Appendix
All Solutions to the Problems
The solution to problems in variational methods is not unique, here give only the
most basic solution, individual problems are given two kinds of solutions.
Chapter 1: Solutions to the Problems in Preliminaries
1.1 Prove = ϕϕψ + 2∇ϕ · ∇ψ + ψψϕ.
Proof The divergence of gradient of product of two functions is
= ∇ · ∇(ϕψ) = ∇ · (ψ∇ϕ + ϕ∇ψ)
= ψ∇ · ∇ϕ + ∇ψ · ∇ϕ + ϕ∇ · ∇ψ + ∇ϕ · ∇ψ
= ψψϕ + 2∇ϕ · ∇ψ + ϕϕψ
Quod erat demonstrandum.
1.2 Prove ∇ · ( pa + q b) = p∇ · a + q∇ · b, where, both p and q are constants.
Proof Let a = a x i + a y j + a z k, b = b x i + b y j + b z k, then
pa + q b = ( pa x + qb x )i + ( pa y + qb y ) j + ( pa z + qb z )k
Therefore
∇ · ( pa + q b) =
∂( pa x + qb x )
∂ x
+
∂( pa y + qb y )
∂ y
+
∂( pa z + qb z )
∂z
= p
∂a x
∂ x
+
∂a y
∂ y
+
∂a z
∂z
+ q
∂b x
∂ x
+
∂b y
∂ y
+
∂b z
∂z
= p∇ · a + q∇ · b
Quod erat demonstrandum.
© Beijing Institute of Technology Press and Springer Nature Singapore Pte Ltd. 2021
D. Lao and S. Zhao, Fundamental Theories and Their Applications of the Calculus
of Variations, https://doi.org/10.1007/978-981-15-6070-5
751