Gradient Divergence Curl Calculator
Our free calculus calculator solves gradient divergence curl problems. Get worked examples, visual aids, and downloadable results.
Formula
div(F) = dF1/dx + dF2/dy + dF3/dz; curl(F) = nabla x F
For a linear vector field F = (ax+by+cz, dx+ey+fz, gx+hy+iz), the divergence is a+e+i (trace of the Jacobian), the curl is (h-f, c-g, d-b), and the Jacobian determinant measures local volume scaling.
Worked Examples
Example 1: Electromagnetic Field Analysis
Problem: Given vector field F = (2x + y, -x + 3y + z, -2y + 4z), compute gradient, divergence, and curl at point (1, 2, 3).
Solution: Field at (1,2,3): F = (2(1)+2, -(1)+3(2)+3, -2(2)+4(3)) = (4, 8, 8)\nDivergence: dF1/dx + dF2/dy + dF3/dz = 2 + 3 + 4 = 9 (source present)\nCurl: (dF3/dy - dF2/dz, dF1/dz - dF3/dx, dF2/dx - dF1/dy)\n = (-2 - 1, 0 - 0, -1 - 1) = (-3, 0, -2)\nCurl magnitude: sqrt(9 + 0 + 4) = 3.606\nJacobian determinant: 2(12-(-2)) - 1(-4-0) + 0 = 28 + 4 = 32
Result: Divergence = 9 (source) | Curl = (-3, 0, -2) with magnitude 3.606 | Not irrotational, not solenoidal
Example 2: Conservative Field Verification
Problem: Check if F = (2x, 2y, 2z) is conservative and solenoidal at point (1, 1, 1).
Solution: Coefficients: a=2, b=0, c=0, d=0, e=2, f=0, g=0, h=0, i=2\nDivergence: 2 + 2 + 2 = 6 (not solenoidal, acts as source)\nCurl: (0-0, 0-0, 0-0) = (0, 0, 0) (irrotational = conservative!)\nThis means F = grad(phi) where phi = x^2 + y^2 + z^2.\nField at (1,1,1): F = (2, 2, 2), magnitude = 2*sqrt(3) = 3.464\nJacobian determinant: 2(4-0) - 0 + 0 = 8
Result: Curl = (0,0,0): Field is conservative. Divergence = 6: Not solenoidal. Potential function: phi = x^2 + y^2 + z^2
Frequently Asked Questions
What is the gradient of a scalar field and what does it represent?
The gradient of a scalar field is a vector that points in the direction of the greatest rate of increase of the function at any given point. Its magnitude equals the rate of change in that direction. Mathematically, for a scalar function f(x,y,z), the gradient is the vector (df/dx, df/dy, df/dz). Think of a topographic map where the scalar field represents altitude: the gradient at any point tells you the steepest uphill direction and how steep the slope is. The gradient is always perpendicular to level curves (contour lines) of the function. In physics, the gradient relates forces to potential energy, as the force equals the negative gradient of potential energy.
What does divergence measure in a vector field?
Divergence measures the net outward flux per unit volume at a point in a vector field, essentially quantifying how much the field is spreading out or converging at that location. A positive divergence means the field acts as a source (vectors spread outward), while negative divergence indicates a sink (vectors converge inward). Zero divergence means the field is incompressible or solenoidal. Mathematically, div(F) = dF1/dx + dF2/dy + dF3/dz. In fluid dynamics, divergence of the velocity field tells you whether fluid is being created or destroyed at a point. In electromagnetism, the divergence of the electric field is proportional to the charge density (Gauss law).
What is the curl of a vector field and when is it important?
The curl of a vector field measures the rotational tendency or circulation density at each point. It produces a new vector whose direction is the axis of rotation and whose magnitude indicates the rotation strength. Mathematically, curl(F) = (dF3/dy - dF2/dz, dF1/dz - dF3/dx, dF2/dx - dF1/dy). In fluid mechanics, the curl of the velocity field gives the vorticity, which describes local spinning motions. In electromagnetism, the curl of the electric field equals the negative time derivative of the magnetic field (Faraday law), and the curl of the magnetic field relates to current density (Ampere law). A field with zero curl everywhere is called irrotational or conservative.
How do gradient, divergence, and curl relate to Maxwell equations?
Maxwell four equations of electromagnetism are elegantly expressed using these vector calculus operators. Gauss law for electricity states that div(E) = rho/epsilon_0, relating the divergence of the electric field to charge density. Gauss law for magnetism states div(B) = 0, meaning the magnetic field is always solenoidal. Faraday law states curl(E) = -dB/dt, connecting the curl of the electric field to changing magnetic fields. Ampere-Maxwell law states curl(B) = mu_0*J + mu_0*epsilon_0*dE/dt, relating the curl of the magnetic field to current density and changing electric fields. These operators thus form the mathematical foundation of all electromagnetic theory.
What is the relationship between divergence theorem and Stokes theorem?
The divergence theorem (Gauss theorem) and Stokes theorem are both generalizations of the fundamental theorem of calculus to higher dimensions. The divergence theorem converts a volume integral of divergence into a surface integral of flux: the integral of div(F) over a volume V equals the integral of F dot n over the bounding surface S. Stokes theorem converts a surface integral of curl into a line integral: the integral of curl(F) dot dS over surface S equals the line integral of F dot dr around the boundary curve C. Both theorems relate an integral over a region to an integral over its boundary, and both are special cases of the generalized Stokes theorem from differential forms.
How is the Laplacian operator related to gradient and divergence?
The Laplacian operator is the divergence of the gradient, written as div(grad(f)) or nabla-squared f. For a scalar field f(x,y,z), the Laplacian equals the sum of second partial derivatives: d2f/dx2 + d2f/dy2 + d2f/dz2. The Laplacian measures how much the value of f at a point deviates from the average value in its neighborhood. It appears in many fundamental equations of physics: the heat equation (df/dt = k nabla^2 f), the wave equation (d2f/dt2 = c^2 nabla^2 f), and Laplace equation (nabla^2 f = 0) which describes steady-state potential fields. A function satisfying Laplace equation is called harmonic and has no local extrema in its interior.