 Sample Theorem If $A \\subseteq B$ and $B \\subseteq A$, then $A = B$ Proof: Let $x \\in A$. Since $A \\subseteq B$, we have $x \\in B$ by definition of subset. Therefore, every element of $A$ is in $B$. Now, let $y \\in B$. Since $B \\subseteq A$, we have $y \\in A$ by definition. Therefore, every element of $B$ is in $A$. Since $A \\subseteq B$ and $B \\subseteq A... From: tao2 Learn more:

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📖 Graph A graph $G$ is a triple $(V(G), E(G), \\psi_G)$ where $V(G)$ is a set of vertices, $E(G)$ is a set of edges, and $\\psi_G$ associates each edge with an unordered pair of vertices. A simple graph has no loops or multiple edges. From: Introduction to Graph Theory Learn more:

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📐 Rank-Nullity Theorem For a linear transformation $T: V \\to W$ with $V$ finite-dimensional, $\\dim(V) = \\dim(\\ker T) + \\dim(\\mathrm{im}\\, T)$. Proof: Let $\\{v_1, \\ldots, v_k\\}$ be a basis for $\\ker T$ and extend to a basis $\\{v_1, \\ldots, v_k, u_1, \\ldots, u_r\\}$ for $V$. Then $\\{Tu_1, \\ldots, Tu_r\\}$ is a basis for $\\mathrm{im}\\, T$, giving $\\dim V = k + r$. From: adv_linalg Learn more:

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📐 Finite-Dimensional One-to-One Characterization Let $T: V \\to W$ be linear with $\\dim V = n$. The following are equivalent: (a) $T$ is one-to-one. (b) If $e_1, \\ldots, e_p$ are independent in $V$, then $T(e_1), \\ldots, T(e_p)$ are independent. (c) $\\dim T(V) = n$. (d) If $\\{e_1, \\ldots, e_n\\}$ is a basis for $V$, then $\\{T(e_1), \\ldots, T(e_n)\\}$ is a basis for $T(V)$. From: calc2 Learn more:

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📐 Pascal Pressure applied to an enclosed fluid is transmitted undiminished to every portion of the fluid and the walls of the containing vessel From: Men of Mathematics Learn more:

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An interactive journey through 2500 years of mathematical history with E.T. Bell

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📖 Implicit Function An implicit function is one where the relationship between variables is implied by an equation but the dependent variable is not directly isolated, such as $2x - y - 7 = 0$. From: Beginner Calculus Learn more:

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📐 Number of Divisors Formula If $n = p_1^{e_1} \\cdots p_k^{e_k}$, then $\\tau(n) = (e_1 + 1)(e_2 + 1) \\cdots (e_k + 1)$. Proof: Each divisor is determined by choosing exponents $d_1, \\ldots, d_k$. For $d_i$, there are $e_i + 1$ choices: $0, 1, 2, \\ldots, e_i$. By the multiplication principle, total divisors = $(e_1 + 1)(e_2 + 1) \\cdots (e_k + 1)$. From: numbers-geometry Learn more:

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📖 SAS Congruence Two triangles are congruent if they have two sides and the included angle equal. From: numbers-geometry Learn more:

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📐 Multivariate Gaussian Distribution The multivariate Gaussian with mean $\\boldsymbol{\\mu}$ and covariance $\\Sigma$ has density $f(\\mathbf{x}) = \\frac{1}{(2\\pi)^{n/2}|\\Sigma|^{1/2}} \\exp\\left(-\\frac{1}{2}(\\mathbf{x}-\\boldsymbol{\\mu})^T\\Sigma^{-1}(\\mathbf{x}-\\boldsymbol{\\mu})\\right)$. From: Linear Algebra Learn more:

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