equation_gpt_train_100_math.jsonl

{"completion": " The Latex Code for equation Diffusion Conduction Equationis <equation>\\\\kappa u_{xx} = u_{t} \\\\\\\\ \\\\frac{\\\\partial \\\\Psi}{\\\\partial t}=\\\\kappa \\\\triangledown^{2} \\\\Psi</equation>", "tag": "math,diffusion,conduction", "prompt": "What is the equation of Diffusion Conduction Equation?"}
{"completion": " The Latex Code for equation Fourier Transformsis <equation>y(t)=\\\\frac{1}{2\\\\pi}\\\\int_{-\\\\infty}^{\\\\infty} \\\\hat{y}(\\\\omega) e^{i\\\\omega t} \\\\mathrm{d} \\\\omega \\\\\\\\ \\\\hat{y}(\\\\omega)=\\\\int_{-\\\\infty}^{\\\\infty} y(t) e^{-i\\\\omega t} \\\\mathrm{d} t</equation>", "tag": "math,fourier transforms", "prompt": "How to write equation Fourier Transforms?"}
{"completion": " The equation for Surface area of a Cuboid is <equation>S = 2lw + 2lh + 2wh</equation>. The meaning of symbol <equation>S</equation> in Surface area of a Cuboid is: Surface.", "tag": "Math,Geometry", "prompt": "What is the meaning of <equation>S</equation> in Surface area of a Cuboid?"}
{"completion": " The Latex Code for equation Diffusion Conduction Equationis <equation>\\\\kappa u_{xx} = u_{t} \\\\\\\\ \\\\frac{\\\\partial \\\\Psi}{\\\\partial t}=\\\\kappa \\\\triangledown^{2} \\\\Psi</equation>", "tag": "math,diffusion,conduction", "prompt": "What is the latex code of equation Diffusion Conduction Equation?"}
{"completion": " The equation for Area of a Trapezoid is <equation>A = \\\\frac{1}{2}(x_1 + x_2 )h</equation>. The meaning of symbol <equation>x_1</equation> in Area of a Trapezoid is: Length of top of trapezoid.", "tag": "Math,Geometry", "prompt": "Help me find the meaning of symbol <equation>x_1</equation> in Area of a Trapezoid?"}
{"completion": " The equation for Circle is: <equation>\\\\left( {x - x_0 } \\\\right)^2 + \\\\left( {y - y_0 } \\\\right)^2 = R^2</equation>. Here is the brief introduction of Circle:\n        Latex code for Circle\n        \n: Coordinates of Circle Center\n: Radius\n\n\nRelated Documents\n\n\nRelated Videos\n\n\n", "tag": "Math,Geometry", "prompt": "Can you explain Circle?"}
{"completion": " The Latex Code for equation Ellipsoidis <equation>\\\\frac{{\\\\left( {x - x_0 } \\\\right)^2 }}{{a^2 }} + \\\\frac{{\\\\left( {y - y_0 } \\\\right)^2 }}{{b^2 }} + \\\\frac{{\\\\left( {z - z_0 } \\\\right)^2 }}{{c^2 }} = 1\\ </equation>", "tag": "Math,Geometry", "prompt": "Show me the latex code of equation Ellipsoid?"}
{"completion": " The related materials of Normal Gaussian Distribution is: ", "tag": "Math,Statistics", "prompt": "Can you help me find related materials of Normal Gaussian Distribution?"}
{"completion": " The equation for Eccentricity of an Ellipse is: <equation>\\\\varepsilon = \\\\frac{{\\\\sqrt {a^2 - b^2 } }}{a}</equation>. Here is the brief introduction of Eccentricity of an Ellipse:\n        Latex code for Eccentricity of an Ellipse.\n        \n: Eccentricity\n\n\nRelated Documents\n\n\nRelated Videos\n\n\n", "tag": "Math,Geometry", "prompt": "Please give me detailed tutorial of Eccentricity of an Ellipse?"}
{"completion": " The Latex Code for equation Area of a Ellipseis <equation>A = \\\\pi r_1 r_2</equation>", "tag": "Math,Geometry", "prompt": "Show me the latex code of equation Area of a Ellipse?"}
{"completion": " The equation for Eccentricity of a Hyperbola is <equation>\\\\varepsilon = \\\\frac{{\\\\sqrt {a^2 + b^2 } }}{a}</equation>. The meaning of symbol <equation>\\varepsilon</equation> in Eccentricity of a Hyperbola is: Eccentricity.", "tag": "Math,Geometry", "prompt": "Can you explain the symbol <equation>\\varepsilon</equation> in Eccentricity of a Hyperbola?"}
{"completion": " The Latex Code for equation Binomial Expansionis <equation>(1+x)^{n}=1+nx+\\\\frac{n(n-1)}{2!}x^{2}+...+ C^{r}_{n}x^{r} +...+x^{n} \\\\\\\\ C^{r}_{n}=\\\\frac{n!}{r!(n-r)!}</equation>", "tag": "math,binomial,expansion", "prompt": "Can you write equation Binomial Expansion?"}
{"completion": " The equation for Poisson Distribution is: <equation>X \\\\sim \\\\pi(\\\\mu) \\\\\\\\f(x)=\\\\frac{\\\\mu^{x}}{x!}e^{-\\\\mu}\\\\\\\\ \\\\text{Poisson Mean} \\\\mu \\\\\\\\  \\\\text{Poisson Variance}\\\\sigma^2=\\\\mu</equation>. Here is the brief introduction of Poisson Distribution:\n        \\mu equals to the probability that an event occurs in a unit time period.\n    ", "tag": "Math,Statistics", "prompt": "I am interested in Poisson Distribution?"}
{"completion": " The equation for Area of a Regular Polygon is <equation>A = \\\\frac{{nsr}}{2} = \\\\frac{{pr}}{2}</equation>. The meaning of symbol <equation>A</equation> in Area of a Regular Polygon is: Area.", "tag": "Math,Geometry", "prompt": "Can you explain the symbol <equation>A</equation> in Area of a Regular Polygon?"}
{"completion": " The Latex Code for equation Cartesian to Spherical Coordinatesis <equation>\\\\begin{array}{*{20}c} {x = R\\\\sin \\\\theta \\\\cos \\\\phi } & {R = \\\\sqrt {x^2 + y^2 + z^2 } } & {} \\\\\\\\ {y = R\\\\sin \\\\theta \\\\sin \\\\phi } & {\\\\phi = \\\\tan ^{ - 1} \\\\left( {\\\\frac{y}{x}} \\\\right)} & {} \\\\\\\\ {z = R\\\\cos \\\\theta } & {\\\\theta = \\\\cos ^{ - 1} \\\\left( {\\\\frac{z}{{\\\\sqrt {x^2 + y^2 + z^2 } }}} \\\\right)} & {} \\\\\\\\ \\\\end{array}</equation>", "tag": "Math,Geometry", "prompt": "Can you write equation Cartesian to Spherical Coordinates?"}
