Interactive 3D Mathematics Lab
Visualize tangent lines, Riemann sums, Oxyz coordinate vectors, matrix transformations, and statistical probability distributions.
Mathematics Learning & Reference Guide for School and University Students
MathLab combines 3D geometry, dynamic calculus, 10 guided solvers, Casio calculator tips, and more than 60 active-recall cards.
Secondary Students (Grades 10–12)
Understand tangents, derivatives, integrals, Oxyz coordinate geometry, and complex numbers while building speed for national entrance exams.
- Ten quantitative solvers with step-by-step solutions and Casio keystroke tips.
- More than 60 active-recall cards across eight core topics.
- Visualize planes, lines, spheres, and spatial relationships in 3D.
University & Engineering Students
Study linear algebra, matrix transformations, differential equations, Fourier series, complex analysis, and probability distributions.
- Inverse matrices, determinants, eigenvalues, and eigenvectors.
- Approximate integrals with Riemann sums and calculate solids of revolution.
- Explore the Gaussian 68–95–99.7 rule and binomial distributions.
Quick Mathematics Reference
Look up trigonometric, derivative, integration-by-parts, Viète, Cauchy–Schwarz, and AM–GM formulas.
- Formulas rendered with standard international KaTeX/LaTeX notation.
- Keyboard shortcuts support efficient navigation and practice.
- Track progress through eight achievements inspired by notable mathematicians.
Key Mathematics Modules & Analytical Geometry
1. Differential Calculus & Tangent Lines
Derivative Definition: $f'(x_0) = \lim_{\Delta x \to 0} \frac{f(x_0 + \Delta x) - f(x_0)}{\Delta x}$
Tangent Line Equation: $y = f'(x_0)(x - x_0) + y_0$
Dynamic geometric interpretation of instantaneous rates of change, critical inflection points, and curve concavity.
2. Integral Calculus & Riemann Sums
Definite Integral: $\int_{a}^{b} f(x) dx = \lim_{n \to \infty} \sum_{i=1}^{n} f(x_i^*) \Delta x = F(b) - F(a)$
Solid of Revolution Volume: $V = \pi \int_{a}^{b} [f(x)]^2 dx$
Partitioning planar areas into left, right, trapezoidal, and midpoint Riemann rectangles with real-time error bounds.
3. 3D Spatial Vector Geometry (Oxyz)
Vector Cross Product: $\vec{a} \times \vec{b} = (a_y b_z - a_z b_y)\vec{i} + (a_z b_x - a_x b_z)\vec{j} + (a_x b_y - a_y b_x)\vec{k}$
Plane Equation: $A(x - x_0) + B(y - y_0) + C(z - z_0) = 0$
3D rotation, normal vectors, dot product projections, and intersection distances between lines and spatial planes.
4. Trigonometry & Unit Circle Dynamics
Euler's Identity: $e^{i\theta} = \cos\theta + i\sin\theta \implies e^{i\pi} + 1 = 0$
Pythagorean Identity: $\sin^2\theta + \cos^2\theta = 1, \quad \tan\theta = \frac{\sin\theta}{\cos\theta}$
Interactive unit circle $(x^2 + y^2 = 1)$ tracing sinusoidal wave harmonics, radian phase shifts, and frequency periodicity.
5. Linear Algebra & Matrix Transformations
2D Rotation Matrix: $R(\theta) = \begin{pmatrix} \cos\theta & -\sin\theta \\ \sin\theta & \cos\theta \end{pmatrix}$
Eigenvalues & Eigenvectors: $A \vec{v} = \lambda \vec{v} \implies \det(A - \lambda I) = 0$
Visualizing linear mappings, determinants as spanned parallelogram areas, shears, scaling, and basis transformations.
6. Probability & Gaussian Normal Distribution
Normal Probability Density: $f(x) = \frac{1}{\sigma \sqrt{2\pi}} e^{-\frac{(x - \mu)^2}{2\sigma^2}}$
Binomial Distribution: $P(X = k) = \binom{n}{k} p^k (1-p)^{n-k}$
Empirical $68-95-99.7$ rule, Central Limit Theorem convergence, Monte Carlo sampling, and standard normal $Z$-scores.
7. Exponential & Logarithmic Functions
Logarithm Laws: $\log_a(x \cdot y) = \log_a x + \log_a y, \quad \log_a(x^\alpha) = \alpha \log_a x$
Change of Base Formula: $\log_a b = \frac{\log_c b}{\log_c a} = \frac{\ln b}{\ln a}, \quad a^{\log_a b} = b$
Exponential growth, radioactive decay halflife, logarithmic scales (pH, Decibel, Richter), and inverse function symmetry.
8. Sequences, Progressions & Limits
Arithmetic & Geometric AP/GP: $u_n = u_1 + (n-1)d, \quad u_n = u_1 \cdot q^{n-1}$
Infinite Series & Euler's e: $S_\infty = \frac{u_1}{1 - q} \, (|q| < 1), \quad \lim_{n \to \infty} \left(1 + \frac{1}{n}\right)^n = e$
Summation formulas, Cauchy convergence criterion, limit indeterminate forms (0/0, inf/inf) and L'Hopital's rule.
9. Complex Numbers & Euler's Polar Form
Algebraic & Modulus Form: $z = a + bi, \quad |z| = \sqrt{a^2 + b^2}, \quad z \cdot \overline{z} = |z|^2$
Euler & De Moivre Formula: $z = r e^{i\theta} = r(\cos\theta + i\sin\theta), \quad z^n = r^n e^{in\theta}$
Complex plane Argand diagram, nth roots of unity forming regular polygons, and AC circuit impedance phasor calculations.
