Introduction to Electrodynamics · Problem 1.61

Vector Analysis — David J. Griffiths

Physics · integral-theorems-corollaries

Problem

Although the gradient, divergence, and curl theorems are the fundamental integral theorems of vector calculus, it is possible to derive a number of corollaries from them. Show that: (a) ∫V(∇T) dτ=∮ST da\int_{V} (\nabla T)\,d\tau = \oint_{S} T\,d\mathbf{a}. [Hint: Let v=cT\mathbf{v} = \mathbf{c}T, where c\mathbf{c} is a constant, in the divergence theorem; use the product rules.] (b) ∫V(∇×v) dτ=−∮Sv×da\int_{V} (\nabla \times \mathbf{v})\,d\tau = -\oint_{S} \mathbf{v} \times d\mathbf{a}. [Hint: Replace v\mathbf{v} by (v×c)(\mathbf{v} \times \mathbf{c}) in the divergence theorem.] (c) ∫V[T∇2U+(∇T)⋅(∇U)] dτ=∮S(T∇U)⋅da\int_{V} [T\nabla^2 U + (\nabla T) \cdot (\nabla U)]\,d\tau = \oint_{S} (T \nabla U) \cdot d\mathbf{a}. [Hint: Let v=T∇U\mathbf{v} = T \nabla U in the divergence theorem.] (d) ∫V(T∇2U−U∇2T) dτ=∮S(T∇U−U∇T)⋅da\int_{V} (T\nabla^2 U - U\nabla^2 T)\,d\tau = \oint_{S} (T \nabla U - U \nabla T) \cdot d\mathbf{a}. [Comment: This is sometimes called Green's second identity; it follows from (c), which is known as Green's identity.] (e) ∫S∇T×da=−∮PT dl\int_{S} \nabla T \times d\mathbf{a} = -\oint_{P} T\,d\mathbf{l}. [Hint: Let v=cT\mathbf{v} = \mathbf{c}T in Stokes' theorem.]

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