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# Curl Calculator + Online Solver With Free Steps

The online **Curl Calculator** is a calculator that allows you to find the **curl** and **divergence** for vectors given to us.

The **Curl Calculator** is a powerful tool used by physicists and engineers to calculate the curl and divergence in fluid mechanics, electromagnetic waves, and elastic theory.

## What Is a Curl Calculator?

**A Curl Calculator is an online calculator used to calculate the curl and divergence for an equation in a vector field.**

The online **Curl Calculator** requires four inputs for it to work. The **Curl Calculator** needs the vector equations for the calculator to work. The **Curl Calculator** also needs you to select the result you want to calculate.

After providing the inputs, the **Curl Calculator** calculates and displays the results in a new separate window. The **Curl Calculator helps** you calculate the **3D cartesian points** of the **curl** and **divergence** of the equation.

## How To Use a Curl Calculator?

To use the **Curl Calculator,** you need to input the vector equation in the calculator and click the “Submit” button on the **Curl Calculator**.

The detailed step-by-step instructions on how to use a **Curl Calculator **are given below:

### Step 1

In the first step, you must enter your **i vector** equation in the first box.

### Step 2

After entering your i vector equation, we move on to input out **j vector** equation in its respective box.

### Step 3

In the third step, you need to input the **k vector** equation into the **Curl Calculator**.

### Step 4

After entering the vector equation, we need to select the type of calculation we need to make. Select curl or divergence from the **dropdown menu** on our **Curl Calculator**.

### Step 5

Once all the inputs have been entered and you have selected the type of calculation you need to perform, click on the** “Submit”** button on the **Curl Calculator**.

The **Curl Calculator** will calculate and display the **curl **and **divergence** points of the equations in a new window.

## How Does a Curl Calculator Work?

A **Curl Calculator** works by using the vector equations as inputs which are represented as $ \vec{F}(x,y,z) = x\hat{i} + y\hat{j} + z\hat{k}$ and calculating the curl and divergence on the equations. The **curl** and **divergence** help us understand the rotations of a **vector field**.

## What Is Divergence in a Vector Field?

**Divergence **is an operation on a vector field that reveals the field’s behavior either toward or away from a point. Locally, the “outflowing-ness” of the vector field at a given moment P is determined by the divergence of the **vector field** $\vec{F}$ in $\mathbb{R}^{2}$ or $\mathbb{R}^{3}$ at that location.

If $\vec{F}$ represents the **velocity** of the fluid, then the divergence of $\vec{F}$ at P indicates the fluid’s amount flowing away from P’s net rate of change over time.

Specifically, the divergence at P is zero if the amount of fluid flowing into P equals the amount flowing out. Keep in mind that the divergence of a vector field is a scalar function rather than a vector field. Using the **gradient operator** as an example below:

\[ \vec{\nabla} = \left \langle \frac{\partial }{\partial x},\frac{\partial }{\partial y},\frac{\partial }{\partial z} \right \rangle \]

Divergence can be written as a dot product as shown below:

\[ div \vec{F} = \vec{\nabla} \cdot \vec{F} \]

However, this notation can be modified such that it is more useful for us. If $ \vec{F} = \left \langle P,Q \right \rangle $ is a vector field $\mathbb{R}^{2}$ and $P_{x}$ and $Q_{y}$ both exist then we can derivative the **divergence** as shown below:

\[ div \vec{F} = P_{x} + Q_{y} \]

\[ div \vec{F} = \frac{\partial{P}}{\partial{x}} + \frac{\partial{Q}}{\partial{y}} \]

\[ div \vec{F} = \vec{\nabla} \cdot \vec{F} \]

## What Is Curl in a Vector Field?

The **curl**, which assesses the **degree of rotation** of a vector field about a point, is the second operation found in a vector field.

Assume that $\vec{F}$ represents the fluid’s velocity field. The likelihood of particles close to P to spin about the axis that points in the direction of this vector is measured by the curl of $\vec{F}$ at point P.

The size of the **curl** vector at P represents how quickly the particles rotate about this axis. Hence, the **spin** of the vector field is measured by the **curl** at a given position.

Visualize inserting a paddlewheel into a fluid at P with the paddlewheel’s axis parallel to the curl vector. The curl measures the paddlewheel’s propensity to rotate.