10. 2D Analytic Geometry & Conic Sections
Ellipse & Hyperbola: $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \, (c^2 = a^2 - b^2), \quad \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$
Parabola Standard Form: $y^2 = 2px \quad (\text{Focus } F(p/2, 0), \text{ Directrix } x = -p/2)$
Foci, eccentricity e = c/a, asymptotes, and tangent lines in Cartesian and polar coordinate systems.
11. Classical Spatial Geometry & Polyhedral Solids
Pyramid & Prism Volumes: $V_{\text{pyramid}} = \frac{1}{3} B \cdot h, \quad V_{\text{prism}} = B \cdot h$
Cone, Cylinder & Sphere: $V_{\text{sphere}} = \frac{4}{3}\pi R^3, \quad S_{\text{sphere}} = 4\pi R^2, \quad V - E + F = 2$
3D perpendicularity, dihedral plane angles, Simson volume ratios, and the 5 Platonic regular convex polyhedra.
12. Combinatorics & Newton's Binomial Theorem
Permutations & Combinations: $P_n = n!, \quad A_n^k = \frac{n!}{(n-k)!}, \quad C_n^k = \binom{n}{k} = \frac{n!}{k!(n-k)!}$
Binomial Expansion: $(a + b)^n = \sum_{k=0}^n \binom{n}{k} a^{n-k} b^k, \quad \sum_{k=0}^n \binom{n}{k} = 2^n$
Multiplication and addition counting principles, Pascal's triangle recursive relations, and Pigeonhole principle.
13. Ordinary Differential Equations (ODE)
First-Order Linear ODE: $y' + P(x)y = Q(x) \implies y = e^{-\int P dx}\left(\int Q e^{\int P dx} dx + C\right)$
Second-Order Harmonic: $y'' + \omega^2 y = 0 \implies y(t) = C_1 \cos(\omega t) + C_2 \sin(\omega t)$
Separation of variables, integrating factors, characteristic equation roots, damped and forced harmonic oscillations.
14. Multivariable Calculus & Gradient Vectors
Gradient Vector: $\nabla f(x, y) = \left(\frac{\partial f}{\partial x}, \frac{\partial f}{\partial y}\right) \quad (\text{Direction of steepest ascent})$
Hessian Matrix Extremum: $\Delta = f_{xx} f_{yy} - (f_{xy})^2 > 0 \implies \text{Extremum at critical point}$
Partial derivatives, directional derivatives, double and triple integrals over 2D/3D domains, and polar coordinate transformations.
15. Classical Inequalities: AM-GM & Cauchy-Schwarz
AM-GM Inequality: $\frac{a_1 + a_2 + \dots + a_n}{n} \ge \sqrt[n]{a_1 a_2 \dots a_n} \quad (\text{Equality iff } a_1 = \dots = a_n)$
Cauchy-Schwarz & Engel: $\left(\sum a_i b_i\right)^2 \le \left(\sum a_i^2\right)\left(\sum b_i^2\right), \quad \sum \frac{x_i^2}{y_i} \ge \frac{(\sum x_i)^2}{\sum y_i}$
Convex function Jensen inequality, Bernoulli inequality, boundary condition analysis, and mathematical olympiad optimization.
16. Numerical Methods & Taylor Series
Taylor Series Expansion: $f(x) = \sum_{n=0}^\infty \frac{f^{(n)}(a)}{n!} (x - a)^n = f(a) + f'(a)(x-a) + \dots$
Newton-Raphson Iteration: $x_{n+1} = x_n - \frac{f(x_n)}{f'(x_n)} \quad (\text{Root-finding convergence})$
Maclaurin series for $e^x, \sin x, \cos x, \ln(1+x)$, polynomial approximation, Simpson's 1/3 rule numerical integration, and error bounds.
17. Discrete Mathematics & Graph Theory
Handshaking Lemma: $\sum_{v \in V} \deg(v) = 2 |E| \implies \text{Number of odd-degree vertices is even}$
Mathematical Induction: $P(1) \text{ is true} \land [\forall k, P(k) \implies P(k+1)] \implies \forall n \ge 1, P(n)$
Eulerian and Hamiltonian graph paths, tree connectivity, shortest path algorithms (Dijkstra), and congruence modular arithmetic.
18. Mathematical Statistics & Regression
Sample Mean & Variance: $\overline{x} = \frac{1}{n}\sum x_i, \quad s^2 = \frac{1}{n-1}\sum (x_i - \overline{x})^2$
Linear Regression Line: $y = ax + b, \quad a = \frac{\text{Cov}(x, y)}{s_x^2} = r \cdot \frac{s_y}{s_x}$
Pearson correlation coefficient r, Student's t-distribution confidence intervals, hypothesis p-values, and unbiased estimators.
19. Vector Spaces & Diagonalization
Characteristic Equation: $\det(A - \lambda I) = 0 \implies \text{Eigenvalues } \lambda_i, \text{ Eigenvectors } \vec{v}_i$
Matrix Diagonalization: $A = P \cdot D \cdot P^{-1} \implies A^k = P \cdot D^k \cdot P^{-1}$
Linear independence, spanning sets, rank-nullity theorem, Gram-Schmidt orthogonalization process, and matrix powers.
20. Fourier Series & Harmonic Wave Analysis
Fourier Series Expansion: $f(x) = \frac{a_0}{2} + \sum_{n=1}^\infty [a_n \cos(nx) + b_n \sin(nx)]$
Fourier Coefficients: $a_n = \frac{1}{\pi} \int_{-\pi}^\pi f(x) \cos(nx) dx, \quad b_n = \frac{1}{\pi} \int_{-\pi}^\pi f(x) \sin(nx) dx$
Trigonometric orthogonal systems, periodic square/sawtooth wave decomposition, Parseval's identity, and signal frequency spectrum.
