## Introduction to Curl Calculator:

Curl Calculator is a wonderful tool that helps you to **find the curl** of a given vector function around a point in a few seconds. It is used to evaluate the rate at which a body rotates around a point in a vector field.

It is very useful in various fields as it can quickly solve the complex problems of vector functions that would take too much time and effort when you perform the calculation manually.

## What is Curl?

Curl is defined as a vector function that is used to **measure the tendency** at which an object is rotating around a specific point in a vector field. It has a magnitude and direction because it does not change the vector calculus.

It is denoted as “” because it is the cross product of gradient and the given function Function F(x,y,z) as ∇F. To find the direction and magnitude of a curl we use the right-hand rule in which the thumb indicates force, and the finger is a vector field that rotates around that direction.

## Formula of Curl:

The **formula of curl** is the cross product of a vector function F with cartesian coordinates in two or three-dimensional space.

$$ Curl \;=\; \nabla \times F \;=\; \left[ \begin{matrix} i & j & k \\ \frac{\partial}{\partial x} & \frac{\partial}{\partial y} & \frac{\partial}{\partial z} \\ F_1 & F_2 & F_3 \\ \end{matrix} \right] $$

$$ F \;=\; F_1 i + F_2 j + F_3 k $$

$$ \nabla \;=\; \frac{\partial}{\partial x} i + \frac{\partial}{\partial y} j + \frac{\partial}{\partial z} k $$

Whereas,

- ∇F is the cross product of function and gradient
- ∇ is the gradient
- F is the function as F(x,y,z)
- ∂/∂x partial derivative with respect to x
- ∂/∂y partial derivative with respect to y
- ∂/∂z partial derivative with respect to z

## How to Calculate the Curl?

For the **calculation of a vector** function, a rotating body curl method is used that gives a systematic way to solve even the complex vector function around a specific point. Let's see how to calculate the curl of a vector function in a stepwise method.

**Step 1**:

Determine the given function F(x,y,z) for the curl vector field.

**Step 2**:

According to the curl function in the vector field, you need to take the cross product of a given function with a gradient such as,

$$ Curl \;=\; \nabla \times F $$

**Step 3**:

To make a determinate matrix you need the given vector function and partial derivative inside the determinate,

$$ Curl \;=\; \nabla \times F \;=\; \left[ \begin{matrix} i & j & k \\ \frac{\partial}{\partial x} & \frac{\partial}{\partial y} & \frac{\partial}{\partial z} \\ F_1 & F_2 & F_3 \\ \end{matrix} \right] $$

**Step 4**:

Add the vector function value in the determinate matrix and solve it.

**Step 5**:

After simplification, you get the solution of the determinant matrix of a curl vector of a given function.

## Solved Example of Curl:

A practical **example** of a curl function gives you an idea about the calculation process of a vector function in a vector field.

**Example**:

$$ Curl\; \vec{F} \;for\; \vec{F} \;=\; (3x + 2z^2) \vec{i} + \frac{x^3 y^2}{z} \vec{j} - (z - 7x) \vec{k} $$

**Solution**:

The given function is,

$$ \vec{F} \;=\; (3x + 2z^2) \vec{i} + \frac{x^3 y^2}{z} \vec{j} - (z - 7x) \vec{k} $$

The curl of a given function F is,

$$ Curl\; \vec{F} \;=\; \nabla \times \vec{F} $$

For the cross product of a given function, make a matrix in the direction of i,j and k which is given as:

$$ curl\; \vec{F} \;=\; \nabla \times \vec{F} \;=\; \left[ \begin{matrix} \vec{i} & \vec{j} & \vec{k} \\ \frac{\partial}{\partial x} & \frac{\partial}{\partial y} & \frac{\partial}{\partial z} \\ 3x + 2z^2 & \frac{x^3 y^2}{z} & 7x - z \\ \end{matrix} \right] $$

Solve the matrix of determinate form,

$$ \frac{\partial}{\partial y}(7x - z) \vec{i} + \frac{\partial}{\partial z}(3x + 2z^2) \vec{j} + \frac{\partial}{\partial x} \biggr(\frac{x^3 y^2}{z} \biggr) \vec{k} - \frac{\partial}{\partial y}(3x + 2z^2) \vec{k} - \frac{\partial}{\partial x}(7x - z)\vec{j} - \frac{\partial}{\partial z} \biggr(\frac{x^3 y^2}{z} \biggr) \vec{i} $$

Simplify the above expression,

$$ 4z \vec{j} + \frac{3x^2 y^2}{z}\vec{k} - 7 \vec{j} + \frac{x^3 y^2}{z^2} \vec{i} $$

Rewrite the above expression as per the position vector direction i,j,k, the solution of given vector function is,

$$ =\; \frac{x^3 y^2}{z^2} \vec{i} + (4z - 7)\vec{j} + \frac{3x^2 y^2}{z} \vec{k} $$

## How to Use the Curl of Vector Field Calculator?

The curl f calculator has an easy-to-use interface; you don't need to become an expert to use it for curl function calculation. You just need to enter your problem of vector function to find a curl vector field solution in a simplified method. Follow our guidelines before using it because using it. These guidelines are:

- Enter the vector function F(x,y,z) for the curl field in the next input box.
- Review the given function value before hitting the calculate button to start the evaluation process in the curl.
- Click the “Calculate” button to get the solution of your given vector curl function problem for solution.
- If you are trying out our curl f calculator for the first time then you must use the load example to learn more about curl vector function calculation.
- Click on the “
**Recalculate**” button to get a new page for finding more example solutions of curl function problems.

## Outcome from the Curl of a Vector Calculator:

Curl Calculator gives you the **solution** to a given vector function question when you add the input into it. It provides you with solutions of the curl function. It may be included as:

**Result Option**:

When you click on the result option the curl of the vector field calculator gives you a solution of the given vector function matrix

**Possible Steps**:

It provides you with a solution where all the calculations of the curl vector function are mentioned, just click on this option.

## Advantages of Using the Curl F Calculator:

The curl of a vector calculator provides several advantages whenever you use it to calculate the curl problems for the vector function F and get solutions immediately. These advantages are:

- The curl of a vector field calculator is a
**free-of-cost tool**that enables you to use it for free to find the determinant matrix of a vector function problem solution without any spending. - It is an adaptable tool that can solve various types of vector functions to find the curl problem solution.
- Our curl vector calculator helps you to get a stronghold on the curl function concept for matrix when you use it for practice by solving more examples.
- It is a swift tool that helps you to find the solution of a given curl function in 3-D space in a couple of minutes.
- The vector curl calculator is a reliable tool as it provides accurate solutions whenever you use it to calculate the determinant matrix for curl function without any error.
- Curl of the vector field calculator provides the solution without imposing any conditions which means you can use it multiple times, whenever you use it.