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.
