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Divergence Calculator

Write down the coordinates of the vector field and the tool will readily compute its divergence, showing detailed computations.

$\mathbf{\vec{F}}\left(x,y,z\right)$

$\left(x_{0}, y_{0}, z_{0}\right)$ (optional)

$x_{0}$:

$y_{0}$:

$z_{0}$:

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Divergence Calculator

Our online divergence calculator computes the divergence of a vector field, giving a scalar measure of how much the field spreads out or converges at a point. Like curl, divergence is widely used in physics, fluid dynamics, and engineering.

What is Divergence?

Definition: Divergence is a vector operator that measures the net "source" or "sink" at a point in a vector field. The result is a scalar value that indicates whether the field is expanding or compressing locally.

Types of Divergence

positive divergence

Positive Divergence: Flux spreads outward from a point, acting as a source.
Negative Divergence: Flux converges inward to a point, acting as a sink.
Zero Divergence: Incoming flux equals outgoing flux; the field is neither expanding nor contracting.

negative divergence

zero divergence

You can instantly evaluate each type using our free divergence calculator.

Divergence Formula

Mathematically, divergence is expressed as:

Divergence = ∇ · A

Where the del operator is defined as:

$$ \nabla = \left(\frac{\partial}{\partial x}, \frac{\partial}{\partial y}, \frac{\partial}{\partial z}\right) $$

For a vector field A = (P, Q, R), the divergence is:

$$ \text{Div} \, \vec{A} = \frac{\partial P}{\partial x} + \frac{\partial Q}{\partial y} + \frac{\partial R}{\partial z} $$

How to Calculate Divergence

Example 1

Find the divergence of:

$$ \vec{A} = (\cos(x^2), \sin(xy), 3) $$

Solution:

Apply the formula:

$$ \text{Div} \, \vec{A} = \frac{\partial}{\partial x} (\cos(x^2)) + \frac{\partial}{\partial y} (\sin(xy)) + \frac{\partial}{\partial z} (3) $$

$$ \frac{\partial}{\partial x} (\cos(x^2)) = -2x \sin(x^2) $$
$$ \frac{\partial}{\partial y} (\sin(xy)) = x \cos(xy) $$
$$ \frac{\partial}{\partial z} (3) = 0 $$

Summing the terms:

$$ \text{Div} \, \vec{A} = -2x \sin(x^2) + x \cos(xy) $$

Example 2

Find the divergence of:

$$ \vec{B} = (\sin(x), \cos(y), 2z) $$

Solution:

$$ \text{Div} \, \vec{B} = \frac{\partial}{\partial x} (\sin(x)) + \frac{\partial}{\partial y} (\cos(y)) + \frac{\partial}{\partial z} (2z) $$

$$ \frac{\partial}{\partial x} (\sin(x)) = \cos(x) $$
$$ \frac{\partial}{\partial y} (\cos(y)) = -\sin(y) $$
$$ \frac{\partial}{\partial z} (2z) = 2 $$

Sum the terms:

$$ \text{Div} \, \vec{B} = \cos(x) - \sin(y) + 2 $$

How the Divergence Calculator Works

Input:

Enter the vector field components (P, Q, R).
Optionally, provide coordinate values to evaluate divergence at a specific point.
Click “Calculate”.

Output:

Partial derivatives of each component.
The summation of derivatives to give the divergence.
Step-by-step breakdown for clarity.

FAQs

What is a real-world example of divergence?

Divergence appears in meteorology, for instance, when strong upper-level winds spread out, causing air to rise and create weather patterns.

What does the divergence theorem describe?

It relates the total outward flux of a vector field through a closed surface to the volume integral of divergence inside the surface.

Is divergence the same as flux?

Divergence measures the local flux density at a point, while flux measures the total flow through a surface.

What is curl?

Curl measures the rotation or circulation of a vector field around a point, indicating how the field "twists."

Conclusion

Divergence quantifies the local rate of expansion or contraction in a vector field. Widely used in physics, fluid dynamics, and engineering, our divergence calculator provides accurate, step-by-step solutions to simplify analysis of vector fields.

References

Wikipedia: Gauss's Law
Khan Academy: Understanding Divergence Formula
Lumen Learning: Vector Fields, Curl, and Divergence

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