Divergence Theorem Calculator

Calculator Inputs

Region (rectangular box) — bounds must satisfy x1>x0, y1>y0, z1>z0.

x0

x1

y0

y1

z0

z1

Vector field F = <P,Q,R> with linear components: ax·x + ay·y + az·z + c.

P: coefficient of x (px)

P: coefficient of y (py)

P: coefficient of z (pz)

P: constant (pc)

Q: coefficient of x (qx)

Q: coefficient of y (qy)

Q: coefficient of z (qz)

Q: constant (qc)

R: coefficient of x (rx)

R: coefficient of y (ry)

R: coefficient of z (rz)

R: constant (rc)

Tip: For this tool, linear fields make both integrals exact and fast.

Example Data Table

Try these sample values to see the equality clearly.

Item	Value	Notes
Bounds	x: 0→2, y: 0→1, z: 0→3	Rectangular box volume = 6
P	2x + 1y + 0z + 0	px=2, py=1, pz=0, pc=0
Q	0x + 3y + 0z + 0	qx=0, qy=3, qz=0, qc=0
R	0x + 0y + 4z + 0	rx=0, ry=0, rz=4, rc=0
Expected	∇·F = 2 + 3 + 4 = 9	Volume integral = 9 × 6 = 54

Formula Used

The Divergence Theorem states that the outward flux of a vector field through a closed surface equals the triple integral of its divergence over the enclosed volume:

∯_S F · n dS = ∭_V (∇ · F) dV

F = <P,Q,R> and ∇·F = ∂P/∂x + ∂Q/∂y + ∂R/∂z.
This calculator uses a rectangular box region and linear components for P, Q, and R.
For linear components, divergence is constant and the volume integral becomes (∇·F) × volume.
Surface flux is computed by integrating P, Q, and R over the six faces with outward normals.

How to Use

Enter bounds for x, y, and z to define the box region.
Enter coefficients for P, Q, and R in the linear form.
Click the compute button to calculate both sides.
Review surface flux, volume integral, and the difference.
Export results using the CSV or PDF buttons.

Calculator Inputs

Example Data Table

Formula Used

How to Use

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