Calculus simulators
Limits, derivatives, integrals and the geometry behind them.
Derivative as a Limit
Slide the secant line as h shrinks to zero and watch it settle into the tangent, then zoom in until the curve itself is a straight line, unless there is a corner.
Partial Derivatives
A surface has a slope for every direction you could walk. Freeze one variable and slide a plane through it for ∂f/∂x or ∂f/∂y, then turn the plane to any direction u and watch D_u f = ∇f·u peak along the gradient.
The Fundamental Theorem
Sweep a point across a curve and watch the area behind it grow into a new function, whose slope at every point is the height of the curve. Riemann sums, signed area and the +C included.
Derivative Rules
Sum, product and chain in one place. Stacked graphs make the slopes add, a growing rectangle leaves two strips and a doomed corner, and a step stretched twice multiplies the factors.
Second Derivatives & Curvature
Join the two points either side of you with a straight line. The curve misses the middle of it, and that gap is the second derivative. Halve the spacing and the gap falls to a quarter.
Integration by Parts
Two areas that tile one rectangle: the integral you were asked for and the one you are handed back. Choosing u is choosing which of the two you would rather do.
Differential Equations
An equation for the slope instead of the value. Draw it everywhere and the plane fills with segments; a solution is any curve that stays tangent to them the whole way, and a swarm of particles shows them all at once.
Taylor Series
Add polynomial terms one by one and watch the approximation hug a curve near a point, then fall apart past its radius of convergence, set by the nearest singularity in the complex plane.
Fourier Series
Multiply the signal by one frequency and add up the area. If the wave fits, the product sits above the axis; if it does not, the halves cancel to exactly zero. Every coefficient is that one measurement.
Laplace Transform
Plot |F(s)| as a landscape over the complex plane, where poles become spikes and the Fourier transform is the slice above the imaginary axis.
The Hessian
A surface bends by a different amount in every direction, and four numbers hold all of them. Its eigenvectors are the sharpest and gentlest directions, and their signs decide bowl, dome or saddle.
Lagrange Multipliers
Optimize along a constraint and watch the objective contour kiss the constraint curve, exactly where the two gradients line up.
Line Integrals
Ride a bead along a curve through a field in 3D, adding up only the part of the field that points your way. Then bend the path and see whether the total cared.
Stokes' Theorem
Stretch any surface across a loop and chop it into patches. Neighbours walk every shared edge both ways, so the sum collapses onto the one curve around the edge. Lay it flat and this is Green's theorem; bend it and the answer refuses to move.
Gauss's Divergence Theorem
Fill a lumpy region with little boxes and count what leaks out of each. Every inner wall is shared by two boxes and cancels, leaving only the outer skin.
Divergence & Curl
Two numbers at every point of a flow. Drop in a material patch and its area grows or shrinks with the divergence; drop in a paddle wheel and it spins with the curl.