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 and the derivative born from a limit.
Partial Derivatives
Slide a vertical plane through a surface and what it cuts is one ordinary curve. The slope of its tangent is a partial derivative; turn the plane for the other.
Riemann Sums
Approximate the area under a curve with rectangles, then add more and thinner ones and watch the staircase converge onto the integral.
The Sum Rule
Add two functions and their graphs stack. Stack the heights and you stack the growths, so the slopes simply add, with no leftover term anywhere.
The Product Rule
A product is the area of a rectangle. Growing it adds two strips and one small corner, and only the corner is small enough to vanish in the limit.
The Chain Rule
Each stage stretches a small step by its own factor, so stretching twice multiplies the factors. Watch why f′ has to be read at g(x) and nowhere else.
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.
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.
Fourier Series
Stack rotating circles (epicycles) to build a square, sawtooth or triangle wave from pure sine harmonics.
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.
Scalar Fields & Gradients
Drag a tiny probe through a 3D density cloud. An arrow shows the direction of steepest increase, with the gradient magnitude alongside.
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.
Green's Theorem
Chop a wobbly region into little squares, each carrying its own spin. Every shared edge is walked twice in opposite directions, so only the rim survives.
Stokes' Theorem
Tile a surface with little loops and watch every shared edge cancel in pairs until only the boundary is left. Bend the surface 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
Place a tiny material patch in a vector field and watch it expand near sources, contract near sinks, or deform at zero divergence.
Curl
Drop a paddle wheel into a vector field and drag it around. It spins fast in the core of a differential rotation and barely at all out at the rim.