Consider vector fields v(x) and w(x), where $v(x) = ∇φ(x)$ and $w(x) =( (1 − x^ 2 − z^ 2 ) arcsin y, (1 − x^ 2 − z^ 2 )arctanh(x 2 ), x) $.
Here, $φ(x) = e^ {|x|^ 2}$ and $x = (x, y, z)$.
If $Γ1$ and $Γ2$ are curves described by $x = (cos θ, e− cos2 θ ,sin θ)$, find the values of
(a) (i) $\int_ {Γ1} (3v + w) · dx$, where $0 \leq θ < 2π$.
(ii) $\int_{ Γ2} (∇ ∧ v) · dx$, where $0 \leq θ < π$
(iii) The line integral $\int_{ Γ2} ∇ · [∇ ∧ (|v| ^2w)]ds$, where $0 \leq θ < π$
So these seem like pretty nasty line integrals so I thought that there must be 'tricks' for each of these that would simplify the matter. For the first path integral I realised that, as $\textbf{v}$ is a conservative field, the integral over the closed path of this vanishes. Then in this case, the x and y components of $\textbf{w}$ vanish so the first case simplifies quite nicely 9I get $\pi$ as the final result)
For the second part, since $\textbf{v}$ is conservative its curl is zero, and therefore the answer is zero.
Now I am stuck on the last part. I cannot see how to simplify it at all. I could put in that the x and y components of w are zero, but then I am still stuck. I do not know how to deal with the divergence of a fucntion in x,y and z multiplied by a vector...
It's zero too. The divergence of the curl of any vector field is always zero.