Let the vector field $$\vec{F}\left(\vec{x}\right)=\begin{pmatrix}{x_1^2+2x_3}\\ x_1x_2\\ x_3^2-2x_1\end{pmatrix}$$ Compute the integral $\int _C\vec{F}\left(\vec{x}\right)d\vec{x}$ from the origin to the point $P(1/2/3)$ if $C$ is a straight line from the origin to $P$.
So in the book they only give us answers, but not how to get the answers. I calculated the integral and got
$$\int _0^112t^2-8t$$ which equals to zero, however in the book the answers is $9\frac{2}{3}$. I think the book is wrong because I just don't see how we can get that answer. Or am I wrong?
You have that $F(x,y,z)=(x^2+2z,xy,z^2-2x)$, and you want to evaluate
$$\int_C F(x,y,z)\cdot (dx,dy,dz)$$
where $C$ is the straight line parametrized by $\gamma:[0,1]\to\Bbb R^3,\,t\mapsto t P$ for $P:=(1,2,3)$. Hence
$$\int_C F(x,y,z)\cdot (dx,dy,dz)=\int_0^1 (F\circ \gamma)(t)\cdot \gamma'(t)\, dt\\ =\int_0^1 (t^2+6t)\, dt+2\int_0^12t^2\, dt+3\int_0^1 (9t^2-2t)\, dt\\ =\frac13+3+2\frac23+3\left(3-1\right)=\frac53+9$$