I am given $f(x,y,z) = 7y^2+5e^{xz}+\ln(6(x^2+y^2+z^2))$ where the vectorfield is defined through $F(x,y,z) = \nabla f(x,y,z)$. Now I am asked to calculate the integral of the tangential component of $F$ along the curve$$x = 8\sin(3\theta)\cos(\theta)$$ $$y = 8\sin(3\theta)\sin(\theta)$$ $$z = 8\cos(3\theta)$$.
I'm confused about how to set this integral up, the partial derivatives $\frac{\partial f}{dx}$, $\frac{\partial f}{dy}$ and $\frac{\partial f}{dz}$ yield some nasty expressions, I don't believe I'm supposed to use these in my integral. Any help is appreciated
Note that the curve does not pass through $(0,0,0)$ and the vector field is the gradient of some function $f(x,y,z)$ which does not have any line of singularities inside the curve. (In fact, it only has a point singularity at the origin.)
Therefore, the gradient theorem applies, and the integral along the curve from point $p$ to point $q$ is $f(p)-=f(q)$
Thus the integral of $F \cdot ds$ along the whole curve (where starting and ending points $p$ and $q$ are identical) will be $f(p)-f(p) = 0$.