{"completion": " The equation for Bessel Equation is <equation>x^{2}\\\\frac{\\\\mathrm{d}^{2} y}{\\\\mathrm{d} x^{2}}+x\\\\frac{\\\\mathrm{d} y}{\\\\mathrm{d} x}+(x^{2}-m^{2})y=0 \\\\\\\\\\             J_{m}(x)=\\\\sum^{\\\\infty}_{k=0}\\\\frac{(-1)^{k} (x/2)^{m+2k}}{k!(m+k)!}</equation>. The meaning of symbol <equation>x^{2}\\frac{\\mathrm{d}^{2} y}{\\mathrm{d} x^{2}}+x\\frac{\\mathrm{d} y}{\\mathrm{d} x}+(x^{2}-m^{2})y=0</equation> in Bessel Equation is: Solutions of Bessel equations  are Bessel functions ..", "tag": "math,bessel equation", "prompt": "What is the meaning of the symbol <equation>x^{2}\\frac{\\mathrm{d}^{2} y}{\\mathrm{d} x^{2}}+x\\frac{\\mathrm{d} y}{\\mathrm{d} x}+(x^{2}-m^{2})y=0</equation> in Bessel Equation?"}
{"completion": " The equation for Distance Between Two Points 3D is <equation>d = \\\\sqrt {\\\\left( {x_1 - x_2 } \\\\right)^2 + \\\\left( {y_1 - y_2 } \\\\right)^2 + \\\\left( {z_1 - z_2 } \\\\right)^2 }</equation>. The meaning of symbol <equation>(z1,z2)</equation> in Distance Between Two Points 3D is: Point 3.", "tag": "Math,Geometry", "prompt": "Can you explain the symbol <equation>(z1,z2)</equation> in Distance Between Two Points 3D?"}
{"completion": " The equation for Distance Between Two Points 3D is <equation>d = \\\\sqrt {\\\\left( {x_1 - x_2 } \\\\right)^2 + \\\\left( {y_1 - y_2 } \\\\right)^2 + \\\\left( {z_1 - z_2 } \\\\right)^2 }</equation>. The meaning of symbol <equation>(z1,z2)</equation> in Distance Between Two Points 3D is: Point 3.", "tag": "Math,Geometry", "prompt": "What does the symbol mean <equation>(z1,z2)</equation> in Distance Between Two Points 3D?"}
{"completion": " The Latex Code for equation Area of a Rectangleis <equation>A = lw</equation>", "tag": "Math,Geometry", "prompt": "Please give me some hints on equation Area of a Rectangle?"}
{"completion": " The equation for Surface area of a Sphere is <equation>S = 4\\\\pi r^2</equation>. The meaning of symbol <equation>r</equation> in Surface area of a Sphere is: Radius r.", "tag": "Math,Geometry", "prompt": "What does the symbol mean <equation>r</equation> in Surface area of a Sphere?"}
{"completion": " The related materials of Area of a Square is: ", "tag": "Math,Geometry", "prompt": "Can you help me find related materials of Area of a Square?"}
{"completion": " The equation for gaussian process is: <equation>\\\\log p(y|X) \\\\propto -[y^{T}(K + \\\\sigma^{2}I)^{-1}y+\\\\log|K + \\\\sigma^{2}I|] \\\\\\\\\\ f(X)=\\\\[f(x_{1}),f(x_{2}),...,f(x_{N}))\\\\]^{T} \\\\sim \\\\mathcal{N}(\\\\mu, K_{X,X}) \\\\\\\\\\ f_{*}|X_{*},X,y \\\\sim \\\\mathcal{N}(\\\\mathbb{E}(f_{*}),\\\\text{cov}(f_{*})) \\\\\\\\\\ \\\\text{cov}(f_{*})=K_{X_{*},X_{*}}-K_{X_{*},X}[K_{X,X}+\\\\sigma^{2}I]^{-1}K_{X,X_{*}}</equation>. Here is the brief introduction of gaussian process:\n        Gaussian process assumes that the output of N function are not independent but correlated. It assumes the collection of N function values, represented by N-dimensional vector f, has a joint Gaussian distribution with mean vector and covariance matrix(kernel matrix). The predicted value of n^{*} test values are given by mean and variance as \\mathbb{E}(f_{*}) and \\text{cov}(f_{*}) respectively.\n        See below link Deep Kernel Learning for more details. \n    ", "tag": "math,gaussian process", "prompt": "Please give me detailed tutorial of gaussian process?"}
{"completion": " The equation for Surface area of a cylinder is: <equation>S = 2\\\\pi r^2 + 2\\\\pi rh</equation>. Here is the brief introduction of Surface area of a cylinder:\n        Latex code for Surface area of a cylinder.\n        \n: Surface\n: Constant of Pi\n: Radius r\n: Height\n\n\nRelated Documents\n\n\nRelated Videos\n\n\n", "tag": "Math,Geometry", "prompt": "Can you explain Surface area of a cylinder?"}
{"completion": " The equation for Eigenvalues and Eigenvectors is: <equation>A\\\\mathbf{u}=\\\\lambda\\\\mathbf{u} \\\\\\\\ \\ P_{n}(\\\\lambda)=|A-\\\\lambda I|</equation>. Here is the brief introduction of Eigenvalues and Eigenvectors: The n eigenvalues  and eigenvectors   of an n by n matrix A are the solutions of the equation . The eigenvalues are the zeros of the polynomial of degree n . Additionally,  is called the characteristic equation of the matrix.\n        ", "tag": "math,eigenvalues,eigenvectors", "prompt": "I am interested in Eigenvalues and Eigenvectors?"}
{"completion": " The equation for Gamma Distribution is <equation>\\\\Gamma \\\\left( a \\\\right) = \\\\int\\\\limits_0^\\\\infty {s^{a - 1} } e^{ - s} ds \\\\\\\\\\             P(x) = \\\\frac{x^{\\\\alpha-1} e^{-frac{x}{\\\\theta}}}{\\\\Gamma(\\\\alpha) \\\\theta^{\\\\alpha}} \\\\\\\\\\             \\\\mu = \\\\alpha \\\\theta \\\\\\\\\\             \\\\sigma^{2} = \\\\alpha \\\\theta^{2} \\\\\\\\\\             \\\\gamma_{1} = \\\\frac{2}{\\\\sqrt{\\\\alpha}} \\\\\\\\\\             \\\\gamma_{2} = \\\\frac{6}{\\\\alpha}</equation>. The meaning of symbol <equation>\\Gamma(a)</equation> in Gamma Distribution is: Gamma function of a.", "tag": "Math,Statistics", "prompt": "Can you explain the symbol <equation>\\Gamma(a)</equation> in Gamma Distribution?"}
{"completion": " The equation for Area of a Circle is <equation>A = \\\\pi r^2</equation>. The meaning of symbol <equation>A</equation> in Area of a Circle is: Area.", "tag": "Math,Geometry", "prompt": "What is the meaning of <equation>A</equation> in Area of a Circle?"}
{"completion": " The Latex Code for equation Wave Equationis <equation>u_{tt}=c^{2}u_{xx}</equation>", "tag": "math,wave equation", "prompt": "Can you write equation Wave Equation?"}
