3B (Vector Calculus)

In the Cartesian basis, writing

F = (F_{1}, F_{2}, F_{3})

\nabla \cdot F + = \partial_{i} F_{i} + = \frac{\partial F _{1}}{\partial x} + + \frac{\partial F _{2}}{\partial y} + + \frac{\partial F _{3}}{\partial z}

(1)

defines the divergence, while the curl is

(\nabla \times F)_{i} + = ϵ_{ijk} \partial_{j} F_{k} .

(2)

Equivalently,

\nabla \times F = + (\partial_{y} F_{3} - \partial_{z} F_{2}) e_{x} + + (\partial_{z} F_{1} - \partial_{x} F_{3}) e_{y} + + (\partial_{x} F_{2} - \partial_{y} F_{1}) e_{z} .

(3)

ii

 0  0

Solution

 0  0

Use the Levi-Civita symbol and the product rule. First,

\nabla \cdot (F \times G) + + + = \partial_{i} (ϵ_{ijk} F_{j} G_{k}) = ϵ_{ijk} (\partial_{i} F_{j}) G_{k} + + ϵ_{ijk} F_{j} \partial_{i} G_{k} = G \cdot (\nabla \times F) - F \cdot (\nabla \times G) .

(1)

For the second identity, the contraction

ϵ_{ijk} ϵ_{k l m} + = δ_{i l} δ_{jm} - δ_{im} δ_{j l}

(2)

gives

+ [\nabla \times (F \times G)]_{i} + + + = ϵ_{ijk} \partial_{j} (ϵ_{k l m} F_{l} G_{m}) = \partial_{j} (F_{i} G_{j} - F_{j} G_{i}) = F_{i} \partial_{j} G_{j} - G_{i} \partial_{j} F_{j} + + G_{j} \partial_{j} F_{i} - F_{j} \partial_{j} G_{i} .

(3)

In vector notation this is

+ \nabla \times (F \times G) + = F (\nabla \cdot G) - G (\nabla \cdot F) + + (G \cdot \nabla) F - (F \cdot \nabla) G .

(4)

These are the divergence and curl of a cross product identities.

iii

 0  0

Solution

 0  0

The first identity in part (ii), with

F

and

G

interchanged, says

\nabla \cdot (G \times F) + = F \cdot (\nabla \times G) - G \cdot (\nabla \times F),

(1)

so the proposed identity is missing its final term. For a concrete counterexample, take

G = e_{x}, F = y e_{z} .

(2)

Then

\nabla \times G = 0

, so the proposed left side is zero, whereas

G \times F = - y e_{y}, \nabla \cdot (G \times F) = - 1.

(3)

Thus the identity is false.

iv

 0  0

Solution

 0  0

For

F = (x, y, z)

\nabla \cdot F = 1 + 1 + 1 = 3

(1)

and all cross-partial terms in the curl vanish, so

\nabla \times F = 0 .

(2)

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Past exam of the mathematics course of the University of Cambridge / 2024 / ia / Paper 3 / 3B (Vector Calculus)

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