Skip to content

Gradient, Divergence, and Curl

Mathematics · FE Reference Handbook section

Mathematics
4 formulas
10 exam-style examples
~53 min
All Mathematics lectures

Handbook notes for this section

Definitions and conditions exactly as the handbook states them.

  • The Laplacian of a scalar function φ is

Core formulas for this FE topic

Definitions, applicability, units, assumptions and worked examples for each relation.

Worked exam-style examples

The four ways this section is written on the real exam — thoughts first, then equations, then substitution.

Example 1
Gradient of a scalar function and divergence of its field — Gradient, Divergence, and Curl

Given the scalar value function φ = 4x² + 2yz + 1z², compute the gradient ∇φ at the point (1, 2, 4), its magnitude, and the divergence ∇·(∇φ).

Given

  • ϕ=4x2+2yz+1z2\phi = 4x^{2} + 2yz + 1z^{2}

  • Point (1, 2, 4)

Find

∇φ, |∇φ| and ∇·∇φ

Start with the thinking

  • The gradient collects the partial derivatives; it points toward the steepest increase of the scalar function.
  • Divergence of a gradient is the Laplacian — a scalar, not a vector.

Step-by-step solution

  1. Formula — ∇φ = (∂φ/∂x)i + (∂φ/∂y)j + (∂φ/∂z)k

  2. Partials — ∂φ/∂x = 8x, ∂φ/∂y = 2z, ∂φ/∂z = 2y + 2z

  3. Substituting

    ∇ϕ=8i+8j+12k∇\phi = 8i + 8j + 12k

  4. Magnitude

    ∣∇ϕ∣=(82+82+122)=16.492|∇\phi| = \sqrt(8^{2} + 8^{2} + 12^{2}) = 16.492

  5. Divergence

    ∇⋅∇ϕ=8+0+2=10∇\cdot∇\phi = 8 + 0 + 2 = 10
Answer:
∇ϕ=(8,8,12),∣∇ϕ∣=16.492,∇⋅∇ϕ=10∇\phi = (8, 8, 12), |∇\phi| = 16.492, ∇\cdot∇\phi = 10

Why the other options are there

  • |∇φ| = 28 (components added)
  • ∇·∇φ = 28 (used the gradient values)

Reference: FE Reference Handbook — Mathematics → Gradient, Divergence, and Curl

Example 2
Gradient of a scalar function and divergence of its field — Gradient, Divergence, and Curl (2)

Given the scalar value function φ = 3x² + 6yz + 4z², compute the gradient ∇φ at the point (1, 2, 4), its magnitude, and the divergence ∇·(∇φ).

Given

  • ϕ=3x2+6yz+4z2\phi = 3x^{2} + 6yz + 4z^{2}

  • Point (1, 2, 4)

Find

∇φ, |∇φ| and ∇·∇φ

Start with the thinking

  • The gradient collects the partial derivatives; it points toward the steepest increase of the scalar function.
  • Divergence of a gradient is the Laplacian — a scalar, not a vector.

Step-by-step solution

  1. Formula — ∇φ = (∂φ/∂x)i + (∂φ/∂y)j + (∂φ/∂z)k

  2. Partials — ∂φ/∂x = 6x, ∂φ/∂y = 6z, ∂φ/∂z = 6y + 8z

  3. Substituting

    ∇ϕ=6i+24j+44k∇\phi = 6i + 24j + 44k

  4. Magnitude

    ∣∇ϕ∣=(62+242+442)=50.478|∇\phi| = \sqrt(6^{2} + 24^{2} + 44^{2}) = 50.478

  5. Divergence

    ∇⋅∇ϕ=6+0+8=14∇\cdot∇\phi = 6 + 0 + 8 = 14
Answer:
∇ϕ=(6,24,44),∣∇ϕ∣=50.478,∇⋅∇ϕ=14∇\phi = (6, 24, 44), |∇\phi| = 50.478, ∇\cdot∇\phi = 14

Why the other options are there

  • |∇φ| = 74 (components added)
  • ∇·∇φ = 74 (used the gradient values)

Reference: FE Reference Handbook — Mathematics → Gradient, Divergence, and Curl

Example 3
Gradient of a scalar function and divergence of its field — Gradient, Divergence, and Curl (3)

Given the scalar value function φ = 5x² + 4yz + 4z², compute the gradient ∇φ at the point (4, 1, 2), its magnitude, and the divergence ∇·(∇φ).

Given

  • ϕ=5x2+4yz+4z2\phi = 5x^{2} + 4yz + 4z^{2}

  • Point (4, 1, 2)

Find

∇φ, |∇φ| and ∇·∇φ

Start with the thinking

  • The gradient collects the partial derivatives; it points toward the steepest increase of the scalar function.
  • Divergence of a gradient is the Laplacian — a scalar, not a vector.

