Divergence Calculator

- May 09, 2025
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Calculate the divergence of a vector field. The divergence measures the rate at which "density" exits a given region of space and is a key concept in vector calculus.

div F = ∇·F = ∂F₁/∂x + ∂F₂/∂y + ∂F₃/∂z

Vector Field Input

Display Options

About Divergence

Divergence is a scalar quantity that measures the tendency of a vector field to "flow" away from or towards a point. It is denoted by ∇·F (del dot F) or div F.

Mathematical Definition:

For a vector field F = (F₁, F₂, F₃) in 3D Cartesian coordinates, the divergence is defined as:

div F = ∇·F = ∂F₁/∂x + ∂F₂/∂y + ∂F₃/∂z
Physical Interpretation:

Divergence measures the net flow of a field out of a small volume around a point. At a given point:

  • Positive divergence: The field is flowing outward (source)
  • Negative divergence: The field is flowing inward (sink)
  • Zero divergence: The field has no net outflow or inflow (incompressible)
Divergence in Different Coordinate Systems:

Cartesian Coordinates (x, y, z):

∇·F = ∂F₁/∂x + ∂F₂/∂y + ∂F₃/∂z

Cylindrical Coordinates (r, θ, z):

∇·F = (1/r) · ∂(r·Fr)/∂r + (1/r) · ∂Fθ/∂θ + ∂Fz/∂z

Spherical Coordinates (ρ, θ, φ):

∇·F = (1/ρ²) · ∂(ρ²·Fρ)/∂ρ + (1/(ρ·sin(φ))) · ∂(sin(φ)·Fθ)/∂θ + (1/(ρ·sin(φ))) · ∂Fφ/∂φ
Applications:
  • Electromagnetics: Gauss's law relates electric field divergence to charge density
  • Fluid Dynamics: The divergence of velocity field indicates compressibility of fluids
  • Heat Transfer: The divergence of heat flux describes heat sources or sinks
  • Quantum Mechanics: Current density divergence relates to charge conservation

What is a Divergence Calculator?

A Divergence Calculator is a useful tool that helps you find the divergence of a vector field. The divergence shows how much a vector field spreads out from a point. This concept is especially important in fields like Physics and engineering where understanding the flow of fields is crucial.

Why Use the Divergence Calculator?

This calculator simplifies the process of calculating divergence. Instead of doing complex mathematical operations by hand, you can quickly input your vector field and receive the results instantly. It's handy for students and professionals who work with vector fields regularly.

Understanding Divergence

Divergence measures how much a field is expanding or contracting at any given point. In simpler terms, it tells us whether the vector field is acting like a source (where things flow out) or a sink (where things flow in). Here are some key points:

  • Positive divergence means the field is flowing outwards.
  • Negative divergence means the field is flowing inwards.
  • Zero divergence indicates no net flow.

Inputting Your Vector Field

The calculator allows you to input your vector field in different coordinate systems: Cartesian, cylindrical, or spherical. You can choose dimensions based on your needs. For Cartesian coordinates, you just need to provide the components of your vector field like F₁, F₂, and F₃.

Choosing Coordinate Systems

Understanding the coordinate systems available is key to using the Divergence Calculator effectively. It offers options for:

  • Cartesian coordinates (x, y, z)
  • Cylindrical coordinates (r, θ, z)
  • Spherical coordinates (ρ, θ, φ)

Each system has its own way of representing vector fields, so select the one that suits your problem best. The calculator makes switching between them easy.

Evaluating at Specific Points

You can also evaluate the divergence at specific points in your vector field. This option lets you choose whether you want a symbolic result or a numerical value. For numerical evaluation, simply enter the coordinates where you want the divergence calculated.

Displaying Results and Steps

Once you’ve input your data and calculated the divergence, the results will be shown clearly. The calculator can even display solution steps, which is great for learning and understanding the process. This feature helps you see how the calculations are done step by step.

Applications in Real Life

Divergence has many applications in various fields, such as:

  • Physics: Understanding electric and magnetic fields.
  • Engineering: Analyzing fluid flow and its behaviour.
  • Environmental Science: Studying pollutant spread in air or water.

By using the Divergence Calculator, you can apply these concepts to real-world problems and enhance your understanding of vector fields.