The potential ϕ(x,y,z) for the vector field F(x,y,z)=yzi+xzj+(xy+1)k is ϕ(x,y,z) = xyz+z+2.
To find the line integral ∫CF⋅dr, we need to evaluate the dot product of F and dr along the curve C. Given that r(t) is the parametrization of C, we can express dr as dr = r'(t)dt.
Substituting the values of r(t) into F(x,y,z), we get F(r(t)) = (tz, t, t^2+1). Taking the dot product with dr = r'(t)dt, we have F(r(t))⋅dr = (tz, t, t^2+1)⋅(dx/dt, dy/dt, dz/dt)dt.
Now we substitute the values of r(t) and r'(t) into the dot product expression and integrate it over the given range of t, which is 0≤t≤2. This will give us the value of the line integral ∫CF⋅dr.
Since the specific values of dx/dt, dy/dt, and dz/dt are not provided, we cannot calculate the exact value of the line integral without additional information.
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