Curl of Linear Velocity Calculator

Calculator Input

Enter a linear velocity field in this form:

F = <u, v, w>

First variable name

Second variable name

Third variable name

u coefficient of x

u coefficient of y

u coefficient of z

u constant

v coefficient of x

v coefficient of y

v coefficient of z

v constant

w coefficient of x

w coefficient of y

w coefficient of z

w constant

Point x

Point y

Point z

Field scale multiplier

Decimal precision

Coordinate orientation

Velocity unit

Coordinate unit

Formula Used

For a velocity field F = <u, v, w>, the curl is:

curl F = <∂w/∂y - ∂v/∂z, ∂u/∂z - ∂w/∂x, ∂v/∂x - ∂u/∂y>

For a linear field:

u = a₁x + b₁y + c₁z + d₁
v = a₂x + b₂y + c₂z + d₂
w = a₃x + b₃y + c₃z + d₃

curl F = <b₃ - c₂, c₁ - a₃, a₂ - b₁>

Angular velocity for rigid rotation is one half of curl.

How to Use This Calculator

Write the velocity components as u, v, and w.
Enter coefficients for x, y, and z in each component.
Add constants if the field has offsets.
Enter a point to evaluate velocity there.
Choose units, precision, and coordinate orientation.
Press the calculate button.
Use CSV or PDF buttons to save the report.

Example Data Table

Example	u component	v component	w component	Expected curl
Rotation about z	-3y	3x	0	<0, 0, 6>
Shear field	2z	0	5y	<5, 2, 0>
No rotation	2x	4y	6z	<0, 0, 0>

Understanding Curl of Linear Velocity

Curl measures local rotation inside a vector velocity field. A linear velocity field uses first degree terms. Each component is a straight combination of x, y, and z. Because the field is linear, every partial derivative is constant. That makes the curl constant throughout the region.

Why This Calculator Helps

Many students meet curl while studying vectors, fluids, electromagnetism, and rigid body motion. Manual work is useful, yet mistakes appear quickly when three components are involved. This calculator keeps the coefficient layout visible. It also shows the derivative pattern, the curl vector, the magnitude, and the angular velocity estimate.

Linear Velocity Field

The tool assumes the field has three components. The first component is u. The second component is v. The third component is w. Each component can include coefficients for x, y, and z, plus a constant offset. Constants affect velocity at a point. They do not affect curl, because their derivatives are zero.

Physical Meaning

A nonzero curl means the velocity field has local spinning tendency. In fluid mechanics, curl of velocity is also called vorticity. For a rigid body rotation model, angular velocity equals one half of the curl vector. The direction follows the standard right hand rule unless a left handed option is selected.

Interpreting Results

The calculator returns curl components in the i, j, and k directions. It also returns vector magnitude. Magnitude helps compare rotation strength between fields. The point inputs are used to calculate the velocity at that location. The curl of a linear field stays the same at every point, so the point is mainly for extra context.

Practical Tips

Use consistent units before entering values. If position is in meters and velocity is meters per second, curl is per second. If your coordinate scale is different, adjust coefficients first. Review the displayed formula after each run. Export the CSV for spreadsheets. Download the PDF when you need a clean report for homework, lab notes, or documentation.

Common Checks

A zero curl can still have motion. It only means local rotation is absent. A high curl does not always mean high speed. It means nearby velocity directions or rates change in a twisting pattern around the point.

FAQs

What is curl of linear velocity?

It is the vector that measures local rotation in a linear velocity field. For linear components, the curl is constant because every partial derivative is constant.

Is curl the same as vorticity?

In fluid mechanics, vorticity is the curl of the velocity field. This calculator reports curl, so it can also be read as vorticity when the field describes fluid velocity.

Why does the point not change curl?

A linear field has constant first derivatives. Since curl uses only first partial derivatives, its value stays the same at all points. The point is used for velocity evaluation.

What does zero curl mean?

Zero curl means the field has no local spinning tendency. The field may still have movement, stretching, compression, or translation, but local rotation is absent.

How is angular velocity related to curl?

For rigid body rotation, angular velocity equals one half of the curl vector. This relation is useful in mechanics and flow problems with rotational motion.

Which coordinate orientation should I choose?

Use right handed orientation for standard vector calculus. Select left handed only when your model or coordinate convention specifically uses that orientation.

Can constants change the curl?

No. Constants vanish during differentiation. They can change velocity at a point, but they do not affect the curl of a linear velocity field.

What units does curl use?

Curl uses velocity unit divided by coordinate unit. If velocity is meters per second and distance is meters, curl has units of per second.