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Math Calculators ▶ Curl Calculator

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**Table of Content**

An online curl calculator is specially designed to calculate the curl of any vector field rotating about a point in an area.

If you are interested in understanding the concept of curl, continue to read.

In calculus, a curl of any vector field A is defined as:

**The measure of rotation (****angular velocity****) at a given point in the vector field.**

The curl of a vector field is a vector quantity.

**Magnitude of curl:**

The magnitude of a curl represents the maximum net rotations of the vector field A as the area tends to zero.

**Direction of the curl:**

The direction of a curl is given by the Right-Hand Rule which states that:

**“Curl the fingers of your right hand in the direction of rotation, and stick out your thumb. The vector representing this three-dimensional rotation is, by definition, oriented in the direction of your thumb.”**

With the help of a free curl calculator, you can work for the curl of any vector field under study.

Suppose we have the following function:

**F = P i + Q j + R k**

The curl for the above vector is defined by:

**Curl = âˆ‡ * F**

First we need to define the del operator **âˆ‡** as follows:

$$ \ âˆ‡ = \frac{\partial}{\partial x} * {\vec{i}} + \frac{\partial}{\partial y} * {\vec{y}}+ \frac{\partial}{\partial z} * {\vec{k}} $$

So we have the curl of a vector field as follows:

\(\operatorname{curl} F= \left|\begin{array}{ccc}\mathbf{\vec{i}} & \mathbf{\vec{j}} & \mathbf{\vec{k}}\\ \frac{\partial}{\partial x} & \frac{\partial}{\partial y} & \frac{\partial}{\partial z}\\P & Q & R\end{array}\right|\)

Thus, \( \operatorname{curl}F= \left(\frac{\partial}{\partial y} \left(R\right) – \frac{\partial}{\partial z} \left(Q\right), \frac{\partial}{\partial z} \left(P\right) – \frac{\partial}{\partial x} \left(R\right), \frac{\partial}{\partial x} \left(Q\right) – \frac{\partial}{\partial y} \left(P\right) \right)\)

The same procedure is performed by our free online curl calculator to evaluate the results.

**Rotational Vector:**

A rotational vector is the one whose curl can never be zero.

**For example:**

Spinning motion of an object, angular velocity, angular momentum etc.

**Irrotational Vector:**

A vector with a zero curl value is termed an irrotational vector.

**Curl = âˆ‡ * F = 0**

**For example:**

A fluid in a state of rest, a swing at rest etc.

To understand the concept of curl in more depth, let us consider the following example:

How to find curl of the function given below?

**F = (cos(x), sin(xyz), 6x + 4)**

**Solution:**

The given function is:

**F = (cos(x), sin(xyz), 6x + 4)**

As we know that, the curl is given by the following formula:

**Curl = âˆ‡ * F = 0**

So, we have;

By definition, \( \operatorname{curl}{\left(\cos{\left(x \right)}, \sin{\left(xyz\right)}, 6x+4\right)} = \nabla\times\left(\cos{\left(x \right)}, \sin{\left(xyz\right)}, 6x+4\right)\),

Or equivalently

\(\operatorname{curl}{\left(\cos{\left(x \right)}, \sin{\left(xyz\right)}, 6x+4\right)} = \left|\begin{array}{ccc}\mathbf{\vec{i}} & \mathbf{\vec{j}} & \mathbf{\vec{k}}\\\frac{\partial}{\partial x} &\frac{\partial}{\partial y} & \ {\partial}{\partial z}\\\\cos{\left(x \right)} & \sin{\left(xyz\right)} & 6x+4\end{array}\right|\)

\(\operatorname{curl}{\left(\cos{\left(x \right)}, \sin{\left(xyz\right)}, 6x+4\right)} = \left(\frac{\partial}{\partial y} \left(6x+4\right) – \frac{\partial}{\partial z} \left(\sin{\left(xyz\right)}\right), \frac{\partial}{\partial z} \left(\cos{\left(x \right)}\right) – \frac{\partial}{\partial x} \left(6x+4\right), \frac{\partial}{\partial x}\left(\sin{\left(xyz\right)}\right) – \frac{\partial}{\partial y}\left(\cos{\left(x \right)}\right) \right)\)

After evaluating the partial derivatives, the curl of the vector is given as follows:

$$ \left(-x y \cos{\left(x \right)}, -6, \cos{\left(x \right)}\right) $$

You can also determine the curl by subjecting to free online curl of a vector calculator.

The net rotational movement of a vector field about a point can be determined easily with the help of curl of vector field calculator. We need to know what to do:

**Input:**

- Put the values of x, y and z coordinates of the vector field

Now, if you wish to determine curl for some specific values of coordinates:

- Select the desired value against each coordinate
- Click â€˜calculateâ€™

**Output:**

With help of input values given, the vector curl calculator calculates:

- Curl of the vector field
- Step by step calculations to clarify the concept.

As you know that curl represents the rotational or irrotational character of the vector field, so a 0 curl means that there is no any rotational motion in the field.

A positive curl is always taken counter clockwise while it is negative for anti-clockwise direction. You can assign your function parameters to vector field curl calculator to find the curl of the given vector.

The divergence of a vector is a scalar quantity that measures how a fluid collects or disperses at a particular point.

Curl has a broad use in vector calculus to determine the circulation of the field. Using curl of a vector field calculator is a handy approach for mathematicians that helps you in understanding how to find curl.

Curl provides you with the angular spin of a body about a point having some specific direction. Curl has a wide range of applications in the field of electromagnetism. Apart from the complex calculations, a free online curl calculator helps you to calculate the curl of a vector field instantly.

From the source of Wikipedia: Intuitive interpretation, Descriptive examples, Differential forms, Curl geometrically

From the source of lumen learning: Vector Fields, Conservative Vector Fields, Path Independence, Line Integrals, Fundamental Theorem for Line Integrals, Greenâ€™s Theorem, Curl and Divergence, Parametric Surfaces and Surface Integrals, Surface Integrals of Vector Fields.

From the source of khan academy: Divergence, Interpretation of divergence, Sources and sinks, Divergence in higher dimensions