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

Understanding the Divergence Calculator

The Divergence Calculator is a handy tool for anyone working with vector fields. It helps you calculate the divergence, which is a measure of how much a vector field spreads out from a point. This concept is vital in many fields, including Physics and engineering, where understanding flow and density is crucial.

What is Divergence?

Divergence is a mathematical term that describes the behaviour of vector fields. When we talk about divergence, we refer to how much the field flows out of or into a point. In simple terms, a positive divergence means the field is spreading out, while a negative divergence indicates it’s coming together. Zero divergence means there's no net flow.

Why Use the Divergence Calculator?

This calculator simplifies the process of determining divergence. Instead of manual calculations, you can enter your vector field and let the tool do the heavy lifting. It provides both symbolic results and numerical evaluations at specific points, which is especially useful for students and professionals alike.

Inputting Your Vector Field

To use the Divergence Calculator, you’ll need to input your vector field components. Depending on your requirements, you can choose between different dimensions (2D or 3D) and coordinate systems (Cartesian, cylindrical, or spherical). Here’s what you need to provide:

  • The components of the vector field (e.g., F₁, F₂, F₃).
  • The dimensions of your vector field.
  • The coordinate system you’re working in.

Evaluating at Specific Points

The calculator allows you to evaluate the divergence at specific points in your vector field. You can choose whether to get a symbolic result or a numerical output. This feature is particularly beneficial when you need precise calculations for specific applications.

Understanding the Output

Once you've input your vector field, the calculator will provide a clear output. This includes the divergence value and a display of the vector field itself. Additionally, it can show you the steps taken to arrive at the result, helping you learn and understand the process better.

Applications of Divergence

Divergence has real-world applications in various fields. Here are a few areas where understanding divergence is important:

  • Fluid Dynamics: It helps analyse how fluids move and behave in different environments.
  • Electromagnetics: Divergence relates to how electric fields change in space, which is important for circuit design.
  • Heat Transfer: Measures how heat flows in different materials and conditions.

Begin Using the Divergence Calculator Today

Whether you're a student, a teacher, or a professional in a related field, the Divergence Calculator can make your work easier and faster. With its user-friendly interface and powerful computational abilities, you'll be able to focus on understanding and applying your results rather than getting bogged down in calculation details.