## Introduction to Vector Curl Calculator

Curl calculator is an online vector function calculator that is used to **find the curl of vector field** functions in two or maybe three dimensions.

Our curl of vector field calculator computes the vector function that is rotating about a specific point in one direction in a few seconds. Whether you're solving problems in physics, engineering, or mathematics, this tool empowers you to explore the rotational behavior of vector quantities with ease and precision.

So, our Calculator streamlines the process of analyzing vector fields by efficiently computing their curl. Further, if you would like to find the vector function of multivariable using first order partial differential method in no time, you can use our gradient of line calculator. Our calculator also evaluate the gradient on different points on straight lines whose components are in different variables in partial derivative functions.

## What is a Curl?

Curl is an applied mathematic operator that **measures the rotation field** at a particular point in both 2-D and 3-D systems.

There is only difference between curl and divergence is it has both magnitude and direction in a rotating field at a specific point. You can find the curl function direction or magnitude using the right-hand rule.

So the curl of a vector field is a vector quantity that measures the rotation or circulation of the field around a given point. Further, if you would like to find the divergence of function in a vector field and provides a solution in the scalar function, you can use our convergence divergence calculator. Our calculator also evaluates the vector field whose flux magnitude is in a directionless system in a fraction of a second.

## Formula Used by Curl of Vector Field Calculator

We used the determinate method for the curl field so that it remains in a vector field because it has both magnitude and direction. The **formula** used by the Curl calculator is,

$$ Curl\;F(x,y,z) \;=\; \nabla * F(x,y,z) $$

$$ \nabla * F(x,y,z) \;=\; \biggr( \frac{\partial F_3}{\partial y} - \frac{\partial F_2}{\partial z} \biggr)i - \biggr(\frac{\partial F_3}{\partial x} - \frac{\partial F_1}{\partial z} \biggr)j + \biggr( \frac{\partial F_2}{\partial x} - \frac{\partial F_1}{\partial y} \biggr)k $$

$$ \nabla * F(x,y,z) \;=\; \biggr| \begin{matrix} i & j & k \\ \frac{\partial }{\partial x} & \frac{\partial }{\partial y} & \frac{\partial }{\partial z} \\ F_1 & F_2 & F_3 \\ \end{matrix} $$

Here

- F(x,y,z) is a divergence function of cartesian form in three-dimension
- ∇F= is the curl of a vector field

Related: If you're interested in exploring the Extrema Calculator, which helps identify maxima and minima of functions, you can utilize our local extrema calculator. Our calculator also provides maxima and minima points of a given function while performing each step of calculations.

## Working Behind the Curl F Calculator

The curl of a vector calculator uses the determinate matrix method behind its working process to solve the curl function in a vector field. You can give various kinds of **input values of the curl function** in this calculator. First, you need to enter the curl function value in the input field of the calculator, and it checks the nature of the function.

After checking the curl of a vector field calculator uses the determinate method of 3 by 3 or 2 by 2 as per the given input. Then write the unit vector in the first row of the matrix, at the second row write the partial derivative and in the third row write down the given function in the matrix.

Now the curl vector calculator expands the function along with the unit vector (i,j,k) to find the determinate along cartesian coordinates. Then it remove the row and column in the matrix after multiplying i, j k with **∂R/∂y -∂Q/∂z, ∂Q/∂x -∂P/∂y,∂Q/∂x -∂P/∂y** in 2 by 2 matrix respectively.

Then vector curl calculator combines all the components in the form of i,j, and k to get the solution of the curl function. The same procedure is used for the 3 by 3 matrix while solving the curl function problems. You can get a better understanding with the below example.

Related: for computations involving Puiseux series expansions and singularities, you can use our Puiseux Series Calculator. Our calculator provides efficient solutions for analyzing functions near singular points, aiding in various mathematical analyses.

