Octopus, Maui Island  More Photos Toad Fish and Ray Shark and Eel

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 and a plane)

How to graph position and derivatives vectors for vector functions and oriented curves in 3-Space

How to find arc length and arc length parametrization of curves, step by step interactive tutorial

How to find unit tangent, inward normal, binormal vectors, osculating, rectifying, normal planes, Frenete (TNB) frame. How to find curvature, 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 total differential, linear approximation and tangent planes for multivariable functions

How to find gradients, directional derivatives, phase plane for a heat seeking bug on a heated 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 through 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 with polygons

How to apply change of variables in double integrals, graph transformations and find Jacobians

How to find a differentiable or piecewise linear, one to one transformation of a square into triangle

How to graph vector fields on the sphere in rectangular or spherical coordinates. Six failed attempts in the Hairy Ball theorem including 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. How to find flux of a vector field, Divergence theorem fails for a non simple, closed surface. How to evaluate 3-Space line integrals using Stokes' theorem

Complete Mathematica based  Vector Calculus  course with Mathematica notebooks, code,  fully interactive Mathematica Player and  Mathematica demonstrations,  PDF graphs, notes ↓

How to graph solutions of a linear 2x2 system of differential equations. Construct matrix by elements, real or complex eigenvalues and eigenvectors in the diagonalizable case. Construct a Jordan cell by elements or a double real eigenvalue. Construct center by elements, eigenvalues or a rotation angle. Provides bifurcation and stability diagrams, transformations to normal forms and angles of rotation for elliptic trajectories. Phase plane includes initial conditions and parameters.

How to find the area under a curve. Graphically illustrates the definition of definite integral as a limit of Riemann Sum. Allows user to choose a function, midpoints, left endpoints or right endpoints and the number of subdivision intervals. Determines if  the sum is an underestimate, overestimate or exact value compared to the value of the definite integral and calculates the error.

How to find work required to pump out liquid from a cylindrical or conical tank. Demonstration introduces a definite integral as a limit of the vertical subdivision process into infinitesimaly thin layers. Allows user to choose an arbitrary conical or cylindrical shape of the tank, total height of the water level and the specific water layer used in integration, calculates the integral.

Trigonometry interactive Mathematica Player demonstration  ↑

How to graph six basic trigonometric functions with phase shifts, vertical shifts, change in amplitude and period. Visually illustrates how changing coefficients affects the shape of the graph, minima and maxima and location of vertical asymptotes.  User friendly controls allow to adjust the domain and range and compare different graphs on the same screen.

How to graph iterations of a linear 2x2 mapping. Allows user to construct matrix by elements, real or complex eigenvalues and eigenvectors in the diagonalizable case. Allows to construct a Jordan cell by elements, double real eigenvalue or as a scalar product. Allows user to construct center by elements, complex eigenvalues or a rotation angle. Provides bifurcation diagrams, stability diagram, full range control, transformations to normal forms and angles of rotation for elliptic trajectories.

College Algebra interactive Mathematica Player demonstration →

How to find the inverse function for a large library of basic algebraic and trigonometric functions. Displays all algebraic 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. Works with trigonometric, polynomial, radical, logarithmic and exponential functions.