Paper 3, Section I, A

Vector Calculus | Part IA, 2014

Let F(x)\mathbf{F}(\mathbf{x}) be a vector field defined everywhere on the domain GR3G \subset \mathbb{R}^{3}.

(a) Suppose that F(x)\mathbf{F}(\mathbf{x}) has a potential ϕ(x)\phi(\mathbf{x}) such that F(x)=ϕ(x)\mathbf{F}(\mathbf{x})=\nabla \phi(\mathbf{x}) for xG\mathbf{x} \in G. Show that

γFdx=ϕ(b)ϕ(a)\int_{\gamma} \mathbf{F} \cdot \mathbf{d} \mathbf{x}=\phi(\mathbf{b})-\phi(\mathbf{a})

for any smooth path γ\gamma from a to b\mathbf{b} in GG. Show further that necessarily ×F=0\nabla \times \mathbf{F}=\mathbf{0} on GG.

(b) State a condition for GG which ensures that ×F=0\nabla \times \mathbf{F}=\mathbf{0} implies γFdx\int_{\gamma} \mathbf{F} \cdot \mathbf{d x} is pathindependent.

(c) Compute the line integral γFdx\oint_{\gamma} \mathbf{F} \cdot \mathbf{d} \mathbf{x} for the vector field

F(x)=(yx2+y2xx2+y20)\mathbf{F}(\mathbf{x})=\left(\begin{array}{c} \frac{-y}{x^{2}+y^{2}} \\ \frac{x}{x^{2}+y^{2}} \\ 0 \end{array}\right)

where γ\gamma denotes the anti-clockwise path around the unit circle in the (x,y)(x, y)-plane. Compute ×F\nabla \times \mathbf{F} and comment on your result in the light of (b).

Typos? Please submit corrections to this page on GitHub.