Answer
$\int_C F \ dr= F \cdot \overrightarrow{AB}$
Work Step by Step
For a vector field to be Conservative, $\dfrac{\partial f_1}{\partial y}=\dfrac{\partial f_2}{\partial x}$
We are given that the force field as: $F(x,y, z)=(a, b, c)$
Next, we will find the potential function for the given vector field as:
$\phi(x,y, z)=ax+by+cz$
Therefore, the integral can be expressed as:
$\int_C F \ dr=\phi(B) -\phi(A) \\=F \cdot B-F \cdot A \\=F \cdot (B-A) \\= F \cdot \overrightarrow{AB}$
Thus, it has been proved that $\int_C F \ dr= F \cdot \overrightarrow{AB}$ by using the fundamental theorem.
