### About

# Two dimensional vector fields

This simulation shows 2 dimensional vector fields, more specifically the *xy*
cross section of a 3 dimensional vector field constant in *z*
direction (as with a cylinder of infinite extension).

Shown are flow fields characterized by local velocity components *a_x *in
the* x* direction and *a_y* in the *y* direction. Arrows
with uniform length display the direction of the resulting flow vector.
The size of the vector is indicated qualitatively by color gradation.

When opening the simulation, a field with two vortices is demonstrated.
Two white **text fields** show the formulas of its vector components.
A blue text field shows the **divergence*** *of the field, a
brown field the *3D* coordinates of its** rotation vecto r**

*.*

In a **combobox **one can choose among many predefined fields. The
type of field and the components of its vectors are stated. The second
case is empty for your own data insertions. Alternatively you can edit
data of predefined cases in the a_x and a_y text fields (do not forget
to press the *Enter *button after changes!)

Changing to another predefined function erases old data. If you want to
preserve the result of your own changes, fabricate a *picture,*
(most simply by pressing the *Print* button of your keyboard to
transfer the window data into the temporary store; then paste them into
a suitable document. A more comfortable way is to use a *screenshot
reader*).

A **red test object** lies in the vector field that will follow the
flow vector both in direction and value once the **start** button is
activated. It jumps back to its initial position when it crosses the
limits of the field.** Step **causes one step of movement.

You can **draw the object** with the mouse, and test the field at
every position that way. This gives an impression of direction and value
within the whole field.

You can turn off the vector **arrows** with an option switch and try
to understand the field just from the movement of the test body.

The **zoom** slider changes the scale of coordinates. As the number
of arrows shown is constant, this helps to recognize details. The
default case with two vortices is a good example.

The **arrow length** slider changes the length of arrows,** init**
resets all parameters , **reset_point** resets the test object to the
default position.

**Vector Algebra**

In this simulation the *z* component of the velocity vector is
zero. The vector lies in the *xy*-plane. The rotation vector is
perpendicular to the *xy*-plane; it has just a *z* component.

This results in:

**a*** = (a _{x} , a_{y}, 0)*

div **a** = *∂a _{x}/∂x + ∂a_{y}/∂y
+ 0 = ∂a_{x}/∂x + ∂a_{y}/∂y*

**rot v** = (*0, 0, ∂a _{y}/∂x -∂a_{x}/∂y*)

When is the field **without vortex**: rotation = 0 ?

∂v_{y}/∂x -∂v_{x}/∂y = 0

This holds when *a _{x} = a(x)* and

*a*; the components are functions of their own coordinates only (e.g.

_{y }= a(y)*a*). This includes the case of both components being constants, whose derivative is zero (e.g.

_{x}= x^{2}+3x-1, a_{y }= y^{3}-y^{2}*a*).

_{x}= 1, a_{y }= -4If one component is a function of the other coordinate, then in general the field will have vortices, except when the partial derivatives just compensate, as for

**rot** (*a _{x} = y, a_{y }= x*) = (0, 0, 1 -
1) = (0, 0, 0 ) = 0

When is the field **without source**: divergence = 0**?**

∂a_{x}/∂x = - ∂a_{y}/∂y

The partial derivatives must be equal in absolute value, with opposite sign (sum = zero).

The movement of the **test object** is determined and calculated by
two simple ordinary differential equations:

*dx/dt = a _{x} *

*dy/dt = a _{y}*

**E 1:** Study the default case of the twin vortices. Calculate
rotation and divergence yourself by partial derivation of the formulas
of the components (the "other" coordinate is treated as a constant).

**E 2:** Change the scale with the **zoom** slider and
observe how the visual impression of the field changes with scale.

**E 3: ** Start the test object and follow its path concerning
direction and velocity. Try to understand this by studying the formulas.

**E 4: ** Draw the test object to other points in the field and
test its quantitative structure by the object´s movement.

**E 5: ** Choose fields that have constants as components.
Change numbers and understand how this influences the field. Start the
test object. Does it change its velocity?

**E 6:** Choose fields with components linear in coordinates. Try to
understand the context. Why are divergences not quoted? (localized
limits may occur at sources).

**E 6: **How does the test object move now?** **Which terms
in the formulas lead to acceleration?

**E 7: **Choose the two corresponding last cases of twin sources and
twin vortices and study the differences.

**E 8: **Invent your own formulas for fields, calculate
divergence and rotation, describe the characteristics.

### Translations

Code | Language | Translator | Run | |
---|---|---|---|---|

### Credits

Dieter Roess - WEH- Foundation; Fremont Teng; Loo Kang Wee

### Sample Learning Goals

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### For Teachers

## Two Dimensional Vector Fields JavaScript Simulation Applet HTML5

### Instructions

#### Function Combo Box

#### Arrow Size Slider

#### Draggable Red Ball

#### Toggling Full Screen

#### Play/Pause and Reset Buttons

Research

[text]

### Video

[text]

### Version:

### Other Resources

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### end faq

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- Details
- Written by Fremont
- Parent Category: Pure Mathematics
- Category: 3 Vectors
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