{"completion": " The equation for Cylindrical to Spherical Coordinates is <equation>\\\\begin{array}{*{20}c} {r = R\\\\sin \\\\theta } & {R = \\\\sqrt {r^2 + z^2 } } & {} \\\\\\\\ {z = R\\\\sin \\\\theta } & {\\\\phi = \\\\theta } & {} \\\\\\\\ {\\\\theta = \\\\phi } & {\\\\theta = \\\\tan ^{ - 1} \\\\left( {\\\\frac{r}{z}} \\\\right)} & {} \\\\\\\\ \\\\end{array}</equation>. The meaning of symbol <equation>r</equation> in Cylindrical to Spherical Coordinates is: Radius.", "tag": "Math,Geometry", "prompt": "What is the meaning of <equation>r</equation> in Cylindrical to Spherical Coordinates?"}
{"completion": " The equation for Cartesian to Spherical Coordinates is: <equation>\\\\begin{array}{*{20}c} {x = R\\\\sin \\\\theta \\\\cos \\\\phi } & {R = \\\\sqrt {x^2 + y^2 + z^2 } } & {} \\\\\\\\ {y = R\\\\sin \\\\theta \\\\sin \\\\phi } & {\\\\phi = \\\\tan ^{ - 1} \\\\left( {\\\\frac{y}{x}} \\\\right)} & {} \\\\\\\\ {z = R\\\\cos \\\\theta } & {\\\\theta = \\\\cos ^{ - 1} \\\\left( {\\\\frac{z}{{\\\\sqrt {x^2 + y^2 + z^2 } }}} \\\\right)} & {} \\\\\\\\ \\\\end{array}</equation>. Here is the brief introduction of Cartesian to Spherical Coordinates:\n        Latex code for Cartesian to Spherical Coordinates.\n        \n: Spherical Coordinates\n: Radius\n\n\nRelated Documents\n\n\nRelated Videos\n\n\n", "tag": "Math,Geometry", "prompt": "I am interested in Cartesian to Spherical Coordinates?"}
{"completion": " The equation for Surface area of a Sphere is <equation>S = 4\\\\pi r^2</equation>. The meaning of symbol <equation>r</equation> in Surface area of a Sphere is: Radius r.", "tag": "Math,Geometry", "prompt": "Help me find the meaning of symbol <equation>r</equation> in Surface area of a Sphere?"}
{"completion": " The Latex Code for equation Area of a Parallelogramis <equation>A = bh</equation>", "tag": "Math,Geometry", "prompt": "Please give me some hints on equation Area of a Parallelogram?"}
{"completion": " The equation for Fourier Transforms is: <equation>y(t)=\\\\frac{1}{2\\\\pi}\\\\int_{-\\\\infty}^{\\\\infty} \\\\hat{y}(\\\\omega) e^{i\\\\omega t} \\\\mathrm{d} \\\\omega \\\\\\\\ \\\\hat{y}(\\\\omega)=\\\\int_{-\\\\infty}^{\\\\infty} y(t) e^{-i\\\\omega t} \\\\mathrm{d} t</equation>. Here is the brief introduction of Fourier Transforms: If y(x) is a function defined in the range , then the fourier transform  is defined as above.\n        ", "tag": "math,fourier transforms", "prompt": "Please give me detailed tutorial of Fourier Transforms?"}
{"completion": " The equation for Power Series with Real Variables is <equation>e^{x}=1+x+\\\\frac{x^{2}}{2!}+...+\\\\frac{x^{n}}{n!}+... \\\\\\\\\\ \\\\ln(1+x) = x - \\\\frac{x^{2}}{2} + \\\\frac{x^{3}}{3} + ... + (-1)^{n+1}\\\\frac{x^{n}}{n!} +... \\\\\\\\\\ \\\\cos(x) = \\\\frac{e^{ix}+e^{-ix}}{2}=1-\\\\frac{x^{2}}{2!}+\\\\frac{x^{4}}{4!}-\\\\frac{x^{6}}{6!}+...\\\\\\\\\\ \\\\sin(x) = \\\\frac{e^{ix}-e^{-ix}}{2i}=x-\\\\frac{x^{3}}{3!}+\\\\frac{x^{5}}{5!}+...</equation>. The meaning of symbol <equation>e^{x}=1+x+\\frac{x^{2}}{2!}+...+\\frac{x^{n}}{n!}+...</equation> in Power Series with Real Variables is: .", "tag": "math,power,series", "prompt": "What does the symbol mean <equation>e^{x}=1+x+\\frac{x^{2}}{2!}+...+\\frac{x^{n}}{n!}+...</equation> in Power Series with Real Variables?"}
{"completion": " The equation for Eccentricity of a Hyperbola is: <equation>\\\\varepsilon = \\\\frac{{\\\\sqrt {a^2 + b^2 } }}{a}</equation>. Here is the brief introduction of Eccentricity of a Hyperbola:\n        Latex code for Eccentricity of a Hyperbola.\n        \n: Eccentricity\n\n\nRelated Documents\n\n\nRelated Videos\n\n\n", "tag": "Math,Geometry", "prompt": "What's the meaning of Eccentricity of a Hyperbola?"}
{"completion": " The Latex Code for equation Determinants of a Matrixis <equation>|A|=\\\\sum_{i,j,k,...}\\\\epsilon_{ijk}A_{1i}A_{2j}A_{3k}...</equation>", "tag": "math,determinants", "prompt": "Please give me some hints on equation Determinants of a Matrix?"}
{"completion": " The equation for Area of a Triangle is <equation>A = \\\\frac{1}{2}(x_1 + x_2 )h</equation>. The meaning of symbol <equation>b</equation> in Area of a Triangle is: Length of Bottom of Triangle.", "tag": "Math,Geometry", "prompt": "Can you explain the symbol <equation>b</equation> in Area of a Triangle?"}
{"completion": " The equation for Spherical Harmonics Equation is <equation>[\\\\frac{1}{\\\\sin \\\\theta} \\\\frac{\\\\partial}{\\\\partial \\\\theta}(\\\\sin \\\\theta \\\\frac{\\\\partial}{\\\\partial \\\\theta}) + \\\\frac{1}{\\\\sin^{2} \\\\theta} \\\\frac{\\\\partial^{2}}{\\\\partial \\\\phi^{2}}) ] Y^{m}_{l} + l(l+1) Y^{m}_{l}=0 \\\\\\\\ Y^{m}_{l}(\\\\theta,\\\\phi)=\\\\sqrt{\\\\frac{2l+1}{4 \\\\pi} \\\\frac{(l-|m|)!}{(l+|m|)!}}P^{m}_{l}(\\\\cos \\\\theta) e^{im \\\\phi} \\\\times \\\\begin{cases}(-1)^{m} & m\\\\ge 0 \\\\\\\\ 1 & m <0 \\\\end{cases}</equation>. The meaning of symbol <equation>[\\frac{1}{\\sin \\theta} \\frac{\\partial}{\\partial \\theta}(\\sin \\theta \\frac{\\partial}{\\partial \\theta}) + \\frac{1}{\\sin^{2} \\theta} \\frac{\\partial^{2}}{\\partial \\phi^{2}}) ] Y^{m}_{l} + l(l+1) Y^{m}_{l}=0</equation> in Spherical Harmonics Equation is: .", "tag": "math,spherical harmonics", "prompt": "What does the symbol mean <equation>[\\frac{1}{\\sin \\theta} \\frac{\\partial}{\\partial \\theta}(\\sin \\theta \\frac{\\partial}{\\partial \\theta}) + \\frac{1}{\\sin^{2} \\theta} \\frac{\\partial^{2}}{\\partial \\phi^{2}}) ] Y^{m}_{l} + l(l+1) Y^{m}_{l}=0</equation> in Spherical Harmonics Equation?"}
