Paper 3, Section II, B

Vector Calculus | Part IA, 2017

(a) The time-dependent vector field F\mathbf{F} is related to the vector field B\mathbf{B} by

F(x,t)=B(z)\mathbf{F}(\mathbf{x}, t)=\mathbf{B}(\mathbf{z})

where z=tx\mathbf{z}=t \mathbf{x}. Show that

(x)F=tFt(\mathbf{x} \cdot \nabla) \mathbf{F}=t \frac{\partial \mathbf{F}}{\partial t} \text {. }

(b) The vector fields B\mathbf{B} and A\mathbf{A} satisfy B=×A\mathbf{B}=\boldsymbol{\nabla} \times \mathbf{A}. Show that B=0\boldsymbol{\nabla} \cdot \mathbf{B}=0.

(c) The vector field B\mathbf{B} satisfies B=0\boldsymbol{\nabla} \cdot \mathbf{B}=0. Show that

B(x)=×(D(x)×x)\mathbf{B}(\mathbf{x})=\nabla \times(\mathbf{D}(\mathbf{x}) \times \mathbf{x})


D(x)=01tB(tx)dt\mathbf{D}(\mathbf{x})=\int_{0}^{1} t \mathbf{B}(t \mathbf{x}) d t

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