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Divergence Calculator 

The divergence calculator is a computational tool used in vector calculus. It calculates the divergence by finding the rate of change of each component of the vector field in its corresponding direction and adding those rates together. 

See all the steps involved in calculating the divergence with their explanation below the results. 

How to use this tool?

To utilize this tool, Enter the expressions for the functions in the designated boxes according to their arrangements. Click calculate to know the answer.

Note: Be careful while inputting the functions (i.e. an i vector differentiates with respect to x). If the order of the expressions is disturbed, the result might vary from the accurate answer. 

What is divergence?

Divergence, in the field of vector calculus, can be defined as the scalar result of the dot product of the del operator (∇) and a vector field. It is denoted as ∇ · F (read as "del dot F").

A scalar field, as opposed to a vector field, assigns a scalar (just a number) to every point in space rather than a vector. It provides a measure of a vector field's tendency to originate from or converge upon a given point. 

In physical terms, divergence is often interpreted as the net flow or flux of a vector field through a small volume around a point. In essence, it quantifies how much of the field is originating from (positive divergence) or terminating at (negative divergence) a point in space.

So, while the mathematical definition involves taking derivatives and adding them, the conceptual definition is about the flow of a vector field at a particular point. This flow interpretation is especially relevant in fields like fluid dynamics and electromagnetism.

Divergence formula: 

Let's say we have a three-dimensional vector field F = Fx i + Fy j + Fz k. Here, Fx, Fy, and Fz are the component functions of the field, and i, j, and k are the unit vectors in the x, y, and z directions, respectively.

The divergence of F is then given by:

div F = ∇ · F = ∂Fx/∂x + ∂Fy/∂y + ∂Fz/∂z

The symbol ∂/∂x represents the partial derivative with respect to x. Similarly, ∂/∂y and ∂/∂z are the partial derivatives with respect to y and z, respectively.

Interpretation of divergence:

The divergence at a point can be interpreted as the amount of "flux" exiting or entering the region per unit volume around the point, in the limit as the volume shrinks to zero. 

  1. If the divergence is positive at a point, more vectors are pointing out of the point than into it. We can think of this as a source of the field. 
  2. If the divergence is negative, more vectors are pointing into the point than out of it, and we can think of this as a sink of the field. 
  3. If the divergence is zero, the vector field is neither a source nor a sink, and we say the field is solenoidal.

How to calculate divergence?

Calculating the divergence of a vector field is a straightforward process that involves the concept of partial derivatives. Here is a step-by-step guide on how to calculate it:

Let's say you have a three-dimensional vector field F = Fx i + Fy j + Fz k. This means that the field has components Fx, Fy, and Fz in the x, y, and z directions respectively.

The divergence of the vector field F, denoted as ∇ · F, is defined as follows:

div F = ∇ · F = ∂Fx/∂x + ∂Fy/∂y + ∂Fz/∂z

Now, let's break down the formula:

∂Fx/∂x: This represents the rate at which the x-component of the vector field changes with respect to x. It's calculated by taking the derivative of Fx with respect to x.

∂Fy/∂y: This represents the rate at which the y-component of the vector field changes with respect to y. It's calculated by taking the derivative of Fy with respect to y.

∂F_z/∂z: This represents the rate at which the z-component of the vector field changes with respect to z. It's calculated by taking the derivative of Fz with respect to z.

To calculate the divergence, you'll add these three derivatives together.


Let's consider a simple example. Let F = x² i + y² j + z² k. Then:

∇ · F = ∂F_x/∂x + ∂F_y/∂y + ∂F_z/∂z

= ∂(x²)/∂x + ∂(y²)/∂y + ∂(z²)/∂z

= 2x + 2y + 2z

So, the divergence of the vector field F is 2x + 2y + 2z.

This process can be generalized to any vector field. The key is understanding that divergence is calculated by adding up the rates of change of the field's components in their respective directions.

Applications of Divergence

The concept of divergence has broad applications across many scientific and engineering disciplines. For instance, 

  • In fluid dynamics, the divergence of a velocity field gives us the rate at which density is changing in the fluid. 
  • In electromagnetism, Gauss's law states that the divergence of the electric field is proportional to the electric charge density, and similarly, the divergence of the magnetic field is always zero, reflecting the fact that there are no magnetic monopoles.

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