{"completion": " The Latex Code for equation Heat Equationis <equation>u_{t}={\\\\alpha}^{2}u_{xx}</equation>", "tag": "math,heat equation", "prompt": "Show me the latex code of equation Heat Equation?"}
{"completion": " The equation for Hyperbola is <equation>\\\\frac{{\\\\left( {x - x_0 } \\\\right)^2 }}{{a^2 }} - \\\\frac{{\\\\left( {y - y_0 } \\\\right)^2 }}{{b^2 }} = 1\\ </equation>. The meaning of symbol <equation>b</equation> in Hyperbola is: Length b.", "tag": "Math,Geometry", "prompt": "What does the symbol mean <equation>b</equation> in Hyperbola?"}
{"completion": " The equation for Distance Between Two Points 3D is <equation>d = \\\\sqrt {\\\\left( {x_1 - x_2 } \\\\right)^2 + \\\\left( {y_1 - y_2 } \\\\right)^2 + \\\\left( {z_1 - z_2 } \\\\right)^2 }</equation>. The meaning of symbol <equation>(x1,x2)</equation> in Distance Between Two Points 3D is: Point 1.", "tag": "Math,Geometry", "prompt": "What does the symbol mean <equation>(x1,x2)</equation> in Distance Between Two Points 3D?"}
{"completion": " The equation for Circle is <equation>\\\\left( {x - x_0 } \\\\right)^2 + \\\\left( {y - y_0 } \\\\right)^2 = R^2</equation>. The meaning of symbol <equation>(x_0,y_0)</equation> in Circle is: Coordinates of Circle Center.", "tag": "Math,Geometry", "prompt": "Help me find the meaning of symbol <equation>(x_0,y_0)</equation> in Circle?"}
{"completion": " The equation for Gamma Distribution is <equation>\\\\Gamma \\\\left( a \\\\right) = \\\\int\\\\limits_0^\\\\infty {s^{a - 1} } e^{ - s} ds \\\\\\\\\\             P(x) = \\\\frac{x^{\\\\alpha-1} e^{-frac{x}{\\\\theta}}}{\\\\Gamma(\\\\alpha) \\\\theta^{\\\\alpha}} \\\\\\\\\\             \\\\mu = \\\\alpha \\\\theta \\\\\\\\\\             \\\\sigma^{2} = \\\\alpha \\\\theta^{2} \\\\\\\\\\             \\\\gamma_{1} = \\\\frac{2}{\\\\sqrt{\\\\alpha}} \\\\\\\\\\             \\\\gamma_{2} = \\\\frac{6}{\\\\alpha}</equation>. The meaning of symbol <equation>\\gamma_{1} = \\frac{2}{\\sqrt{\\alpha}}</equation> in Gamma Distribution is: Skewness of Gamma Distribution.", "tag": "Math,Statistics", "prompt": "Can you explain the symbol <equation>\\gamma_{1} = \\frac{2}{\\sqrt{\\alpha}}</equation> in Gamma Distribution?"}
{"completion": " The equation for Area of a Parallelogram is <equation>A = bh</equation>. The meaning of symbol <equation>A</equation> in Area of a Parallelogram is: Area.", "tag": "Math,Geometry", "prompt": "What does the symbol mean <equation>A</equation> in Area of a Parallelogram?"}
{"completion": " The equation for Eccentricity of an Ellipse is <equation>\\\\varepsilon = \\\\frac{{\\\\sqrt {a^2 - b^2 } }}{a}</equation>. The meaning of symbol <equation>\\varepsilon</equation> in Eccentricity of an Ellipse is: Eccentricity.", "tag": "Math,Geometry", "prompt": "What does the symbol mean <equation>\\varepsilon</equation> in Eccentricity of an Ellipse?"}
{"completion": " The Latex Code for equation Arithmetic and Geometric Progressionsis <equation>S_{n}=a+(a+d)+(a+2d)+...+[a+(n-1)d]=\\\\frac{n}{2}[2a+(n-1)d] \\\\\\\\ S_{n}=a+ar+ar^{2}+...+ar^{n-1}=a\\\\frac{1-r^{n}}{1-r}</equation>", "tag": "math,arithmetic,geometric,progressions", "prompt": "What is the equation of Arithmetic and Geometric Progressions?"}
{"completion": " The Latex Code for equation Hyperbolais <equation>\\\\frac{{\\\\left( {x - x_0 } \\\\right)^2 }}{{a^2 }} - \\\\frac{{\\\\left( {y - y_0 } \\\\right)^2 }}{{b^2 }} = 1\\ </equation>", "tag": "Math,Geometry", "prompt": "Can you write equation Hyperbola?"}
{"completion": " The equation for Area of a Triangle is <equation>A = \\\\frac{1}{2}(x_1 + x_2 )h</equation>. The meaning of symbol <equation>A</equation> in Area of a Triangle is: Area.", "tag": "Math,Geometry", "prompt": "What is the meaning of the symbol <equation>A</equation> in Area of a Triangle?"}
{"completion": " The equation for Elliptic Cylinder is: <equation>\\\\frac{{\\\\left( {x - x_0 } \\\\right)^2 }}{{a^2 }} + \\\\frac{{\\\\left( {y - y_0 } \\\\right)^2 }}{{b^2 }} = 1\\ </equation>. Here is the brief introduction of Elliptic Cylinder:\n        Latex code for Elliptic Cylinder.\n        \n: Center Coordinates of Elliptic Cylinder\n: Length of Axis x\n: Length of Axis y\n\n\nRelated Documents\n\n\nRelated Videos\n\n\n", "tag": "Math,Geometry", "prompt": "I am interested in Elliptic Cylinder?"}
{"completion": " The Latex Code for equation Normal Gaussian Distributionis <equation>X \\\\sim \\\\mathcal{N}(\\\\mu,\\\\sigma^2) \\\\\\\\ f(x)=\\\\frac{1}{\\\\sigma\\\\sqrt{2\\\\pi}}\\\\exp{[-\\\\frac{(x-\\\\mu)^{2}}{2\\\\sigma^{2}}]}</equation>", "tag": "Math,Statistics", "prompt": "How to write equation Normal Gaussian Distribution?"}
{"completion": " The equation for Regression Least Squares Fitting is: <equation>y_{i}=\\\\alpha + \\\\beta (x_{i} - \\\\bar{x}) + \\\\epsilon_{i} \\\\\\\\ \\\\hat{\\\\alpha}=\\\\bar{y} \\\\\\\\ \\\\hat{\\\\beta}=\\\\frac{s^{2}_{xy}}{s^{2}_{x}}</equation>. Here is the brief introduction of Regression Least Squares Fitting:\n            To fit a straight line by least squares to n pairs of points (xi,yi), model the observations y by the above equation .\n        ", "tag": "math,regression", "prompt": "I am interested in Regression Least Squares Fitting?"}
