Paper 3, Section II, C

Vector Calculus | Part IA, 2016

(a) Let

F=(z,x,y)\mathbf{F}=(z, x, y)

and let CC be a circle of radius RR lying in a plane with unit normal vector (a,b,c)(a, b, c). Calculate ×F\nabla \times \mathbf{F} and use this to compute CFdx\oint_{C} \mathbf{F} \cdot d \mathbf{x}. Explain any orientation conventions which you use.

(b) Let F:R3R3\mathbf{F}: \mathbb{R}^{3} \rightarrow \mathbb{R}^{3} be a smooth vector field such that the matrix with entries Fjxi\frac{\partial F_{j}}{\partial x_{i}} is symmetric. Prove that CFdx=0\oint_{C} \mathbf{F} \cdot d \mathbf{x}=0 for every circle CR3C \subset \mathbb{R}^{3}.

(c) Let F=1r(x,y,z)\mathbf{F}=\frac{1}{r}(x, y, z), where r=x2+y2+z2r=\sqrt{x^{2}+y^{2}+z^{2}} and let CC be the circle which is the intersection of the sphere (x5)2+(y3)2+(z2)2=1(x-5)^{2}+(y-3)^{2}+(z-2)^{2}=1 with the plane 3x5yz=23 x-5 y-z=2. Calculate CFdx\oint_{C} \mathbf{F} \cdot d \mathbf{x}.

(d) Let F\mathbf{F} be the vector field defined, for x2+y2>0x^{2}+y^{2}>0, by

F=(yx2+y2,xx2+y2,z)\mathbf{F}=\left(\frac{-y}{x^{2}+y^{2}}, \frac{x}{x^{2}+y^{2}}, z\right)

Show that ×F=0\nabla \times \mathbf{F}=\mathbf{0}. Let CC be the curve which is the intersection of the cylinder x2+y2=1x^{2}+y^{2}=1 with the plane z=x+200z=x+200. Calculate CFdx\oint_{C} \mathbf{F} \cdot d \mathbf{x}.

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