LeoMath
中文Log in

Explore

Mathematical experiments computed live in your browser. Each one is tied to a concept's principles and derivation.

Ae1, Ae2A\mathbf e_1,\ A\mathbf e_2

Linear transformation

Drag the basis vectors and watch a matrix stretch, shear, rotate and flip the plane; see the determinant and eigenvectors change with it.

Related concept: Linear maps

yn+1=yn+h Φ(tn,yn)y_{n+1}=y_n+h\,\Phi(t_n,y_n)

Numerical ODE explorer

Change initial values, parameters and step size; compare Euler and RK4 on the same equation and watch the phase portrait.

Related concept: Numerical solutions

lim⁡h→0ah−1h\lim_{h\to0}\frac{a^h-1}{h}

Why eˣ is its own derivative

Drag the base a and the step h; watch when the difference-quotient curve coincides with aˣ.

Related concept: The derivative

f(x0+h)−f(x0)h→f′(x0)\frac{f(x_0+h)-f(x_0)}{h}\to f'(x_0)

Secant becomes tangent

Shrink the distance h between two points and watch the secant slope approach the tangent slope: difference quotient → derivative.

Related concept: The derivative

∑f(ξi) Δxi→∫abf\sum f(\xi_i)\,\Delta x_i\to\int_a^b f

Riemann sums

Increase the number of subintervals and compare how left, right and midpoint sampling approach the same area.

Related concept: The integral

∑k≤nf(k)(a)k!(x−a)k\sum_{k\le n}\frac{f^{(k)}(a)}{k!}(x-a)^k

Taylor approximation

Raise the polynomial order step by step: see it hug the function near the centre, the error grow outward, and where the radius of convergence ends.

Related concept: Taylor expansion