Question

Multivariable calculus. Line integral

Answer 1

Parameterize
`C_R` by

`\mathbf r(t)=(R\cos t,R\sin t)`

with
`0\le t\le2\pi`.

Plug
`\mathbf r` into
`\mathbf G` and compute the differential
`\mathrm d\mathbf r`:

`\mathbf G(\mathbf r(t))=((R\cos t,R\sin t))/(√(R^2-1))`

`\mathrm d\mathbf r=(-R\sin t,R\cos t)\,\mathrm dt`

Plug everything into the integral and evaluate:

`\displaystyle\int_(C_R)\mathbf G\cdot\mathrm d\mathbf r=\int_0^(2\pi)0\,\mathrm dt=0`