Divergence Calculator

Divergence calculator:

Divergence calculator is used to calculate the divergence of the entered vector-valued function and gives a step-by-step solution to the problem.

What is divergence?

The divergence of a function refers to a method of calculation in which we change a vector-valued function into a scalar quantity/value. This process takes place by differentiating the given vector.

The divergence of a vector field or a function is a measure of how much the field or function is "spreading out" from a given point. It is denoted by the symbol ∇·F, where ∇ is the del operator and F is the vector field or function.

Mathematical representation:

Mathematically, the divergence of a vector field F = (F₁, F₂, F₃) is defined as:

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

where ∂F/∂x, ∂F/∂y, and ∂F/∂z are the partial derivatives of F with respect to x, y, and z, respectively.

Types of divergence:

Generally, the divergence is classified into three types:

• Positive divergence
• Negative divergence
• Null / zero divergence

Positive divergence

If the divergence of a vector field at a point is positive, it means that the field is spreading out from that point. This indicates that the flux or flow of the field is moving away from the point. For example, a fluid flowing out of a source point will have a positive divergence at that point.

Negative divergence

If it is negative, it means that the field is flowing inward toward that point. For example, a fluid flowing into a sink or a vortex will have a negative divergence at that point.

Null/zero divergence

If the divergence is zero, it means that the field is neither spreading out nor flowing inward. For example, a fluid flowing uniformly in a straight pipe will have zero divergences at any point along the pipe.                                                                                                                                    

How to find divergence?

The above divergence calculator can be used to find the divergence of the given function. Alternatively, the below example will let you know how to find the divergence manually.

Example

Calculate the divergence of the function (sin(xyz), y³, z²) at the point (1, 2, 5)

Solution

Step 1: Extract the data.

Function (f) = (sin(xyz), y³, z²)

Points = (1, 2, 5)

Step 2: Calculate the partial derivatives.  

∂/∂x (sin(xyz)) = yz cos(xyz)

∂/∂y (y³) = 3y²

∂/∂z (z²) = 2z

Step 3: Add up the partial derivatives.

div(sin(xyz), y³, z²) = 3y² + yz cos(xyz) + 2z

Step 4: Calculate the result at the given points (1, 2, 5)

div(sin(xyz), y³, z²)_{(1, 2, 5)} =10cos(10) + 22