Bernd Schröder

Animations

 

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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. Older animations have descriptions beside them, newer ones have a descriptive paragraph on screen.

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

The trajectory of a vector valued function - My best analogy is the vapor trail behind a jet engine. This animation shows how a trajectory is traced (without airplanes).

Tangent of a 3d trajectory

velocity and acceleration on an ellipse (not planetary motion)

osculating circle for an ellipse

osculating circle for a spiral

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

Differential Equations

The Dirac Delta Function.

Partial Differential Equations

Plucked string - This animation shows how a frictionless violin string that is plucked in the middle would oscillate. The animation is derived from theoretical predictions using partial differential equations (see my differential equations book). 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. 

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. (Same initial condition as for the plucked string.)

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". (Same initial condition as for the traveling wave.)

Vibrating drum membrane 1 - This animation shows a vibrating drum membrane in a mode corresponding to the zeroeth Bessel function.

Vibrating drum membrane 1 - This animation shows a vibrating drum membrane in a mode corresponding to the first Bessel function.