Step-by-step solution

  1. Formula — ∇φ = (∂φ/∂x)i + (∂φ/∂y)j + (∂φ/∂z)k

  2. Partials — ∂φ/∂x = 10x, ∂φ/∂y = 4z, ∂φ/∂z = 4y + 8z

  3. Substituting

    ∇ϕ=40i+8j+20k∇\phi = 40i + 8j + 20k

  4. Magnitude

    ∣∇ϕ∣=(402+82+202)=45.431|∇\phi| = \sqrt(40^{2} + 8^{2} + 20^{2}) = 45.431

  5. Divergence

    ∇⋅∇ϕ=10+0+8=18∇\cdot∇\phi = 10 + 0 + 8 = 18
Answer:
∇ϕ=(40,8,20),∣∇ϕ∣=45.431,∇⋅∇ϕ=18∇\phi = (40, 8, 20), |∇\phi| = 45.431, ∇\cdot∇\phi = 18

Why the other options are there

  • |∇φ| = 68 (components added)
  • ∇·∇φ = 68 (used the gradient values)

Reference: FE Reference Handbook — Mathematics → Gradient, Divergence, and Curl

Example 4
Gradient of a scalar function and divergence of its field — Gradient, Divergence, and Curl (4)

Given the scalar value function φ = 2x² + 6yz + 2z², compute the gradient ∇φ at the point (1, 2, 3), its magnitude, and the divergence ∇·(∇φ).

Given

  • ϕ=2x2+6yz+2z2\phi = 2x^{2} + 6yz + 2z^{2}

  • Point (1, 2, 3)

Find

∇φ, |∇φ| and ∇·∇φ

Start with the thinking

  • The gradient collects the partial derivatives; it points toward the steepest increase of the scalar function.
  • Divergence of a gradient is the Laplacian — a scalar, not a vector.

Step-by-step solution

  1. Formula — ∇φ = (∂φ/∂x)i + (∂φ/∂y)j + (∂φ/∂z)k

  2. Partials — ∂φ/∂x = 4x, ∂φ/∂y = 6z, ∂φ/∂z = 6y + 4z

  3. Substituting

    ∇ϕ=4i+18j+24k∇\phi = 4i + 18j + 24k

  4. Magnitude

    ∣∇ϕ∣=(42+182+242)=30.265|∇\phi| = \sqrt(4^{2} + 18^{2} + 24^{2}) = 30.265

  5. Divergence

    ∇⋅∇ϕ=4+0+4=8∇\cdot∇\phi = 4 + 0 + 4 = 8
Answer:
∇ϕ=(4,18,24),∣∇ϕ∣=30.265,∇⋅∇ϕ=8∇\phi = (4, 18, 24), |∇\phi| = 30.265, ∇\cdot∇\phi = 8

Why the other options are there

  • |∇φ| = 46 (components added)
  • ∇·∇φ = 46 (used the gradient values)

Reference: FE Reference Handbook — Mathematics → Gradient, Divergence, and Curl

Example 5
Gradient of a scalar function and divergence of its field — Gradient, Divergence, and Curl (5)

Given the scalar value function φ = 2x² + 4yz + 1z², compute the gradient ∇φ at the point (1, 4, 2), its magnitude, and the divergence ∇·(∇φ).

Given

  • ϕ=2x2+4yz+1z2\phi = 2x^{2} + 4yz + 1z^{2}

  • Point (1, 4, 2)

Find

∇φ, |∇φ| and ∇·∇φ

Start with the thinking

  • The gradient collects the partial derivatives; it points toward the steepest increase of the scalar function.
  • Divergence of a gradient is the Laplacian — a scalar, not a vector.

Step-by-step solution

  1. Formula — ∇φ = (∂φ/∂x)i + (∂φ/∂y)j + (∂φ/∂z)k

  2. Partials — ∂φ/∂x = 4x, ∂φ/∂y = 4z, ∂φ/∂z = 4y + 2z

  3. Substituting

    ∇ϕ=4i+8j+20k∇\phi = 4i + 8j + 20k

  4. Magnitude

    ∣∇ϕ∣=(42+82+202)=21.909|∇\phi| = \sqrt(4^{2} + 8^{2} + 20^{2}) = 21.909

  5. Divergence

    ∇⋅∇ϕ=4+0+2=6∇\cdot∇\phi = 4 + 0 + 2 = 6
Answer:
∇ϕ=(4,8,20),∣∇ϕ∣=21.909,∇⋅∇ϕ=6∇\phi = (4, 8, 20), |∇\phi| = 21.909, ∇\cdot∇\phi = 6

Why the other options are there

  • |∇φ| = 32 (components added)
  • ∇·∇φ = 32 (used the gradient values)

Reference: FE Reference Handbook — Mathematics → Gradient, Divergence, and Curl

Example 6
Gradient of a scalar function and divergence of its field — Gradient, Divergence, and Curl (6)

Given the scalar value function φ = 5x² + 2yz + 1z², compute the gradient ∇φ at the point (1, 1, 3), its magnitude, and the divergence ∇·(∇φ).

