Paper 3, Section I, B

Vector Calculus | Part IA, 2019

Apply the divergence theorem to the vector field u(x)=aϕ(x)\mathbf{u}(\mathbf{x})=\mathbf{a} \phi(\mathbf{x}) where a\mathbf{a} is an arbitrary constant vector and ϕ\phi is a scalar field, to show that

VϕdV=SϕdS\int_{V} \nabla \phi d V=\int_{S} \phi d \mathbf{S}

where VV is a volume bounded by the surface SS and dSd \mathbf{S} is the outward pointing surface element.

Verify that this result holds when ϕ=x+y\phi=x+y and VV is the spherical volume x2+x^{2}+ y2+z2a2y^{2}+z^{2} \leqslant a^{2}. [You may use the result that dS=a2sinθdθdϕ(sinθcosϕ,sinθsinϕ,cosθ)d \mathbf{S}=a^{2} \sin \theta d \theta d \phi(\sin \theta \cos \phi, \sin \theta \sin \phi, \cos \theta), where θ\theta and ϕ\phi are the usual angular coordinates in spherical polars and the components of dSd \mathbf{S} are with respect to standard Cartesian axes.]

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