## Introduction to Divergence Calculator:

Divergence calculator is an amazing online tool that helps you to **find the divergence** of a given vector function.It is used to simplify the process of evaluating the vector function in the complex vector field.

Divergence theorem calculator is a useful resource for students, and researchers as it is used in various fields like fluid mechanics in compression or expansion of fluid, in physics and engineering it is used to know the behavior of a flux in a vector field.

## What is Divergence?

Divergence is an operator that is used to measure the rate of change of a vector field at a point in vector calculus. It is used to convert a vector field into a scalar field where a flux of a magnetic field changes into a directionless system.

It is used the gradient operator **∇** nebula and vector field **F(F1, F2, F3), as ∇ . F** it becomes a scalar quantity. Divergence is also called the dot product of two or three functions that depend on a given dimension space.

## Formula of Divergence:

The **formula of divergence** has a vector function **F **has cartesian coordinates **(x,y,z) **in three dimensions as **F**(fi, fj, fk) and **∇ **is the gradient of function F.

$$ Div \;=\; \nabla . F \;=\; \frac{\partial F_1}{\partial x} + \frac{\partial F_2}{\partial y} + \frac{\partial F_3}{\partial z} $$

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

- ∂F1/∂x is the partial derivative of F1 with respect to x.
- ∂F2/∂y is the partial derivative of F2 with respect to y.
- ∂F3/∂z is the partial derivative of F3 with respect to z.

## How to Calculate the Divergence?

For the **calculation** of the divergence, you need a vector function in cartesian coordinates in 2D or 3D space. It gives an easy approach to solve complex vector functions without any difficulty, especially for higher order function.

**Step 1**:

Identify the given vector function F(F1,F2,F3).

**Step 2**:

Find the dot product of a given function F with the gradient such as ∇ . F

**Step 3**:

Apply the partial derivative of a given function F with respect to its cartesian coordinates (x,y,z).

**Step 4**:

After getting a solution of a partial derivative, to know whether the result is convergence or divergence calculator over a vector field. Here are the three conditions that give you more clarity about the behavior of flux.

$$ \nabla. F > 0 $$

If the divergence test calculator function gives a positive value that shows the flux in a vector field is going outwards from a specific point.

$$ \nabla . F < 0 $$

When you get a negative divergence function value then your flux is going inward directionlessly on the vector field.

$$ \nabla .F \;=\; 0 $$

When you have the divergence function which is equal to zero, no change has occurred neither the flux moves inward nor the flux moves outward in a scalar field. It means the number of inward or outward flux are same in a vector field.

## Solved Example of Divergence:

The solved **example of divergence** gives you conceptual clarity about the calculation process of the convergence divergence calculator.

**Example**:

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

**Solution**:

The given vector function is,

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

Take the partial derivative of a given vector for cartesian coordinates points with respect to x, y, and z such as:

$$ div\; \vec{F} \;=\; \nabla . \vec{F} \;=\; \frac{\partial}{\partial x}(x^2 y) + \frac{\partial}{\partial y}(3x - z^3) + \frac{\partial}{\partial z}(4y^2) $$

After taking the first derivative calculation you get a scalar quantity,

$$ \nabla . \vec{F} \;=\; \frac{\partial}{\partial x}(x^2 y) + \frac{\partial}{\partial y}(3x - z^3) + \frac{\partial}{\partial z}(4y^2) \;=\; 2xy $$

As the given function has a positive value so the given quantity is coverage but in outward direction to a specific point.

## How to Use the Divergence Test Calculator?

Divergence of vector field calculator is an easy-to-use tool, so you can easily use it to evaluate the divergence of the given vector function. Before adding the input value in the series divergence calculator for the solutions, you must follow some simple steps. These steps are:

- Enter the given vector function for the divergence in the vector field.
- Recheck your input value for the divergence function before hitting the calculate button to start the calculation process in the divergence.
- Click on the “
**Calculate**” button to get the desired result of your given divergence problem. - If you want to try out our divergence theorem calculator to check its accuracy in solution, use the load example to get more understanding.
- Click on the “Recalculate” button to get a new page for solving more divergence questions.

## Final Result of Divergence Convergence Calculator:

The divergence test calculator gives you the **solution** to a given divergence function problem when you add the input value in it. It provides you solutions. It may contain as:

**Result Option**:

You can click on the result option as it provides you with a solution of vector function questions for finding the divergence of a flux.

**Possible Step**:

When you click on the possible steps option it provides you with the solution of the divergence function problem where all calculation steps are included.

## Benefits of Divergence of Vector Field Calculator:

The divergence theorem calculator gives you multiple **benefits** whenever you use it to calculate the divergence vector problems and to get their solutions. These benefits are:

- Our convergence divergence calculator saves the time and effort that you consume in solving vector function questions for divergence and get solutions in a few seconds.
- It is a free-of-cost tool that provides you with a solution of a given divergence question of a vector field to find the convergence of flux without paying a single penny.
- It is an adaptive tool that allows you to find the behavior of flux in (x,y,z) from the given vector field system for divergence.
- You can use this divergence test calculator for practice to get familiar with this concept of divergence for a vector function easily.
- The divergence convergence calculator is a trustworthy tool that provides you with accurate solutions as per your input to calculate the divergence function problem.