Given

  • ϕ=5x2+2yz+1z2\phi = 5x^{2} + 2yz + 1z^{2}

  • Point (1, 1, 3)

Find

∇φ, |∇φ| and ∇·∇φ

Start with the thinking

  • The gradient collects the partial derivatives; it points toward the steepest increase of the scalar function.
  • Divergence of a gradient is the Laplacian — a scalar, not a vector.

Step-by-step solution

  1. Formula — ∇φ = (∂φ/∂x)i + (∂φ/∂y)j + (∂φ/∂z)k

  2. Partials — ∂φ/∂x = 10x, ∂φ/∂y = 2z, ∂φ/∂z = 2y + 2z

  3. Substituting

    ∇ϕ=10i+6j+8k∇\phi = 10i + 6j + 8k

  4. Magnitude

    ∣∇ϕ∣=(102+62+82)=14.142|∇\phi| = \sqrt(10^{2} + 6^{2} + 8^{2}) = 14.142

  5. Divergence

    ∇⋅∇ϕ=10+0+2=12∇\cdot∇\phi = 10 + 0 + 2 = 12
Answer:
∇ϕ=(10,6,8),∣∇ϕ∣=14.142,∇⋅∇ϕ=12∇\phi = (10, 6, 8), |∇\phi| = 14.142, ∇\cdot∇\phi = 12

Why the other options are there

  • |∇φ| = 24 (components added)
  • ∇·∇φ = 24 (used the gradient values)

Reference: FE Reference Handbook — Mathematics → Gradient, Divergence, and Curl

Example 7
Gradient of a scalar function and divergence of its field — Gradient, Divergence, and Curl (7)

Given the scalar value function φ = 3x² + 5yz + 3z², compute the gradient ∇φ at the point (3, 1, 3), its magnitude, and the divergence ∇·(∇φ).

Given

  • ϕ=3x2+5yz+3z2\phi = 3x^{2} + 5yz + 3z^{2}

  • Point (3, 1, 3)

Find

∇φ, |∇φ| and ∇·∇φ

Start with the thinking

  • The gradient collects the partial derivatives; it points toward the steepest increase of the scalar function.
  • Divergence of a gradient is the Laplacian — a scalar, not a vector.

Step-by-step solution

  1. Formula — ∇φ = (∂φ/∂x)i + (∂φ/∂y)j + (∂φ/∂z)k

  2. Partials — ∂φ/∂x = 6x, ∂φ/∂y = 5z, ∂φ/∂z = 5y + 6z

  3. Substituting

    ∇ϕ=18i+15j+23k∇\phi = 18i + 15j + 23k

  4. Magnitude

    ∣∇ϕ∣=(182+152+232)=32.833|∇\phi| = \sqrt(18^{2} + 15^{2} + 23^{2}) = 32.833

  5. Divergence

    ∇⋅∇ϕ=6+0+6=12∇\cdot∇\phi = 6 + 0 + 6 = 12
Answer:
∇ϕ=(18,15,23),∣∇ϕ∣=32.833,∇⋅∇ϕ=12∇\phi = (18, 15, 23), |∇\phi| = 32.833, ∇\cdot∇\phi = 12

Why the other options are there

  • |∇φ| = 56 (components added)
  • ∇·∇φ = 56 (used the gradient values)

Reference: FE Reference Handbook — Mathematics → Gradient, Divergence, and Curl

Example 8
Gradient of a scalar function and divergence of its field — Gradient, Divergence, and Curl (8)

Given the scalar value function φ = 5x² + 6yz + 1z², compute the gradient ∇φ at the point (4, 4, 4), its magnitude, and the divergence ∇·(∇φ).

Given

  • ϕ=5x2+6yz+1z2\phi = 5x^{2} + 6yz + 1z^{2}

  • Point (4, 4, 4)

Find

∇φ, |∇φ| and ∇·∇φ

Start with the thinking

  • The gradient collects the partial derivatives; it points toward the steepest increase of the scalar function.
  • Divergence of a gradient is the Laplacian — a scalar, not a vector.