{"completion": " The equation for Elliptic Cylinder is <equation>\\\\frac{{\\\\left( {x - x_0 } \\\\right)^2 }}{{a^2 }} + \\\\frac{{\\\\left( {y - y_0 } \\\\right)^2 }}{{b^2 }} = 1\\ </equation>. The meaning of symbol <equation>(x_0,y_0)</equation> in Elliptic Cylinder is: Center Coordinates of Elliptic Cylinder.", "tag": "Math,Geometry", "prompt": "What is the meaning of the symbol <equation>(x_0,y_0)</equation> in Elliptic Cylinder?"}
{"completion": " The Latex Code for equation Cylindrical to Spherical Coordinatesis <equation>\\\\begin{array}{*{20}c} {r = R\\\\sin \\\\theta } & {R = \\\\sqrt {r^2 + z^2 } } & {} \\\\\\\\ {z = R\\\\sin \\\\theta } & {\\\\phi = \\\\theta } & {} \\\\\\\\ {\\\\theta = \\\\phi } & {\\\\theta = \\\\tan ^{ - 1} \\\\left( {\\\\frac{r}{z}} \\\\right)} & {} \\\\\\\\ \\\\end{array}</equation>", "tag": "Math,Geometry", "prompt": "What is the latex code of equation Cylindrical to Spherical Coordinates?"}
{"completion": " The equation for Power Series with Real Variables is <equation>e^{x}=1+x+\\\\frac{x^{2}}{2!}+...+\\\\frac{x^{n}}{n!}+... \\\\\\\\\\ \\\\ln(1+x) = x - \\\\frac{x^{2}}{2} + \\\\frac{x^{3}}{3} + ... + (-1)^{n+1}\\\\frac{x^{n}}{n!} +... \\\\\\\\\\ \\\\cos(x) = \\\\frac{e^{ix}+e^{-ix}}{2}=1-\\\\frac{x^{2}}{2!}+\\\\frac{x^{4}}{4!}-\\\\frac{x^{6}}{6!}+...\\\\\\\\\\ \\\\sin(x) = \\\\frac{e^{ix}-e^{-ix}}{2i}=x-\\\\frac{x^{3}}{3!}+\\\\frac{x^{5}}{5!}+...</equation>. The meaning of symbol <equation>\\cos(x) = \\frac{e^{ix}+e^{-ix}}{2}=1-\\frac{x^{2}}{2!}+\\frac{x^{4}}{4!}-\\frac{x^{6}}{6!}+...</equation> in Power Series with Real Variables is: .", "tag": "math,power,series", "prompt": "What is the meaning of the symbol <equation>\\cos(x) = \\frac{e^{ix}+e^{-ix}}{2}=1-\\frac{x^{2}}{2!}+\\frac{x^{4}}{4!}-\\frac{x^{6}}{6!}+...</equation> in Power Series with Real Variables?"}
{"completion": " The equation for Area of a Trapezoid is <equation>A = \\\\frac{1}{2}(x_1 + x_2 )h</equation>. The meaning of symbol <equation>x_1</equation> in Area of a Trapezoid is: Length of top of trapezoid.", "tag": "Math,Geometry", "prompt": "What is the meaning of the symbol <equation>x_1</equation> in Area of a Trapezoid?"}
{"completion": " The equation for Legendre Equation is <equation>(1-x^{2})\\\\frac{\\\\mathrm{d}^{2} y}{\\\\mathrm{d} x^{2}}-2x\\\\frac{\\\\mathrm{d} y}{\\\\mathrm{d} x}+l(l+1)y=0 \\\\\\\\\\ P_{l}(x)=\\\\frac{1}{2^{l}l!}(\\\\frac{\\\\mathrm{d}}{\\\\mathrm{d} x})^{l}(x^2-1)^{l}\\\\\\\\\\ P_{l}(x)=\\\\frac{1}{l}[(2l-1)xP_{l-1}(x)-(l-1)P_{l-2}(x)]</equation>. The meaning of symbol <equation>P_{l}(x)=\\frac{1}{2^{l}l!}(\\frac{\\mathrm{d}}{\\mathrm{d} x})^{l}(x^2-1)^{l}</equation> in Legendre Equation is: Solutions of Legendre equations are Legendre polynomials .", "tag": "math,legendre equation", "prompt": "Help me find the meaning of symbol <equation>P_{l}(x)=\\frac{1}{2^{l}l!}(\\frac{\\mathrm{d}}{\\mathrm{d} x})^{l}(x^2-1)^{l}</equation> in Legendre Equation?"}
{"completion": " The equation for Elliptic Paraboloid is <equation>\\\\frac{{\\\\left( {x - x_0 } \\\\right)^2 }}{{a^2 }} + \\\\frac{{\\\\left( {y - y_0 } \\\\right)^2 }}{{b^2 }} = \\\\frac{{\\\\left( {z - z_0 } \\\\right)}}{c}\\ </equation>. The meaning of symbol <equation>a</equation> in Elliptic Paraboloid is: Length of Axis x.", "tag": "Math,Geometry", "prompt": "What is the meaning of the symbol <equation>a</equation> in Elliptic Paraboloid?"}
{"completion": " The Latex Code for equation Area of a Equilateral Triangleis <equation>A = \\\\frac{{h^2 \\\\sqrt 3 }}{3}</equation>", "tag": "Math,Geometry", "prompt": "Please give me some hints on equation Area of a Equilateral Triangle?"}
{"completion": " The Latex Code for equation Area of a Squareis <equation>A = x^2</equation>", "tag": "Math,Geometry", "prompt": "Can you write equation Area of a Square?"}
{"completion": " The equation for Elliptic Paraboloid is <equation>\\\\frac{{\\\\left( {x - x_0 } \\\\right)^2 }}{{a^2 }} + \\\\frac{{\\\\left( {y - y_0 } \\\\right)^2 }}{{b^2 }} = \\\\frac{{\\\\left( {z - z_0 } \\\\right)}}{c}\\ </equation>. The meaning of symbol <equation>c</equation> in Elliptic Paraboloid is: Length of Axis z.", "tag": "Math,Geometry", "prompt": "Can you explain the symbol <equation>c</equation> in Elliptic Paraboloid?"}
{"completion": " The Latex Code for equation Laplace Transformis <equation>\\\\bar{y}(s)=\\\\mathcal{L}(y(t))=\\\\int^{\\\\infty}_{0} e^{-st} y(t) \\\\mathrm{d}t</equation>", "tag": "math,laplace transform", "prompt": "What is the latex code of equation Laplace Transform?"}
{"completion": " The equation for Hyperbola is <equation>\\\\frac{{\\\\left( {x - x_0 } \\\\right)^2 }}{{a^2 }} - \\\\frac{{\\\\left( {y - y_0 } \\\\right)^2 }}{{b^2 }} = 1\\ </equation>. The meaning of symbol <equation>(x_0,y_0)</equation> in Hyperbola is: Center Point.", "tag": "Math,Geometry", "prompt": "Help me find the meaning of symbol <equation>(x_0,y_0)</equation> in Hyperbola?"}
{"completion": " The Latex Code for equation Cartesian to Polar Coordinatesis <equation>\\\\begin{array}{*{20}c} {x = r\\\\cos \\\\theta } & {r = \\\\sqrt {x^2 + y^2 } } \\\\\\\\ {y = r\\\\sin \\\\theta } & {\\\\theta = \\\\tan ^{ - 1} \\\\left( {\\\\frac{y}{x}} \\\\right)} \\\\\\\\ \\\\end{array}</equation>", "tag": "Math,Geometry", "prompt": "Show me the latex code of equation Cartesian to Polar Coordinates?"}
{"completion": " The equation for Plane is <equation>Ax + By + Cz + D = 0</equation>. The meaning of symbol <equation>A</equation> in Plane is: Coefficient of x.", "tag": "Math,Geometry", "prompt": "What is the meaning of <equation>A</equation> in Plane?"}