## Example of Curl Problem:

An example of the curl function to know the manual calculation is given. The curl f calculator will give you accurate solutions in seconds but knowing the manual way of calculation is also important so the example is given,

### Example:

Determine the curl of the following vector field,

$$ \vec{F} \;=\; x^3 y^2 \vec{i} + x^2 y^3 z^4 \vec{j} + x^2 z^2 \vec{k} $$

**Solution:**

Using the formula,

$$ curl \; \vec{F} \;=\; \nabla * \vec{F}(x,y,z) \;=\; \biggr| \begin{matrix} \vec{i} & \vec{j} & \vec{k} \\ \frac{\partial }{\partial x} & \frac{\partial }{\partial y} & \frac{\partial }{\partial z} \\ P & Q & R \\ \end{matrix} $$

$$ \biggr( \frac{\partial F}{\partial y} - \frac{\partial Q}{\partial z} \biggr) \vec{i} + \biggr( \frac{\partial P}{\partial z} - \frac{\partial R}{\partial x} \biggr) \vec{j} + \biggr( \frac{\partial Q}{\partial x} - \frac{\partial P}{\partial y} \biggr) \vec{k} $$

Now apply to the situation,

$$ curl \; \vec{F} \;=\; \nabla * \vec{F}(x,y,z) \;=\; \biggr| \begin{matrix} \vec{i} & \vec{j} & \vec{k} \\ \frac{\partial }{\partial x} & \frac{\partial }{\partial y} & \frac{\partial }{\partial z} \\ x^3 y^2 & x^2 y^3 z^4 & x^2 z^2 \\ \end{matrix} $$

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

$$ \biggr( 0 - 4x^2 y^3 z^3 \biggr)\vec{i} + \biggr(0 - 2xz^2 \biggr)\vec{j} + \biggr(2xy^3 z^4 - 2x^3 y \biggr)\vec{k} $$

So, the curl is

$$ \biggr( -4x^2 y^3 z^3 \biggr)\vec{i} + \biggr( -2xz^2 \biggr) \vec{j} + \biggr(2xy^3 z^4 - 2x^3 y \biggr) \vec{k} $$

Additionally, for calculations involving Maclaurin series expansions, consider using our mclaurin series calculator. Our calculator computes the Maclaurin series expansion of functions, facilitating analysis and approximation in mathematical contexts.

## How to Use Curl Calculator?

The curl of a vector calculator has a user-friendly design so that you can **use** it to calculate the Curl function value in less than a minute.

Before adding the input value to the calculator, you must abide by some simple steps so that you get a wonderful experience during the calculation process. These steps are:

- Enter the Curl function F(x,y,z) in the input box
- Add the points of cartesian coordinates in the relevant field if it is present in question otherwise, you can skip it and empty this box.
- Click the “Calculate” button to get the desired result of your given Curl function
- If you want to try out our curl of vector field calculator first then you can use the load example that gives you better clarity about its working procedure.
- Click on the “Recalculate” button to get a new page for solving more curl problems in a vector field.

Further, for computations involving Laurent series expansions and complex analytic functions, you should use our laurent expansion calculator. Our calculator offers efficient solutions for analyzing functions near singular points, aiding in various mathematical analyses.

## Final Result of Curl of a Vector Calculator

The vector curl calculator gives you the **solution of the curl problem** in a rotating field when you add the input to it. It provides you with solutions in a step-wise process in less than a minute. It may contain as

- Result option gives you a solution for the Curl function
- Possible step option provides you with all the steps of the problem of the Curl vector function along with its direction

Related: for computations involving Taylor series expansions and polynomial approximations, you can use our taylor approximation calculator. Our calculator offers efficient solutions for analyzing functions through their Taylor series representations, aiding in various mathematical analyses.

## Benefits of Using Curl of a Vector Field Calculator

The curl f calculator will give you tons of **benefits** whenever you use it to calculate the Curl problem in a vector field. Its benefits are:

- Our curl vector calculator saves your time from doing lengthy and complex calculations of determinants matrix problems for the solution of curl value problems.
- It is a free-of-cost tool as it does not demand for fee so you can use it for free to find the curl value problems.
- It is a handy tool as it can operate on different devices (laptop, desktop, mobile) through internet connectivity.
- It has a simple layout so everyone can easily use it to calculate the curl examples quickly. You do not need to become an expert to use our curl of vector field calculator.
- It is a reliable tool as it always provides you with accurate solutions of curl function problems in no time.
- The vector Curl calculator is an educational tool that is very helpful for students who want to learn the curl concept of vector analysis without going to any tutor.

Further, experience the benefits of our curl Calculator and explore a wealth of mathematical tools on our All Calculators page. Simplify your computations and elevate your mathematical prowess with ease.