Calculus in more than one dimension: functions become landscapes, slopes become gradients, and areas become volumes — 33 interactive demonstrations, most of them in 3D. Swing vectors around to feel the dot and cross products, and ride a curve with its moving frame. Roll wheels to trace cycloids and epicycloids, and sweep a ray round to draw polar roses and the areas they enclose. Slice surfaces to find their slopes, zoom in until they become planes, and climb to a peak along the gradient. Stack prisms under a surface, warp grids to see the Jacobian, and watch high-dimensional balls empty out. Release particles into vector fields, drop paddle wheels into a flow, and trace a shape with a planimeter, a machine that measures area by walking round the edge. Then meet the great theorems of Green, Stokes and Gauss, which all say the same thing: what happens inside is recorded on the boundary.
See also: Differential Calculus and Integral Calculus for calculus in one variable, Linear Algebra for the matrices behind the Jacobian, Electromagnetism for vector calculus at work, and Differential Geometry for the curvature of curves and surfaces.
Arrows in three dimensions: the dot product that measures alignment, the cross product that measures area, and the lines and planes they describe
A point moving through space carries a frame with it. Tangent, normal and binormal, the circle that best fits a curve, and Kepler’s planets
Curves traced by a moving point: rolling wheels, roses and spirals in polar coordinates, and the area they sweep
Landscapes z = f(x, y) and their contour maps, level surfaces in space, and limits that depend on how you approach
Slice a surface to find its slope in each direction, zoom in until it becomes a plane, and point downhill with the directional derivative
The gradient points uphill. Climb to peaks, tell maxima from saddles, and find the best point on a constraint with Lagrange multipliers
Volume under a surface by stacking prisms, slicing it either way round, and integrating in polar coordinates
How a map stretches area, the Jacobian that measures it, spherical coordinates, and why balls in high dimensions are nearly empty
Arrows at every point: flows, the work done along a path, and fields where the path does not matter
Paddle wheels that measure swirl, tiny circulations that cancel, and the planimeter, a machine that measures area by walking round it
Flux through surfaces, sources inside a balloon, and the theorem that lets any surface with the same rim give the same answer