{"completion": " The Latex Code for equation Area of a Rectangleis <equation>A = lw</equation>", "tag": "Math,Geometry", "prompt": "What is the equation of Area of a Rectangle?"}
{"completion": " The equation for Laplace Equation is: <equation>u_{xx} = 0</equation>. Here is the brief introduction of Laplace Equation:\n\n: Second Order Partial Derivative of function u(x,t) over x.\n\n", "tag": "math,laplace equation", "prompt": "Can you explain Laplace Equation?"}
{"completion": " The equation for Power Series with Real Variables is <equation>e^{x}=1+x+\\\\frac{x^{2}}{2!}+...+\\\\frac{x^{n}}{n!}+... \\\\\\\\\\ \\\\ln(1+x) = x - \\\\frac{x^{2}}{2} + \\\\frac{x^{3}}{3} + ... + (-1)^{n+1}\\\\frac{x^{n}}{n!} +... \\\\\\\\\\ \\\\cos(x) = \\\\frac{e^{ix}+e^{-ix}}{2}=1-\\\\frac{x^{2}}{2!}+\\\\frac{x^{4}}{4!}-\\\\frac{x^{6}}{6!}+...\\\\\\\\\\ \\\\sin(x) = \\\\frac{e^{ix}-e^{-ix}}{2i}=x-\\\\frac{x^{3}}{3!}+\\\\frac{x^{5}}{5!}+...</equation>. The meaning of symbol <equation>e^{x}</equation> in Power Series with Real Variables is: Taylor Series Expansion of .", "tag": "math,power,series", "prompt": "Help me find the meaning of symbol <equation>e^{x}</equation> in Power Series with Real Variables?"}
{"completion": " The equation for Area of a Triangle is <equation>A = \\\\frac{1}{2}(x_1 + x_2 )h</equation>. The meaning of symbol <equation>h</equation> in Area of a Triangle is: Height.", "tag": "Math,Geometry", "prompt": "What is the meaning of the symbol <equation>h</equation> in Area of a Triangle?"}
{"completion": " The equation for Ellipsoid is <equation>\\\\frac{{\\\\left( {x - x_0 } \\\\right)^2 }}{{a^2 }} + \\\\frac{{\\\\left( {y - y_0 } \\\\right)^2 }}{{b^2 }} + \\\\frac{{\\\\left( {z - z_0 } \\\\right)^2 }}{{c^2 }} = 1\\ </equation>. The meaning of symbol <equation>b</equation> in Ellipsoid is: Length of Axis b.", "tag": "Math,Geometry", "prompt": "What does the symbol mean <equation>b</equation> in Ellipsoid?"}
{"completion": " The Latex Code for equation Gamma Distributionis <equation>\\\\Gamma \\\\left( a \\\\right) = \\\\int\\\\limits_0^\\\\infty {s^{a - 1} } e^{ - s} ds \\\\\\\\\\             P(x) = \\\\frac{x^{\\\\alpha-1} e^{-frac{x}{\\\\theta}}}{\\\\Gamma(\\\\alpha) \\\\theta^{\\\\alpha}} \\\\\\\\\\             \\\\mu = \\\\alpha \\\\theta \\\\\\\\\\             \\\\sigma^{2} = \\\\alpha \\\\theta^{2} \\\\\\\\\\             \\\\gamma_{1} = \\\\frac{2}{\\\\sqrt{\\\\alpha}} \\\\\\\\\\             \\\\gamma_{2} = \\\\frac{6}{\\\\alpha}</equation>", "tag": "Math,Statistics", "prompt": "Show me the latex code of equation Gamma Distribution?"}
{"completion": " The equation for Area of a Trapezoid is <equation>A = \\\\frac{1}{2}(x_1 + x_2 )h</equation>. The meaning of symbol <equation>A</equation> in Area of a Trapezoid is: Area.", "tag": "Math,Geometry", "prompt": "What does the symbol mean <equation>A</equation> in Area of a Trapezoid?"}
{"completion": " The equation for Power Series with Real Variables is <equation>e^{x}=1+x+\\\\frac{x^{2}}{2!}+...+\\\\frac{x^{n}}{n!}+... \\\\\\\\\\ \\\\ln(1+x) = x - \\\\frac{x^{2}}{2} + \\\\frac{x^{3}}{3} + ... + (-1)^{n+1}\\\\frac{x^{n}}{n!} +... \\\\\\\\\\ \\\\cos(x) = \\\\frac{e^{ix}+e^{-ix}}{2}=1-\\\\frac{x^{2}}{2!}+\\\\frac{x^{4}}{4!}-\\\\frac{x^{6}}{6!}+...\\\\\\\\\\ \\\\sin(x) = \\\\frac{e^{ix}-e^{-ix}}{2i}=x-\\\\frac{x^{3}}{3!}+\\\\frac{x^{5}}{5!}+...</equation>. The meaning of symbol <equation>\\ln(1+x) = x - \\frac{x^{2}}{2} + \\frac{x^{3}}{3} + ... + (-1)^{n+1}\\frac{x^{n}}{n!} +....</equation> in Power Series with Real Variables is: .", "tag": "math,power,series", "prompt": "What is the meaning of the symbol <equation>\\ln(1+x) = x - \\frac{x^{2}}{2} + \\frac{x^{3}}{3} + ... + (-1)^{n+1}\\frac{x^{n}}{n!} +....</equation> in Power Series with Real Variables?"}
{"completion": " The equation for Binomial Expansion is: <equation>(1+x)^{n}=1+nx+\\\\frac{n(n-1)}{2!}x^{2}+...+ C^{r}_{n}x^{r} +...+x^{n} \\\\\\\\ C^{r}_{n}=\\\\frac{n!}{r!(n-r)!}</equation>. Here is the brief introduction of Binomial Expansion:\n\nIf n is a positive integer the series terminates, the term is .  is combination symbol, which denotes the number of different ways in which an unordered sample of r objects can be selected from a set of n objects without replacement. \n\n", "tag": "math,binomial,expansion", "prompt": "Can you explain Binomial Expansion?"}
{"completion": " The equation for Circle is <equation>\\\\left( {x - x_0 } \\\\right)^2 + \\\\left( {y - y_0 } \\\\right)^2 = R^2</equation>. The meaning of symbol <equation>R</equation> in Circle is: Radius.", "tag": "Math,Geometry", "prompt": "What is the meaning of <equation>R</equation> in Circle?"}
{"completion": " The equation for Ellipse is <equation>\\\\frac{{\\\\left( {x - x_0 } \\\\right)^2 }}{{a^2 }} + \\\\frac{{\\\\left( {y - y_0 } \\\\right)^2 }}{{b^2 }} = 1\\ </equation>. The meaning of symbol <equation>a</equation> in Ellipse is: Length of Axis a.", "tag": "Math,Geometry", "prompt": "Help me find the meaning of symbol <equation>a</equation> in Ellipse?"}
{"completion": " The equation for Area of a Trapezoid is: <equation>A = \\\\frac{1}{2}(x_1 + x_2 )h</equation>. Here is the brief introduction of Area of a Trapezoid:\n        Latex code for Area of a Trapezoid.\n        \n: Area\n: Length of top of trapezoid\n: Length of bottom of trapezoid\n: Height\n\n\nRelated Documents\n\n\nRelated Videos\n\n\n", "tag": "Math,Geometry", "prompt": "What's the meaning of Area of a Trapezoid?"}
