Divergence Calculator – Vector Field Divergence
Calculate the divergence of a 3D vector field
How to Use
- Enter coefficients for the P component (coefficient of x, y, z)
- Enter coefficients for the Q component (coefficient of x, y, z)
- Enter coefficients for the R component (coefficient of x, y, z)
- Enter the point (x, y, z) where you want to evaluate divergence
- Click calculate to see the divergence result
What is Divergence?
Divergence is a vector operator that measures the magnitude of a vector field's source or sink at a given point. In other words, it tells you how much a vector field is 'spreading out' or 'converging' at that point.
For a 3D vector field F(x,y,z) = (P, Q, R), the divergence is defined as: div F = ∂P/∂x + ∂Q/∂y + ∂R/∂z
Physical Interpretation
Divergence has important physical interpretations:
- Positive divergence: The field is acting as a source (fluid flowing outward)
- Negative divergence: The field is acting as a sink (fluid flowing inward)
- Zero divergence: The field is incompressible (volume is conserved)
- In fluid dynamics: divergence measures the rate of expansion or compression
- In electromagnetism: divergence relates to charge density (Gauss's law)
How to Calculate Divergence
To calculate divergence of a vector field F = (P, Q, R):
- Take the partial derivative of P with respect to x: ∂P/∂x
- Take the partial derivative of Q with respect to y: ∂Q/∂y
- Take the partial derivative of R with respect to z: ∂R/∂z
- Add these three partial derivatives: div F = ∂P/∂x + ∂Q/∂y + ∂R/∂z
For example, if F(x,y,z) = (2x, 3y, 4z), then div F = 2 + 3 + 4 = 9.
Applications of Divergence
- Fluid dynamics: modeling incompressible flow
- Electromagnetism: Gauss's law and Maxwell's equations
- Heat transfer: analyzing heat flux and temperature distribution
- Computer graphics: simulating fluid and smoke effects
- Weather modeling: analyzing atmospheric pressure systems
- Engineering: stress analysis and material deformation
The Divergence Theorem
The divergence theorem (also called Gauss's theorem) relates the flux of a vector field through a closed surface to the divergence of the field in the volume enclosed by the surface:
∫∫∫ (div F) dV = ∫∫ F · n dS
This theorem is fundamental in physics and engineering, connecting local properties (divergence) to global properties (flux through a boundary).
Frequently Asked Questions
- What does positive divergence mean?
- Positive divergence means the vector field is acting as a source at that point - the field vectors are pointing outward, like fluid flowing out of a source. The magnitude tells you how strong the source is.
- What is the difference between divergence and curl?
- Divergence measures how much a vector field spreads out or converges (producing a scalar), while curl measures how much it rotates (producing a vector). Divergence uses ∇·F notation, curl uses ∇×F.
- Can divergence be negative?
- Yes, negative divergence indicates the field is acting as a sink - vectors are pointing inward toward that point. For example, a drain in a fluid flow would have negative divergence.
- What does zero divergence mean?
- Zero divergence means the field is incompressible or divergence-free at that point. The amount of field flowing into any small volume equals the amount flowing out. This is important for modeling incompressible fluids and magnetic fields.