SimReal: Mathematics - Vector Calculus - Curl |

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Today it's possible (by the help of ultrasound and mathematical tools) to construct the vector field representing the bloodstream in the heart.

Ultrasound use two important topics from physics: Doppler effect and Beat.

Imagine the two vector fields in the simulation windows illustrate the blood stream in the left ventricle of two hearts. It's important that the bloodstream don't make any curl (circulation) inside the heart, but instead is pumped through the body.

Imagine you are a doctor and look at the two vector fields in the simulation windows above. I think you will agree it's not easy to see a clear difference in the two vector fields.

But one of the vector fields has a curl, the order is curl free. If you only had one of the vector fields in front of you, it's not easy to tell if it's curl free or not.

Here is an example where mathematics can help deciding if the bloodstream is okay or not.

Try to find out by yourself in the two interactive simulations above which vector field is curl free. Use the big scrollbar down in the simulation window and the checkboxes in the control window.

As a help you can study three simulations where it's easy to decide if there is curl or not.

The application is still not computing av numerical value of the curl (it will follow later). Later you will find here a short tutorial with some ideas and a mathematical solution of this problem.

Curl is used in many other practical situations, here in the computation of a paddle wheel.

Here theory about curl.

Now try the two simulations above:

You are now informed that the vector field to the left has a negative k-component of the curl: curl F = [0,0,-1].

Just keep the two checboxes P (point) and Ft (tangential component of F) on.

Draw the big scrollbar from the start to the end and convince yuorself that the curl vector must have a negative k-component.

Do the same with the simulation to the right (here the vector field is curl free).

Compare the two simulations and convince yourself that the curl of the vector field to the right cannot have so much negative k-component.

SimReal is now developed in a new version that can be running on PC, laptop, tablet, mobile phone, TV, ... .

Here is a link to the new version of this example.