When $\vec{F}\left \langle P, Q, R \right \rangle$ is in a vector field $\mathbb{R}^{3}$ we can write the curl equation as shown below:

\[ \vec{F} = (R_{y}-Q_{z})\hat{i} + (P_{z}-R_{x})\hat{j} + (Q_{x}-P_{y})\hat{k} \]

\[ \vec{F} = \left ( \frac{\partial{R}}{\partial{y}} – \frac{\partial{Q}}{\partial{Z}} \right )\hat{i} + \left ( \frac{\partial{P}}{\partial{z}} – \frac{\partial{R}}{\partial{x}} \right )\hat{j} + \left ( \frac{\partial{Q}}{\partial{x}} – \frac{\partial{P}}{\partial{y}} \right )\hat{k} \]

To simply the equation above and remember it for later use it can be written as the **determinant** of $\vec{\nabla} \cdot \vec{F}$ as shown below:

\[ \begin{vmatrix}

\hat{i} &\hat{j} &\hat{k} \\

\frac{\partial}{\partial{x}}&\frac{\partial}{\partial{y}} &\frac{\partial}{\partial{z}} \\

P &Q &R

\end{vmatrix} \]

The determinant of this matrix is:

\[ \vec{F}=(R_{y} – Q_{z}) \hat{i} – (P_{z}-R_{x}) \hat{j} + (Q_{x}-P_{y}) \hat{k} = (R_{y} – Q_{z}) \hat{i} + (P_{z} – R_{x}) \hat{j} + (Q_{x}-P_{y}) \hat{k} \]

## Solved Examples

The **Curl Calculator** provides an instant solution for calculating the curl and divergence values in a vector field.

Here are some examples solved using a** Curl Calculator**:

### Solved Example 1

A college student has to find the curl and divergence of the following equation:

\[ \vec{F}(P,Q,R) = \left \langle x^{2}z , e^{y}+z , xyz \right \rangle \]

Using the **Curl Calculator,** find both the **curl** and** divergence** of the vector field equation.

### Solution

Using the **Curl Calculator**, we instantly calculated the **curl** and **divergence** of the provided equations. First, we need to input the i vector equation into the calculator, which is $x^{2}$ in our case. Next, we enter the j vector equation which is $e^{y} + z$. After entering both the inputs, we plug in our xyz vector equation into the k box,

After entering all our inputs, we select the dropdown menu and select the **“Curl”** mode.

Finally, we click the **“Submit”** button and display our results in another window. We then change the mode on our Curl Calculator to **“Divergence,”** allowing the calculator to find the divergence.

The results from the Curl Calculator are seen below:

Curl:

\[ curl\left \{ x^{2}, e^{y} + z , xyz \right \} = (x z-1, -yz, 0) \]

Divergence:

\[ div\left \{ x^{2}, e^{y} + z , xyz \right \} = x(y+2)+e^{y} \]

### Solved Example 2

While researching electromagnetism a physicist comes across the following equation:

\[ \vec{F}(P,Q,R) = \left \langle x^{2} + y^{2}, \sin{y^{2}, xz} \right \rangle \]

To complete his research, the physicist needs to find the curl and divergence of the point in the vector field. Find the **curl** and **divergence** of the equation using the **Curl Calculator**.

### Solution

To solve this problem, we can use the **Curl Calculator**. We start by plugging in the first vector equation $x^{2} + y^{2}$ into the i box. After adding the first input, we add our second input $\sin{y^{2}}$ into the j box. Finally, in the k box we enter our last vector equation, xz.

After we plugged in all our inputs, we first select the** “Curl”** mode on our **Curl Calculator** and click the **“Submit”** button. We repeated this process and select the **“Divergence”** mode the second time. The curl and divergence results are displayed in a new window.

The results produced from the **Curl Calculator **are shown below:

Curl:

\[ curl\left \{ x^{2} + y^{2}, \sin{y^{2}}, xz \right \} = (-1,-z,y(\cos{(x)}\sin^{y-1}{(x)}-2)) \]

Divergence:

\[ div\left \{ x^{2} + y^{2}, \sin{y^{2}}, xz \right \} = \sin^{y}{x}\log{(sin{(x)})+3x} \]

### Solved Example 3

Consider the following equation:

\[ \vec{F}(P,Q,R) = \left \langle y^{2+}z^{3}, \cos^{y},e^{z}+y \right \rangle \]

Using the **Curl Calculator,** find the **curl** and **divergence** points in the vector field.

### Solution

To solve the equation, we simply enter our vector equation $y^{2+}z^{3}$ in the i position.

Subsequently, we enter the next two inputs $ \cos^{y} $ and $e^{z}+y$ into the j and k positions respectively.

Once we are done entering our equations, we select the “Curl” mode on our Curl Calculator and click the “Submit” button. This step is repeated, but we change the mode to “Divergence.”

The **Curl Calculator** displays the Curl and Divergence values in a new window. The result is shown below:

Curl:

\[ curl\left \{ y^{2+}z^{3}, \cos^{y},e^{z}+y \right \} = (1,3z^{2},y(-\sin{(x)}\cos^{y-1}{(x)}-2)) \]

Divergence:

\[ div\left \{ y^{2+}z^{3}, \cos^{y},e^{z}+y \right \} = \cos^{y}{(x)}\log{(\cos{(x)})}+e^{z} \]