How to find the angle between vectors, use scalar, vector, triple products for areas and volumes.
How to find parametric equations for curve of intersection of surfaces (cone, cylinder, quadrics).
How to graph position and derivatives vectors for vector functions and oriented curves in 3-space.
How to find arc length and arc length parameterization of curves, step by step interactive tutorial.
Unit tangent, normal, binormal vectors, osculating, rectifying, normal planes, Frenete (TNB) frame.
How curvature affect the shape of the curve, find radius of curvature, osculating circle and evolute.
How to find velocity, speed, acceleration, vector and scalar normal and tangential components.
How to find and compare directional and general limits for functions of 2 variables, Plucker cone.
How to find differential, linear approximation and tangent planes for multivariable functions.
How to find gradients and directional derivatives, phase plane for a heat seeking bug on a plate.
How to solve optimization problems for multivariable functions and use extreme value theorem.
How to use quadratic model of bifurcation from minimum to maximum and cylindrical surface.
How to graph parametric surfaces, find unit normals and surface area, library of surfaces: torus,
cylinders, cones, spheres, astroidal sphere, helicoid, pseudosphere, Klein bottle, Mobius strip.
How to set up limits of integration in double and triple integrals, volumes of 3-space regions.
How to find the centroid of a 2-space region using double integrals, demonstration for polygons.
How to apply change of variables in double integrals, graph transformations and find Jacobians.
Describes a differentiable or piecewise linear, one to one transformation of a square into triangle.
How to graph vector fields on the sphere in rectangular and spherical coordinates, tangent fields.
Six failed attempts in the Hairy Ball theorem, non vanishing tangent vector field with one cowlick.
How to graph vector fields on the torus, vector fields, phase flows and single trajectories for torus
wrap with closed and dense trajectories. Find the flux of a vector field, closed or capped surfaces.
How Divergence theorem fails for a non-simple, closed surface. Line integrals and Stokes' theorem.
Quantum and classical Particles »
Free particle, linear potential, harmonic and aharmonic oscillators and nonlinear pendulum, minimizing wave packets, potential barrier, quantum tunnelling, reflection and transition.
Surfaces, Metric and Geodesics »
How to graph geodesics between points on a surface or the geodesic in a given direction.
How to compare local and global geodesics, simple examples for a cone and a polynomial surface.
How to compare spherical distances along parallels and great arcs, geodesics on a sphere.
How to find an isometric parameterization of a cone and graph cone geodesics.
Linear Algebra and Differential Equations »
How to graph phase curves and iterations of dynamical systems. Construct by elements or eigenvalues and eigenvectors, diagonalizable, shear, center, focus. Bifurcation, stability, normal forms.
Geometric and matrix description of shears, skew or orthonormal bases.
Matrices commuting with a given matrix, phase plane diagrams.
Lorentz, Galilean transforms, twins paradox, space time diagrams. Matrix calculator.
How to find the area under a curve, definite integral as a limit of a Riemann Sum, choose midpoints, left or right endpoints, the number of subdivisions. Interactive limits and L'Hospital's rule. Optimization, inscribed cylinder, carry the longest pipe around the corner. Work required to pump out liquid from cylindrical or conical tank, choose tank shape and water level.
How to graph ellipses, hyperbolas and parabolas. One-parametric family of conics bifurcating from an ellipse through a parabola to a hyperbola. Calculates minor and major axes, foci, asymptotes and vertices as functions of parameter. Cone cut by a plane z=1+ax+by, 3-space graphs of ellipses, hyperbolas and parabolas as conic sections. Follow the changes in the plane of parameters.
College Algebra »
How to find the inverse function for a large library of algebraic, trigonometric, logarithmic and exponential functions. Displays all solutions, graphs the function and its inverse showing the reflection. Uses the vertical line test to restrict the domain to ensure one to one property.
Cylinder cut by a plane z=ax+by, 3-space graphs of ellipses as cylindrical sections.
How to graph six basic trigonometric functions with phase and vertical shifts, change in amplitude and period. How changing coefficients affects the shape of the graph, minima and maxima, vertical asymptotes. Adjust the domain and range and compare graphs on the same screen.
Add vectors, multiply by a scalar, normalize and translate to standard position, calculate components and norm.