Divergence and Curl Calculator: Divergence and Curl of a 3D Vector Field at a Point

Divergence and curl of a vector field at a point.

Calculated from standard textbook formulas, with every step shown. The formulas are fixed, so the working does not need checking: what matters is the numbers you type in. Use these calculators only if you are competent to choose the inputs (loads, material strength, dimensions, supports) and check your units and assumptions. For learning and checking your own working: design and sign-off are for a qualified engineer. Drafting Central does not store the numbers you type.

How this calculator works

Type P, Q and R as expressions in x, y and z.

Enter the point.

The calculator gives the divergence, the curl by components and the size of the curl.

Formulas used

  • \nabla\cdot\mathbf F = \frac{\partial P}{\partial x} + \frac{\partial Q}{\partial y} + \frac{\partial R}{\partial z}
  • \nabla\times\mathbf F = \left(\frac{\partial R}{\partial y} - \frac{\partial Q}{\partial z},\ \frac{\partial P}{\partial z} - \frac{\partial R}{\partial x},\ \frac{\partial Q}{\partial x} - \frac{\partial P}{\partial y}\right)

Worked example

With the default numbers in the calculator above:

  1. The field at the point: (2, 6, 3)
  2. Divergence of the field: 6
  3. Curl x-component: -2
  4. Curl y-component: -3
  5. Curl z-component: -1
  6. Size of the curl: 3.74166

Change the numbers in the calculator to see the full working, every step, for your own case.

A second worked example

A different case, with these numbers: P(x, y, z): the x-component of the field 2*x*y; Q(x, y, z): the y-component of the field x^2; R(x, y, z): the z-component of the field 0.

  1. The field at the point: (4, 1, 0)
  2. Divergence of the field: 4
  3. Curl x-component: 0
  4. Curl y-component: 0
  5. Curl z-component: 0
  6. Size of the curl: 0

Common questions

What does the divergence measure?

How much the field spreads out from a point. Positive means a source, negative means a sink, and zero means the flow neither spreads nor gathers.

What does the curl measure?

How much the field rotates around a point. A field with zero curl everywhere is conservative.

How do I write a field with a missing component?

Type 0 for that component. For a 2D field, leave z out of every expression and set R to 0.

Related calculators

Got a question about this? Ask an engineering question and get a plain-language answer with references.

Further reading: Any vector calculus text (divergence and curl).

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