Cal11 calculator
Search calculator...

7 Calculate The Divergence of The Following Vector Field

Reviewed by Calculator Editorial Team

Divergence is a fundamental concept in vector calculus that measures how much a vector field spreads out from a given point. It's calculated by taking the partial derivatives of the vector field components and summing them. This calculation is essential in physics, engineering, and fluid dynamics for analyzing fields like fluid flow, electric fields, and gravitational forces.

What is divergence?

Divergence is a scalar value that describes the net rate at which a vector field's source strength is distributed over an infinitesimal volume around a given point. In simpler terms, it measures how much the vector field is spreading out from a point.

Divergence is positive when the vector field is spreading out (like a source), negative when it's converging (like a sink), and zero when the flow is incompressible (like in an ideal fluid).

Divergence formula

Divergence in Cartesian coordinates

For a vector field F = (P, Q, R), the divergence is calculated as:

∇·F = ∂P/∂x + ∂Q/∂y + ∂R/∂z

Divergence in cylindrical coordinates

For a vector field F = (Fr, Fθ, Fz), the divergence is:

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

Divergence in spherical coordinates

For a vector field F = (Fr, Fθ, Fφ), the divergence is:

∇·F = (1/r²)(∂(r²Fr)/∂r) + (1/(r sinθ))(∂(Fθ sinθ)/∂θ) + (1/(r sinθ))(∂Fφ/∂φ)

How to calculate divergence

  1. Identify the vector field components (P, Q, R) in Cartesian coordinates or their equivalents in other coordinate systems.
  2. Take the partial derivative of each component with respect to its corresponding coordinate.
  3. Sum the partial derivatives to get the divergence.
  4. For cylindrical or spherical coordinates, apply the appropriate coordinate-specific formulas.

Note

When calculating divergence, ensure all components of the vector field are differentiable and that the coordinate system is properly accounted for in the partial derivatives.

Example calculation

Let's calculate the divergence of the vector field F = (2xy, xz, yz) in Cartesian coordinates.

  1. Identify the components: P = 2xy, Q = xz, R = yz
  2. Calculate partial derivatives:
    • ∂P/∂x = 2y
    • ∂Q/∂y = 0 (since Q doesn't depend on y)
    • ∂R/∂z = y
  3. Sum the partial derivatives: ∇·F = 2y + 0 + y = 3y

The divergence of this vector field is 3y, which means the field spreads out more as y increases.

Applications of divergence

Divergence has numerous applications in physics and engineering:

  • Fluid dynamics: Measures how much fluid is being created or destroyed at a point
  • Electromagnetism: Describes charge density in Maxwell's equations
  • Heat transfer: Analyzes energy conservation in thermal systems
  • Continuum mechanics: Studies deformation and stress in materials
  • Weather forecasting: Helps model atmospheric pressure changes

FAQ

What does a positive divergence mean?

A positive divergence indicates that the vector field is spreading out from the point, like a source. It suggests the creation of something (fluid, charge, etc.) at that point.

What does a negative divergence mean?

A negative divergence indicates that the vector field is converging toward the point, like a sink. It suggests the destruction or absorption of something at that point.

Can divergence be zero?

Yes, zero divergence means the vector field has no net spreading or converging at that point. This is common in incompressible flows and conservative fields.

How is divergence different from curl?

Divergence measures the spreading or converging of a vector field, while curl measures the rotation or swirling of the field. Both are fundamental in vector calculus.