{"completion": " The equation for Cartesian to Polar Coordinates is <equation>\\\\begin{array}{*{20}c} {x = r\\\\cos \\\\theta } & {r = \\\\sqrt {x^2 + y^2 } } \\\\\\\\ {y = r\\\\sin \\\\theta } & {\\\\theta = \\\\tan ^{ - 1} \\\\left( {\\\\frac{y}{x}} \\\\right)} \\\\\\\\ \\\\end{array}</equation>. The meaning of symbol <equation>r</equation> in Cartesian to Polar Coordinates is: Radius.", "tag": "Math,Geometry", "prompt": "What does the symbol mean <equation>r</equation> in Cartesian to Polar Coordinates?"}
{"completion": " The equation for Cylindrical to Spherical Coordinates is: <equation>\\\\begin{array}{*{20}c} {r = R\\\\sin \\\\theta } & {R = \\\\sqrt {r^2 + z^2 } } & {} \\\\\\\\ {z = R\\\\sin \\\\theta } & {\\\\phi = \\\\theta } & {} \\\\\\\\ {\\\\theta = \\\\phi } & {\\\\theta = \\\\tan ^{ - 1} \\\\left( {\\\\frac{r}{z}} \\\\right)} & {} \\\\\\\\ \\\\end{array}</equation>. Here is the brief introduction of Cylindrical to Spherical Coordinates:\n        Latex code for Cylindrical to Spherical Coordinates.\n        \n: Spherical Coordinates\n: Radius\n\n\nRelated Documents\n\n\nRelated Videos\n\n\n", "tag": "Math,Geometry", "prompt": "Please give me detailed tutorial of Cylindrical to Spherical Coordinates?"}
{"completion": " The equation for Area of a Circle is <equation>A = \\\\pi r^2</equation>. The meaning of symbol <equation>r</equation> in Area of a Circle is: Radius.", "tag": "Math,Geometry", "prompt": "What is the meaning of the symbol <equation>r</equation> in Area of a Circle?"}
{"completion": " The equation for Cartesian to Polar Coordinates is: <equation>\\\\begin{array}{*{20}c} {x = r\\\\cos \\\\theta } & {r = \\\\sqrt {x^2 + y^2 } } \\\\\\\\ {y = r\\\\sin \\\\theta } & {\\\\theta = \\\\tan ^{ - 1} \\\\left( {\\\\frac{y}{x}} \\\\right)} \\\\\\\\ \\\\end{array}</equation>. Here is the brief introduction of Cartesian to Polar Coordinates:\n        Latex code for Cartesian to Polar Coordinates.\n        \n: Polar Coordinates\n: Radius\n\n\nRelated Documents\n\n\nRelated Videos\n\n\n", "tag": "Math,Geometry", "prompt": "I am interested in Cartesian to Polar Coordinates?"}
{"completion": " The equation for Spherical Harmonics Equation is <equation>[\\\\frac{1}{\\\\sin \\\\theta} \\\\frac{\\\\partial}{\\\\partial \\\\theta}(\\\\sin \\\\theta \\\\frac{\\\\partial}{\\\\partial \\\\theta}) + \\\\frac{1}{\\\\sin^{2} \\\\theta} \\\\frac{\\\\partial^{2}}{\\\\partial \\\\phi^{2}}) ] Y^{m}_{l} + l(l+1) Y^{m}_{l}=0 \\\\\\\\ Y^{m}_{l}(\\\\theta,\\\\phi)=\\\\sqrt{\\\\frac{2l+1}{4 \\\\pi} \\\\frac{(l-|m|)!}{(l+|m|)!}}P^{m}_{l}(\\\\cos \\\\theta) e^{im \\\\phi} \\\\times \\\\begin{cases}(-1)^{m} & m\\\\ge 0 \\\\\\\\ 1 & m <0 \\\\end{cases}</equation>. The meaning of symbol <equation>Y^{m}_{l}(\\theta,\\phi)</equation> in Spherical Harmonics Equation is: Spherical Harmonics Solution :.", "tag": "math,spherical harmonics", "prompt": "Can you explain the symbol <equation>Y^{m}_{l}(\\theta,\\phi)</equation> in Spherical Harmonics Equation?"}
{"completion": " The equation for Area of a Rhombus is: <equation>A = \\\\frac{{x_1 x_2 }}{2}</equation>. Here is the brief introduction of Area of a Rhombus:\n        Latex code for Area of a Rhombus.\n        \n: Area\n\n\nRelated Documents\n\n\nRelated Videos\n\n\n", "tag": "Math,Geometry", "prompt": "What's the meaning of Area of a Rhombus?"}
{"completion": " The Latex Code for equation Cartesian to Polar Coordinatesis <equation>\\\\begin{array}{*{20}c} {x = r\\\\cos \\\\theta } & {r = \\\\sqrt {x^2 + y^2 } } \\\\\\\\ {y = r\\\\sin \\\\theta } & {\\\\theta = \\\\tan ^{ - 1} \\\\left( {\\\\frac{y}{x}} \\\\right)} \\\\\\\\ \\\\end{array}</equation>", "tag": "Math,Geometry", "prompt": "Can you write equation Cartesian to Polar Coordinates?"}
{"completion": " The equation for Complex Numbers is <equation>z=x+iy=r(\\\\cos \\\\theta + i \\\\sin \\\\theta)=re^{i(\\\\theta+2n\\\\pi)} \\\\\\\\\\ z^{*}=x-iy=r(\\\\cos \\\\theta - i \\\\sin \\\\theta)=re^{-i\\\\theta} \\\\\\\\\\ zz^{*}=|z|^{2}=x^{2}+y^{2}</equation>. The meaning of symbol <equation>z^{*}</equation> in Complex Numbers is: The complex conjugate of z..", "tag": "math,complex numbers", "prompt": "Can you explain the symbol <equation>z^{*}</equation> in Complex Numbers?"}
{"completion": " The equation for Area of a Triangle is <equation>A = \\\\frac{1}{2}(x_1 + x_2 )h</equation>. The meaning of symbol <equation>b</equation> in Area of a Triangle is: Length of Bottom of Triangle.", "tag": "Math,Geometry", "prompt": "Help me find the meaning of symbol <equation>b</equation> in Area of a Triangle?"}
{"completion": " The equation for Spiral of Archimedes is <equation>r = a \\\\theta</equation>. The meaning of symbol <equation>a</equation> in Spiral of Archimedes is: a.", "tag": "Math,Geometry", "prompt": "What does the symbol mean <equation>a</equation> in Spiral of Archimedes?"}
{"completion": " The equation for Area of a Sector is <equation>A = \\\\frac{{\\\\theta r^2 }}{2}</equation>. The meaning of symbol <equation>A</equation> in Area of a Sector is: Area.", "tag": "Math,Geometry", "prompt": "What is the meaning of the symbol <equation>A</equation> in Area of a Sector?"}
{"completion": " The equation for Wave Equation is <equation>u_{tt}=c^{2}u_{xx}</equation>. The meaning of symbol <equation>u_{tt}</equation> in Wave Equation is: Second Order Partial Derivative of function u(x,t) over time t..", "tag": "math,wave equation", "prompt": "Can you explain the symbol <equation>u_{tt}</equation> in Wave Equation?"}
