Animations

Many concepts in mathematics are dynamic. These concepts are easier to explain if the dynamic nature, the motion, could be shown in class. This is a perfect place for the use of animations. This page contains animations that I use in class to briefly highlight a concept. The newer ones have a quick explanation incorporated. For the older ones, comments as to what the animation does are included.

I usually talk about an animation for at most five minutes as I describe the underlying idea. Some animations I revisit several times in a class. For example, the "secant lines approach a tangent line" animation can be shown early in calculus when the motivation for limits is needed and then later on be revisited when derivatives are formally defined. Other animations, specifically the vibrating strings and drum membranes can be used to show how mathematics and science truly predict reality.

All animations are produced with MathCAD. This page is designed for colleagues who do not have MathCAD as well as for myself (I don't want to re-create these animations for every class). If you have MathCAD, you are welcome to use the tools on the MathCAD page to animate functions of your choice.

To download, right click and choose "save as". All files are uncompressed .avi files. Some of them are quite large (over 5MB). It is best to first download a file and then view it. Plug-ins slow down the process and sometimes also display the animation at the wrong size, which leads to de-rezzing or illegible text.

All animations are also linked into the pdf versions of my books. So for a text that seamlessly integrates the animations and MathCAD, check out my books page.

Limits
Convergence: epsilon-delta for tan(x)/x
Divergence: epsilon-delta for sin(1/x), sequence approach for sin(1/x), sequence approach for jump function

Derivatives
Tangent approached by secants
zooms: differentiable function, absolute value function
Tangent sliding along the function
Parameter dependent functions: ax/(a+x²), x⁴+ax², normal distribution (σ=1, μ varies), normal distribution (μ=0, σ varies), exponential distribution

Riemann sums
sine on [0,π/2]: left endpoints, right endpoints, midpoints
x² on [0,3]: right endpoints

Multivariable Calculus
Tangent of a 3d trajectory
velocity and acceleration on an ellipse (not planetary motion)
osculating circle for an ellipse
mixed partial derivatives: side view, front view
multivariable Riemann sum
Fubini's Theorem: x-axis refined faster than y-axis, y-axis refined faster than x-axis
explanation of the surface integral for vector fields

Numerics
Newton's method stuck at a point
numerical integration: trapezoidal rule for sine on [0,π/2], left endpoints, right endpoints, midpoints
Taylor polynomials of the sine function
Finite radius of convergence

Statistics
Families of densities: normal distribution (σ=1, μ varies), normal distribution (μ=0, σ varies), exponential distribution

T-distribution approaches the standard normal distribution

Older animations that have not yet been updated.

The trajectory of a vector valued function When explaining the trajectory of a vector valued function, an animation says more than a thousand words. My best analogy is the vapor trail behind a jet engine. This animation shows how a trajectory is traced (without airplanes). The animation was generated with traject.mcd which can be found on the MathCAD page.

The tangent line of a vector valued function This animation shows the tangent line to a spiral as it moves along the length of the curve. The animation was generated with the file vvtlslide.mcd

osculating circle 1
osculating circle 2
The geometric interpretation of the curvature is the osculating circle, which is the circle whose radius is the reciprocal of the curvature and which "kisses" (=osculates?) the function at the point. These animations show the osculating circle for the ellipse (3cos(t),2sin(t)) and for a two dimensional spiral. The animations have been generated with 2dpar.mcd which can be found on the MathCAD page.

Heat distribution 1 This animation shows how a straight rod for which the ends are kept at constant temperature cools off. Initially the rod is hottest in the middle and heat drops linearly towards the ends. This animation can be generated with little work using the program written in the partial differential equations project of my book. Since I keep assigning this project, no code is posted.

Heat distribution 2 This animation shows how a straight rod for which the ends are kept at constant temperature cools off. Initially the heat distribution is a "little triangle in the middle". This animation can be generated with little work using the program written in the partial differential equations project of my book. Since I keep assigning this project, no code is posted.

Plucked string This animation shows how a frictionless violin string that is plucked in the middle would oscillate. This animation can be generated with little work using the program written in the partial differential equations project of my book. Since I keep assigning this project, no code is posted.
The animation is derived from theoretical predictions using partial differential equations. The theory is verified very nicely through Loren Winters' high speed images of plucked strings.

Traveling wave This animation shows how a certain initial condition induces two traveling triangles on a frictionless violin string. This animation can be generated with little work using the program written in the partial differential equations project of my book. Since I keep assigning this project, no code is posted.

Vibrating drum membrane 1 This animation shows a vibrating drum membrane in a mode corresponding to the zeroeth Bessel function. Since this is a project in my book and since my code is nothing worth looking at, no code is posted. If anyone knows of a place where actual high speed images of vibrating drum membranes are posted, please let me know.

Vibrating drum membrane 2 This animation shows a vibrating drum membrane in a mode corresponding to the first Bessel function. Since this is a project in my book and since my code is nothing worth looking at, no code is posted. If anyone knows of a place where actual high speed images of vibrating drum membranes are posted, please let me know.