## Introduction to Convergence Divergence Calculator

The convergence or divergence calculator is an online scalar function that helps you to **find the divergence of function** in a vector field and provides a solution in the scalar function.

Our divergence test calculator evaluates the vector field whose flux magnitude is in a directionless system in a fraction of a second. Whether you're working on calculus problems, studying sequences and series, or conducting mathematical research, this calculator is a valuable resource for exploring the convergence properties of various mathematical expressions

Our Calculator evaluates the divergence of a vector field at a specified point or over a region in space. Additionally, if you would like to find the curl of vector field functions in two or maybe three dimensions, you can use our curl of a vector field calculator. Our calculator also streamlines the process of analyzing vector fields by efficiently computing their curl.

## What is a Divergence?

Divergence is a mathematical operator of vector function that determines the behavior of flux (both inward or outward director) in a scalar field in three dimensions.

It is a vector function that multiplies with a gradient vector function and gives a scalar function. It is represented as **div **or **∇. F **in vector analysis.

In vector calculus, divergence is a measure of the tendency of a vector field to spread out from or converge towards a specific point in space. Further, if you would like to find the vector function of multivariable using first order partial differential method in no time, you should use our gradient formula calculator. Our calculator also evaluate the gradient on different points on straight lines whose components are in different variables in partial derivative functions.

## Formula Behind Divergence Test Calculator:

**Divergence formula** depends on the vector function along with cartesian coordinates in three dimensions. The series divergence calculator uses the following formula to solve the vector functions,

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

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

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

Whereas,

- F: vector function of cartesian coordinates
- Fx: vector component along the x-axis
- Fy: vector component along the y-axis
- Fz: vector component along the z-axis
- ∇: gradient operator
- ∇.F: divergence of a vector function

Related: for further analysis of functions to identify extremal points such as maxima and minima, you can utilize our extreme point calculator. Our calculator assists in determining critical points and classifying them as local extrema, aiding in optimization and critical point analysis.

## Working Process of Divergence Theorem Calculator:

The divergence of vector field calculator uses the easiest method with the help of partial derivatives of first-order or second-order because it has built-in programming to solve various types of vector field functions instantly. You just enter the input function and the rest of the work will **automatically** done in it and it provides you with the solution in a detailed process.

When you add the input value to the divergence convergence calculator, it starts the calculating process after checking the nature of the vector function field. Then it takes the partial derivatives of function F(x,y,z) according to its given variables.

In the end, it **adds the point value** (if points are given at the input field) of cartesian coordinates in the divergence function to see the velocity of the fluid at a point. Divergence function has three cases that define the rate of change in flux. These cases are:

**Case 1: ∇.V>0**

Your divergence function gives a positive answer then which means the flux in a vector field is going outwards

**Case 2: ∇ .V< 0**

If you have a negative divergence function value then your flux on the vector field is going inward without having any direction.

**Case 3: ∇ .V = 0**

When you get the divergence function is equal to zero it means no change has occurred. The number of inward fluxes in the convergence divergence calculator is equal to the outward flux.

Let's understand the workings of divergence theorem calculator with the help of an example which is given below,

Additionally, for further mathematical analysis, you can use our maclaurin series expansion calculator, which assists in determining polynomial approximations for functions around a fixed point. This calculator is invaluable for various mathematical applications and calculations.

## Example of Divergence Derivative Problem:

To get a better understanding of divergence, an **example** with a solution is given below.

### Example :

Determine the divergence of F at (0,2,-1), if,

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

**Solution:**

The divergence of F is:

$$ \frac{\partial }{\partial x} (e^x) + \frac{\partial }{\partial y}(yz) - \frac{\partial }{\partial z}(yz^2) \;=\; e^x + z - 2yz $$

The divergence at (0,2,-1) is:

$$ e^0 -1 + 4 \;=\; 4 $$

Further, for additional mathematical tools and computations, you can utilize our Puiseux Series Calculator, offering functionalities for analyzing Puiseux series expansions and related concepts.

## How to Use the Divergence Calculator?

The divergence of vector field calculator has a simple interface so that you can use it to calculate the vector function into the scalar function in less than a minute.

Before entering the input function to the divergence test calculator, you must follow some **simple steps** so that you do not face any type of difficulty during the calculation process. These steps are:

- Enter the function value f(x,y,z) in its respective input box.
- Add the given divergence coordinates (if you have coordinates points) value in the relevant box.
- Review your scalar field function before hitting the calculate button to start the evaluation process.
- Click the “Calculate” button to get the result of your given divergence function.
- If you want to try out our divergence function calculator first time then you can use the load example to see the calculator's working process which gives you better clarity.
- Click on the “Recalculate” button to get a new page for solving more vector function problems

Additionally, for computations involving Laurent series expansions and singularities analysis, you should use our laurent series calculator. Our calculator provides efficient solutions for analyzing functions near singular points, aiding in various mathematical analyses.

## Result from the Divergence of Vector Field Calculator:

The convergence or divergence calculator gives you the **solution of a given divergence** problem when you add the input function into it. It provides you with solutions in a complete process in less than a minute. It may contain as:

- Result option gives you a solution for the vector field function
- Possible step section provides you solution with all the evaluation steps of the divergence function in a vector field.

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

## Benefits of Using the Divergence Convergence Calculator:

The divergence theorem calculator will give you tons of **benefits** whenever you use it to calculate the divergence function problems from a vector field to a scalar field in three dimensions. These advantages are:

- It is a free-of-cost tool so you can use it to find the divergence of function in real time.
- It is a versatile tool that allows you to get the solution of various types of divergence functions
- You can use this divergence of vector field calculator to practice more examples so that you grasp this vector field function concept easily and quickly.
- Our divergence test calculator saves your time and effort from doing complex derivation calculations of the divergence problem
- It is a reliable tool that provides you with accurate solutions every time whenever you use it to calculate divergence examples.
- The convergence divergence calculator provides a solution with a complete process in a step-by-step method so that you get a better understanding of the divergence function.

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