Step-by-step solution

  1. Formula — ∇φ = (∂φ/∂x)i + (∂φ/∂y)j + (∂φ/∂z)k

  2. Partials — ∂φ/∂x = 10x, ∂φ/∂y = 6z, ∂φ/∂z = 6y + 2z

  3. Substituting

    ∇ϕ=40i+24j+32k∇\phi = 40i + 24j + 32k

  4. Magnitude

    ∣∇ϕ∣=(402+242+322)=56.569|∇\phi| = \sqrt(40^{2} + 24^{2} + 32^{2}) = 56.569

  5. Divergence

    ∇⋅∇ϕ=10+0+2=12∇\cdot∇\phi = 10 + 0 + 2 = 12
Answer:
∇ϕ=(40,24,32),∣∇ϕ∣=56.569,∇⋅∇ϕ=12∇\phi = (40, 24, 32), |∇\phi| = 56.569, ∇\cdot∇\phi = 12

Why the other options are there

  • |∇φ| = 96 (components added)
  • ∇·∇φ = 96 (used the gradient values)

Reference: FE Reference Handbook — Mathematics → Gradient, Divergence, and Curl

Example 9
Gradient of a scalar function and divergence of its field — Gradient, Divergence, and Curl (9)

Given the scalar value function φ = 2x² + 4yz + 3z², compute the gradient ∇φ at the point (4, 3, 3), its magnitude, and the divergence ∇·(∇φ).

Given

  • ϕ=2x2+4yz+3z2\phi = 2x^{2} + 4yz + 3z^{2}

  • Point (4, 3, 3)

Find

∇φ, |∇φ| and ∇·∇φ

Start with the thinking

  • The gradient collects the partial derivatives; it points toward the steepest increase of the scalar function.
  • Divergence of a gradient is the Laplacian — a scalar, not a vector.

Step-by-step solution

  1. Formula — ∇φ = (∂φ/∂x)i + (∂φ/∂y)j + (∂φ/∂z)k

  2. Partials — ∂φ/∂x = 4x, ∂φ/∂y = 4z, ∂φ/∂z = 4y + 6z

  3. Substituting

    ∇ϕ=16i+12j+30k∇\phi = 16i + 12j + 30k

  4. Magnitude

    ∣∇ϕ∣=(162+122+302)=36.056|∇\phi| = \sqrt(16^{2} + 12^{2} + 30^{2}) = 36.056

  5. Divergence

    ∇⋅∇ϕ=4+0+6=10∇\cdot∇\phi = 4 + 0 + 6 = 10
Answer:
∇ϕ=(16,12,30),∣∇ϕ∣=36.056,∇⋅∇ϕ=10∇\phi = (16, 12, 30), |∇\phi| = 36.056, ∇\cdot∇\phi = 10

Why the other options are there

  • |∇φ| = 58 (components added)
  • ∇·∇φ = 58 (used the gradient values)

Reference: FE Reference Handbook — Mathematics → Gradient, Divergence, and Curl

Example 10
Gradient of a scalar function and divergence of its field — Gradient, Divergence, and Curl (10)

Given the scalar value function φ = 6x² + 3yz + 5z², compute the gradient ∇φ at the point (2, 2, 3), its magnitude, and the divergence ∇·(∇φ).

Given

  • ϕ=6x2+3yz+5z2\phi = 6x^{2} + 3yz + 5z^{2}

  • Point (2, 2, 3)

Find

∇φ, |∇φ| and ∇·∇φ

Start with the thinking

  • The gradient collects the partial derivatives; it points toward the steepest increase of the scalar function.
  • Divergence of a gradient is the Laplacian — a scalar, not a vector.

Step-by-step solution

  1. Formula — ∇φ = (∂φ/∂x)i + (∂φ/∂y)j + (∂φ/∂z)k

  2. Partials — ∂φ/∂x = 12x, ∂φ/∂y = 3z, ∂φ/∂z = 3y + 10z

  3. Substituting

    ∇ϕ=24i+9j+36k∇\phi = 24i + 9j + 36k

  4. Magnitude

    ∣∇ϕ∣=(242+92+362)=44.193|∇\phi| = \sqrt(24^{2} + 9^{2} + 36^{2}) = 44.193

  5. Divergence

    ∇⋅∇ϕ=12+0+10=22∇\cdot∇\phi = 12 + 0 + 10 = 22
Answer:
∇ϕ=(24,9,36),∣∇ϕ∣=44.193,∇⋅∇ϕ=22∇\phi = (24, 9, 36), |∇\phi| = 44.193, ∇\cdot∇\phi = 22

Why the other options are there

  • |∇φ| = 69 (components added)
  • ∇·∇φ = 69 (used the gradient values)

Reference: FE Reference Handbook — Mathematics → Gradient, Divergence, and Curl

© 2026 Dr. Steve Efe. Civil Engineering Capstone Studio. All rights reserved.