{"completion": " The equation for Cylindrical to Spherical Coordinates is <equation>\\\\begin{array}{*{20}c} {r = R\\\\sin \\\\theta } & {R = \\\\sqrt {r^2 + z^2 } } & {} \\\\\\\\ {z = R\\\\sin \\\\theta } & {\\\\phi = \\\\theta } & {} \\\\\\\\ {\\\\theta = \\\\phi } & {\\\\theta = \\\\tan ^{ - 1} \\\\left( {\\\\frac{r}{z}} \\\\right)} & {} \\\\\\\\ \\\\end{array}</equation>. The meaning of symbol <equation>r</equation> in Cylindrical to Spherical Coordinates is: Radius.", "tag": "Math,Geometry", "prompt": "Can you explain the symbol <equation>r</equation> in Cylindrical to Spherical Coordinates?"}
{"completion": " The equation for Cartesian to Polar Coordinates is <equation>\\\\begin{array}{*{20}c} {x = r\\\\cos \\\\theta } & {r = \\\\sqrt {x^2 + y^2 } } \\\\\\\\ {y = r\\\\sin \\\\theta } & {\\\\theta = \\\\tan ^{ - 1} \\\\left( {\\\\frac{y}{x}} \\\\right)} \\\\\\\\ \\\\end{array}</equation>. The meaning of symbol <equation>r</equation> in Cartesian to Polar Coordinates is: Radius.", "tag": "Math,Geometry", "prompt": "Can you explain the symbol <equation>r</equation> in Cartesian to Polar Coordinates?"}
{"completion": " The equation for Circle is <equation>\\\\left( {x - x_0 } \\\\right)^2 + \\\\left( {y - y_0 } \\\\right)^2 = R^2</equation>. The meaning of symbol <equation>(x_0,y_0)</equation> in Circle is: Coordinates of Circle Center.", "tag": "Math,Geometry", "prompt": "Can you explain the symbol <equation>(x_0,y_0)</equation> in Circle?"}
{"completion": " The equation for Area of a Triangle is <equation>A = \\\\frac{1}{2}(x_1 + x_2 )h</equation>. The meaning of symbol <equation>b</equation> in Area of a Triangle is: Length of Bottom of Triangle.", "tag": "Math,Geometry", "prompt": "What is the meaning of the symbol <equation>b</equation> in Area of a Triangle?"}
{"completion": " The Latex Code for equation Eccentricity of a Hyperbolais <equation>\\\\varepsilon = \\\\frac{{\\\\sqrt {a^2 + b^2 } }}{a}</equation>", "tag": "Math,Geometry", "prompt": "How to write equation Eccentricity of a Hyperbola?"}
{"completion": " The equation for Binomial Expansion is: <equation>(1+x)^{n}=1+nx+\\\\frac{n(n-1)}{2!}x^{2}+...+ C^{r}_{n}x^{r} +...+x^{n} \\\\\\\\ C^{r}_{n}=\\\\frac{n!}{r!(n-r)!}</equation>. Here is the brief introduction of Binomial Expansion:\n\nIf n is a positive integer the series terminates, the term is .  is combination symbol, which denotes the number of different ways in which an unordered sample of r objects can be selected from a set of n objects without replacement. \n\n", "tag": "math,binomial,expansion", "prompt": "I am interested in Binomial Expansion?"}
{"completion": " The equation for Hyperbola is <equation>\\\\frac{{\\\\left( {x - x_0 } \\\\right)^2 }}{{a^2 }} - \\\\frac{{\\\\left( {y - y_0 } \\\\right)^2 }}{{b^2 }} = 1\\ </equation>. The meaning of symbol <equation>a</equation> in Hyperbola is: Length a.", "tag": "Math,Geometry", "prompt": "Can you explain the symbol <equation>a</equation> in Hyperbola?"}
{"completion": " The equation for Surface area of a cube is <equation>S = 6x^2</equation>. The meaning of symbol <equation>S</equation> in Surface area of a cube is: Surface.", "tag": "Math,Geometry", "prompt": "Help me find the meaning of symbol <equation>S</equation> in Surface area of a cube?"}
{"completion": " The equation for Circle is <equation>\\\\left( {x - x_0 } \\\\right)^2 + \\\\left( {y - y_0 } \\\\right)^2 = R^2</equation>. The meaning of symbol <equation>R</equation> in Circle is: Radius.", "tag": "Math,Geometry", "prompt": "Help me find the meaning of symbol <equation>R</equation> in Circle?"}
{"completion": " The equation for Distance Between Two Points 2D is <equation>d = \\\\sqrt {\\\\left( {x_1 - x_2 } \\\\right)^2 + \\\\left( {y_1 - y_2 } \\\\right)^2 }</equation>. The meaning of symbol <equation>(y1,y2)</equation> in Distance Between Two Points 2D is: Point 2.", "tag": "Math,Geometry", "prompt": "Can you explain the symbol <equation>(y1,y2)</equation> in Distance Between Two Points 2D?"}
{"completion": " The equation for Power Series with Real Variables is <equation>e^{x}=1+x+\\\\frac{x^{2}}{2!}+...+\\\\frac{x^{n}}{n!}+... \\\\\\\\\\ \\\\ln(1+x) = x - \\\\frac{x^{2}}{2} + \\\\frac{x^{3}}{3} + ... + (-1)^{n+1}\\\\frac{x^{n}}{n!} +... \\\\\\\\\\ \\\\cos(x) = \\\\frac{e^{ix}+e^{-ix}}{2}=1-\\\\frac{x^{2}}{2!}+\\\\frac{x^{4}}{4!}-\\\\frac{x^{6}}{6!}+...\\\\\\\\\\ \\\\sin(x) = \\\\frac{e^{ix}-e^{-ix}}{2i}=x-\\\\frac{x^{3}}{3!}+\\\\frac{x^{5}}{5!}+...</equation>. The meaning of symbol <equation>\\ln(1+x)</equation> in Power Series with Real Variables is: Taylor Series Expansion of .", "tag": "math,power,series", "prompt": "What is the meaning of <equation>\\ln(1+x)</equation> in Power Series with Real Variables?"}
{"completion": " The equation for Cartesian to Cylindrical Coordinates is <equation>\\\\begin{array}{*{20}c} {x = r\\\\cos \\\\theta } & {r = \\\\sqrt {x^2 + y^2 } } & {} \\\\\\\\ {y = r\\\\sin \\\\theta } & {\\\\theta = \\\\tan ^{ - 1} \\\\left( {\\\\frac{y}{x}} \\\\right)} & {} \\\\\\\\ {z = z} & {z = z} & {} \\\\\\\\ \\\\end{array}</equation>. The meaning of symbol <equation>(x,y,z)</equation> in Cartesian to Cylindrical Coordinates is: Cylindrical Coordinates.", "tag": "Math,Geometry", "prompt": "What is the meaning of <equation>(x,y,z)</equation> in Cartesian to Cylindrical Coordinates?"}
{"completion": " The Latex Code for equation Area of a Triangleis <equation>A = \\\\frac{1}{2}(x_1 + x_2 )h</equation>", "tag": "Math,Geometry", "prompt": "Please give me some hints on equation Area of a Triangle?"}